Provided by: liblapack-doc-man_3.6.0-2ubuntu2_all 
NAME
complex_eig - complex
Functions
subroutine cbdt01 (M, N, KD, A, LDA, Q, LDQ, D, E, PT, LDPT, WORK, RWORK, RESID)
CBDT01
subroutine cbdt02 (M, N, B, LDB, C, LDC, U, LDU, WORK, RWORK, RESID)
CBDT02
subroutine cbdt03 (UPLO, N, KD, D, E, U, LDU, S, VT, LDVT, WORK, RESID)
CBDT03
subroutine cchkbb (NSIZES, MVAL, NVAL, NWDTHS, KK, NTYPES, DOTYPE, NRHS, ISEED, THRESH,
NOUNIT, A, LDA, AB, LDAB, BD, BE, Q, LDQ, P, LDP, C, LDC, CC, WORK, LWORK, RWORK,
RESULT, INFO)
CCHKBB
subroutine cchkbd (NSIZES, MVAL, NVAL, NTYPES, DOTYPE, NRHS, ISEED, THRESH, A, LDA, BD,
BE, S1, S2, X, LDX, Y, Z, Q, LDQ, PT, LDPT, U, VT, WORK, LWORK, RWORK, NOUT, INFO)
CCHKBD
subroutine cchkbk (NIN, NOUT)
CCHKBK
subroutine cchkbl (NIN, NOUT)
CCHKBL
subroutine cchkec (THRESH, TSTERR, NIN, NOUT)
CCHKEC
program cchkee
CCHKEE
subroutine cchkgg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, TSTDIF, THRSHN, NOUNIT, A,
LDA, B, H, T, S1, S2, P1, P2, U, LDU, V, Q, Z, ALPHA1, BETA1, ALPHA3, BETA3, EVECTL,
EVECTR, WORK, LWORK, RWORK, LLWORK, RESULT, INFO)
CCHKGG
subroutine cchkgk (NIN, NOUT)
CCHKGK
subroutine cchkgl (NIN, NOUT)
CCHKGL
subroutine cchkhb (NSIZES, NN, NWDTHS, KK, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA,
SD, SE, U, LDU, WORK, LWORK, RWORK, RESULT, INFO)
CCHKHB
subroutine cchkhs (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, T1, T2,
U, LDU, Z, UZ, W1, W3, EVECTL, EVECTR, EVECTY, EVECTX, UU, TAU, WORK, NWORK, RWORK,
IWORK, SELECT, RESULT, INFO)
CCHKHS
subroutine cchkst (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, AP, SD, SE,
D1, D2, D3, D4, D5, WA1, WA2, WA3, WR, U, LDU, V, VP, TAU, Z, WORK, LWORK, RWORK,
LRWORK, IWORK, LIWORK, RESULT, INFO)
CCHKST
subroutine cckcsd (NM, MVAL, PVAL, QVAL, NMATS, ISEED, THRESH, MMAX, X, XF, U1, U2, V1T,
V2T, THETA, IWORK, WORK, RWORK, NIN, NOUT, INFO)
CCKCSD
subroutine cckglm (NN, NVAL, MVAL, PVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, X,
WORK, RWORK, NIN, NOUT, INFO)
CCKGLM
subroutine cckgqr (NM, MVAL, NP, PVAL, NN, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, AQ,
AR, TAUA, B, BF, BZ, BT, BWK, TAUB, WORK, RWORK, NIN, NOUT, INFO)
CCKGQR
subroutine cckgsv (NM, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, U, V,
Q, ALPHA, BETA, R, IWORK, WORK, RWORK, NIN, NOUT, INFO)
CCKGSV
subroutine ccklse (NN, MVAL, PVAL, NVAL, NMATS, ISEED, THRESH, NMAX, A, AF, B, BF, X,
WORK, RWORK, NIN, NOUT, INFO)
CCKLSE
subroutine ccsdts (M, P, Q, X, XF, LDX, U1, LDU1, U2, LDU2, V1T, LDV1T, V2T, LDV2T, THETA,
IWORK, WORK, LWORK, RWORK, RESULT)
CCSDTS
subroutine cdrges (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q,
LDQ, Z, ALPHA, BETA, WORK, LWORK, RWORK, RESULT, BWORK, INFO)
CDRGES
subroutine cdrges3 (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q,
LDQ, Z, ALPHA, BETA, WORK, LWORK, RWORK, RESULT, BWORK, INFO)
CDRGES3
subroutine cdrgev (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q,
LDQ, Z, QE, LDQE, ALPHA, BETA, ALPHA1, BETA1, WORK, LWORK, RWORK, RESULT, INFO)
CDRGEV
subroutine cdrgev3 (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, S, T, Q,
LDQ, Z, QE, LDQE, ALPHA, BETA, ALPHA1, BETA1, WORK, LWORK, RWORK, RESULT, INFO)
CDRGEV3
subroutine cdrgsx (NSIZE, NCMAX, THRESH, NIN, NOUT, A, LDA, B, AI, BI, Z, Q, ALPHA, BETA,
C, LDC, S, WORK, LWORK, RWORK, IWORK, LIWORK, BWORK, INFO)
CDRGSX
subroutine cdrgvx (NSIZE, THRESH, NIN, NOUT, A, LDA, B, AI, BI, ALPHA, BETA, VL, VR, ILO,
IHI, LSCALE, RSCALE, S, STRU, DIF, DIFTRU, WORK, LWORK, RWORK, IWORK, LIWORK, RESULT,
BWORK, INFO)
CDRGVX
subroutine cdrvbd (NSIZES, MM, NN, NTYPES, DOTYPE, ISEED, THRESH, A, LDA, U, LDU, VT,
LDVT, ASAV, USAV, VTSAV, S, SSAV, E, WORK, LWORK, RWORK, IWORK, NOUNIT, INFO)
CDRVBD
subroutine cdrves (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, HT, W,
WT, VS, LDVS, RESULT, WORK, NWORK, RWORK, IWORK, BWORK, INFO)
CDRVES
subroutine cdrvev (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, H, W, W1,
VL, LDVL, VR, LDVR, LRE, LDLRE, RESULT, WORK, NWORK, RWORK, IWORK, INFO)
CDRVEV
subroutine cdrvsg (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, B, LDB, D,
Z, LDZ, AB, BB, AP, BP, WORK, NWORK, RWORK, LRWORK, IWORK, LIWORK, RESULT, INFO)
CDRVSG
subroutine cdrvst (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NOUNIT, A, LDA, D1, D2, D3,
WA1, WA2, WA3, U, LDU, V, TAU, Z, WORK, LWORK, RWORK, LRWORK, IWORK, LIWORK, RESULT,
INFO)
CDRVST
subroutine cdrvsx (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NIUNIT, NOUNIT, A, LDA, H,
HT, W, WT, WTMP, VS, LDVS, VS1, RESULT, WORK, LWORK, RWORK, BWORK, INFO)
CDRVSX
subroutine cdrvvx (NSIZES, NN, NTYPES, DOTYPE, ISEED, THRESH, NIUNIT, NOUNIT, A, LDA, H,
W, W1, VL, LDVL, VR, LDVR, LRE, LDLRE, RCONDV, RCNDV1, RCDVIN, RCONDE, RCNDE1, RCDEIN,
SCALE, SCALE1, RESULT, WORK, NWORK, RWORK, INFO)
CDRVVX
subroutine cerrbd (PATH, NUNIT)
CERRBD
subroutine cerrec (PATH, NUNIT)
CERREC
subroutine cerred (PATH, NUNIT)
CERRED
subroutine cerrgg (PATH, NUNIT)
CERRGG
subroutine cerrhs (PATH, NUNIT)
CERRHS
subroutine cerrst (PATH, NUNIT)
CERRST
subroutine cget02 (TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB, RWORK, RESID)
CGET02
subroutine cget10 (M, N, A, LDA, B, LDB, WORK, RWORK, RESULT)
CGET10
subroutine cget22 (TRANSA, TRANSE, TRANSW, N, A, LDA, E, LDE, W, WORK, RWORK, RESULT)
CGET22
subroutine cget23 (COMP, ISRT, BALANC, JTYPE, THRESH, ISEED, NOUNIT, N, A, LDA, H, W, W1,
VL, LDVL, VR, LDVR, LRE, LDLRE, RCONDV, RCNDV1, RCDVIN, RCONDE, RCNDE1, RCDEIN, SCALE,
SCALE1, RESULT, WORK, LWORK, RWORK, INFO)
CGET23
subroutine cget24 (COMP, JTYPE, THRESH, ISEED, NOUNIT, N, A, LDA, H, HT, W, WT, WTMP, VS,
LDVS, VS1, RCDEIN, RCDVIN, NSLCT, ISLCT, ISRT, RESULT, WORK, LWORK, RWORK, BWORK,
INFO)
CGET24
subroutine cget35 (RMAX, LMAX, NINFO, KNT, NIN)
CGET35
subroutine cget36 (RMAX, LMAX, NINFO, KNT, NIN)
CGET36
subroutine cget37 (RMAX, LMAX, NINFO, KNT, NIN)
CGET37
subroutine cget38 (RMAX, LMAX, NINFO, KNT, NIN)
CGET38
subroutine cget51 (ITYPE, N, A, LDA, B, LDB, U, LDU, V, LDV, WORK, RWORK, RESULT)
CGET51
subroutine cget52 (LEFT, N, A, LDA, B, LDB, E, LDE, ALPHA, BETA, WORK, RWORK, RESULT)
CGET52
subroutine cget54 (N, A, LDA, B, LDB, S, LDS, T, LDT, U, LDU, V, LDV, WORK, RESULT)
CGET54
subroutine cglmts (N, M, P, A, AF, LDA, B, BF, LDB, D, DF, X, U, WORK, LWORK, RWORK,
RESULT)
CGLMTS
subroutine cgqrts (N, M, P, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK,
LWORK, RWORK, RESULT)
CGQRTS
subroutine cgrqts (M, P, N, A, AF, Q, R, LDA, TAUA, B, BF, Z, T, BWK, LDB, TAUB, WORK,
LWORK, RWORK, RESULT)
CGRQTS
subroutine cgsvts3 (M, P, N, A, AF, LDA, B, BF, LDB, U, LDU, V, LDV, Q, LDQ, ALPHA, BETA,
R, LDR, IWORK, WORK, LWORK, RWORK, RESULT)
CGSVTS3
subroutine chbt21 (UPLO, N, KA, KS, A, LDA, D, E, U, LDU, WORK, RWORK, RESULT)
CHBT21
subroutine chet21 (ITYPE, UPLO, N, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK, RWORK,
RESULT)
CHET21
subroutine chet22 (ITYPE, UPLO, N, M, KBAND, A, LDA, D, E, U, LDU, V, LDV, TAU, WORK,
RWORK, RESULT)
CHET22
subroutine chkxer (SRNAMT, INFOT, NOUT, LERR, OK)
CHKXER
subroutine chpt21 (ITYPE, UPLO, N, KBAND, AP, D, E, U, LDU, VP, TAU, WORK, RWORK, RESULT)
CHPT21
subroutine chst01 (N, ILO, IHI, A, LDA, H, LDH, Q, LDQ, WORK, LWORK, RWORK, RESULT)
CHST01
subroutine clarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
CLARFY
subroutine clarhs (PATH, XTYPE, UPLO, TRANS, M, N, KL, KU, NRHS, A, LDA, X, LDX, B, LDB,
ISEED, INFO)
CLARHS
subroutine clatm4 (ITYPE, N, NZ1, NZ2, RSIGN, AMAGN, RCOND, TRIANG, IDIST, ISEED, A, LDA)
CLATM4
logical function clctes (Z, D)
CLCTES
logical function clctsx (ALPHA, BETA)
CLCTSX
subroutine clsets (M, P, N, A, AF, LDA, B, BF, LDB, C, CF, D, DF, X, WORK, LWORK, RWORK,
RESULT)
CLSETS
subroutine csbmv (UPLO, N, K, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CSBMV
subroutine csgt01 (ITYPE, UPLO, N, M, A, LDA, B, LDB, Z, LDZ, D, WORK, RWORK, RESULT)
CSGT01
logical function cslect (Z)
CSLECT
subroutine cstt21 (N, KBAND, AD, AE, SD, SE, U, LDU, WORK, RWORK, RESULT)
CSTT21
subroutine cstt22 (N, M, KBAND, AD, AE, SD, SE, U, LDU, WORK, LDWORK, RWORK, RESULT)
CSTT22
subroutine cunt01 (ROWCOL, M, N, U, LDU, WORK, LWORK, RWORK, RESID)
CUNT01
subroutine cunt03 (RC, MU, MV, N, K, U, LDU, V, LDV, WORK, LWORK, RWORK, RESULT, INFO)
CUNT03
Detailed Description
This is the group of complex LAPACK TESTING EIG routines.
Function Documentation
subroutine cbdt01 (integer M, integer N, integer KD, complex, dimension( lda, * ) A, integer
LDA, complex, dimension( ldq, * ) Q, integer LDQ, real, dimension( * ) D, real, dimension(
* ) E, complex, dimension( ldpt, * ) PT, integer LDPT, complex, dimension( * ) WORK, real,
dimension( * ) RWORK, real RESID)
CBDT01
Purpose:
CBDT01 reconstructs a general matrix A from its bidiagonal form
A = Q * B * P'
where Q (m by min(m,n)) and P' (min(m,n) by n) are unitary
matrices and B is bidiagonal.
The test ratio to test the reduction is
RESID = norm( A - Q * B * PT ) / ( n * norm(A) * EPS )
where PT = P' and EPS is the machine precision.
Parameters:
M
M is INTEGER
The number of rows of the matrices A and Q.
N
N is INTEGER
The number of columns of the matrices A and P'.
KD
KD is INTEGER
If KD = 0, B is diagonal and the array E is not referenced.
If KD = 1, the reduction was performed by xGEBRD; B is upper
bidiagonal if M >= N, and lower bidiagonal if M < N.
If KD = -1, the reduction was performed by xGBBRD; B is
always upper bidiagonal.
A
A is COMPLEX array, dimension (LDA,N)
The m by n matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Q
Q is COMPLEX array, dimension (LDQ,N)
The m by min(m,n) unitary matrix Q in the reduction
A = Q * B * P'.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,M).
D
D is REAL array, dimension (min(M,N))
The diagonal elements of the bidiagonal matrix B.
E
E is REAL array, dimension (min(M,N)-1)
The superdiagonal elements of the bidiagonal matrix B if
m >= n, or the subdiagonal elements of B if m < n.
PT
PT is COMPLEX array, dimension (LDPT,N)
The min(m,n) by n unitary matrix P' in the reduction
A = Q * B * P'.
LDPT
LDPT is INTEGER
The leading dimension of the array PT.
LDPT >= max(1,min(M,N)).
WORK
WORK is COMPLEX array, dimension (M+N)
RWORK
RWORK is REAL array, dimension (M)
RESID
RESID is REAL
The test ratio: norm(A - Q * B * P') / ( n * norm(A) * EPS )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cbdt02 (integer M, integer N, complex, dimension( ldb, * ) B, integer LDB, complex,
dimension( ldc, * ) C, integer LDC, complex, dimension( ldu, * ) U, integer LDU, complex,
dimension( * ) WORK, real, dimension( * ) RWORK, real RESID)
CBDT02
Purpose:
CBDT02 tests the change of basis C = U' * B by computing the residual
RESID = norm( B - U * C ) / ( max(m,n) * norm(B) * EPS ),
where B and C are M by N matrices, U is an M by M orthogonal matrix,
and EPS is the machine precision.
Parameters:
M
M is INTEGER
The number of rows of the matrices B and C and the order of
the matrix Q.
N
N is INTEGER
The number of columns of the matrices B and C.
B
B is COMPLEX array, dimension (LDB,N)
The m by n matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
C
C is COMPLEX array, dimension (LDC,N)
The m by n matrix C, assumed to contain U' * B.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
U
U is COMPLEX array, dimension (LDU,M)
The m by m orthogonal matrix U.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M).
WORK
WORK is COMPLEX array, dimension (M)
RWORK
RWORK is REAL array, dimension (M)
RESID
RESID is REAL
RESID = norm( B - U * C ) / ( max(m,n) * norm(B) * EPS ),
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cbdt03 (character UPLO, integer N, integer KD, real, dimension( * ) D, real,
dimension( * ) E, complex, dimension( ldu, * ) U, integer LDU, real, dimension( * ) S,
complex, dimension( ldvt, * ) VT, integer LDVT, complex, dimension( * ) WORK, real RESID)
CBDT03
Purpose:
CBDT03 reconstructs a bidiagonal matrix B from its SVD:
S = U' * B * V
where U and V are orthogonal matrices and S is diagonal.
The test ratio to test the singular value decomposition is
RESID = norm( B - U * S * VT ) / ( n * norm(B) * EPS )
where VT = V' and EPS is the machine precision.
Parameters:
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix B is upper or lower bidiagonal.
= 'U': Upper bidiagonal
= 'L': Lower bidiagonal
N
N is INTEGER
The order of the matrix B.
KD
KD is INTEGER
The bandwidth of the bidiagonal matrix B. If KD = 1, the
matrix B is bidiagonal, and if KD = 0, B is diagonal and E is
not referenced. If KD is greater than 1, it is assumed to be
1, and if KD is less than 0, it is assumed to be 0.
D
D is REAL array, dimension (N)
The n diagonal elements of the bidiagonal matrix B.
E
E is REAL array, dimension (N-1)
The (n-1) superdiagonal elements of the bidiagonal matrix B
if UPLO = 'U', or the (n-1) subdiagonal elements of B if
UPLO = 'L'.
U
U is COMPLEX array, dimension (LDU,N)
The n by n orthogonal matrix U in the reduction B = U'*A*P.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,N)
S
S is REAL array, dimension (N)
The singular values from the SVD of B, sorted in decreasing
order.
VT
VT is COMPLEX array, dimension (LDVT,N)
The n by n orthogonal matrix V' in the reduction
B = U * S * V'.
LDVT
LDVT is INTEGER
The leading dimension of the array VT.
WORK
WORK is COMPLEX array, dimension (2*N)
RESID
RESID is REAL
The test ratio: norm(B - U * S * V') / ( n * norm(A) * EPS )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkbb (integer NSIZES, integer, dimension( * ) MVAL, integer, dimension( * ) NVAL,
integer NWDTHS, integer, dimension( * ) KK, integer NTYPES, logical, dimension( * )
DOTYPE, integer NRHS, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT, complex,
dimension( lda, * ) A, integer LDA, complex, dimension( ldab, * ) AB, integer LDAB, real,
dimension( * ) BD, real, dimension( * ) BE, complex, dimension( ldq, * ) Q, integer LDQ,
complex, dimension( ldp, * ) P, integer LDP, complex, dimension( ldc, * ) C, integer LDC,
complex, dimension( ldc, * ) CC, complex, dimension( * ) WORK, integer LWORK, real,
dimension( * ) RWORK, real, dimension( * ) RESULT, integer INFO)
CCHKBB
Purpose:
CCHKBB tests the reduction of a general complex rectangular band
matrix to real bidiagonal form.
CGBBRD factors a general band matrix A as Q B P* , where * means
conjugate transpose, B is upper bidiagonal, and Q and P are unitary;
CGBBRD can also overwrite a given matrix C with Q* C .
For each pair of matrix dimensions (M,N) and each selected matrix
type, an M by N matrix A and an M by NRHS matrix C are generated.
The problem dimensions are as follows
A: M x N
Q: M x M
P: N x N
B: min(M,N) x min(M,N)
C: M x NRHS
For each generated matrix, 4 tests are performed:
(1) | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
(2) | I - Q' Q | / ( M ulp )
(3) | I - PT PT' | / ( N ulp )
(4) | Y - Q' C | / ( |Y| max(M,NRHS) ulp ), where Y = Q' C.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
The possible matrix types are
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (3), but multiplied by SQRT( overflow threshold )
(7) Same as (3), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U D V, where U and V are orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U D V, where U and V are orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U D V, where U and V are orthogonal and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Rectangular matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
Parameters:
NSIZES
NSIZES is INTEGER
The number of values of M and N contained in the vectors
MVAL and NVAL. The matrix sizes are used in pairs (M,N).
If NSIZES is zero, CCHKBB does nothing. NSIZES must be at
least zero.
MVAL
MVAL is INTEGER array, dimension (NSIZES)
The values of the matrix row dimension M.
NVAL
NVAL is INTEGER array, dimension (NSIZES)
The values of the matrix column dimension N.
NWDTHS
NWDTHS is INTEGER
The number of bandwidths to use. If it is zero,
CCHKBB does nothing. It must be at least zero.
KK
KK is INTEGER array, dimension (NWDTHS)
An array containing the bandwidths to be used for the band
matrices. The values must be at least zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CCHKBB
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
NRHS
NRHS is INTEGER
The number of columns in the "right-hand side" matrix C.
If NRHS = 0, then the operations on the right-hand side will
not be tested. NRHS must be at least 0.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CCHKBB to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is REAL array, dimension
(LDA, max(NN))
Used to hold the matrix A.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 1
and at least max( NN ).
AB
AB is REAL array, dimension (LDAB, max(NN))
Used to hold A in band storage format.
LDAB
LDAB is INTEGER
The leading dimension of AB. It must be at least 2 (not 1!)
and at least max( KK )+1.
BD
BD is REAL array, dimension (max(NN))
Used to hold the diagonal of the bidiagonal matrix computed
by CGBBRD.
BE
BE is REAL array, dimension (max(NN))
Used to hold the off-diagonal of the bidiagonal matrix
computed by CGBBRD.
Q
Q is COMPLEX array, dimension (LDQ, max(NN))
Used to hold the unitary matrix Q computed by CGBBRD.
LDQ
LDQ is INTEGER
The leading dimension of Q. It must be at least 1
and at least max( NN ).
P
P is COMPLEX array, dimension (LDP, max(NN))
Used to hold the unitary matrix P computed by CGBBRD.
LDP
LDP is INTEGER
The leading dimension of P. It must be at least 1
and at least max( NN ).
C
C is COMPLEX array, dimension (LDC, max(NN))
Used to hold the matrix C updated by CGBBRD.
LDC
LDC is INTEGER
The leading dimension of U. It must be at least 1
and at least max( NN ).
CC
CC is COMPLEX array, dimension (LDC, max(NN))
Used to hold a copy of the matrix C.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
max( LDA+1, max(NN)+1 )*max(NN).
RWORK
RWORK is REAL array, dimension (max(NN))
RESULT
RESULT is REAL array, dimension (4)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
INFO
INFO is INTEGER
If 0, then everything ran OK.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
NTESTT The total number of tests performed so far.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far.
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkbd (integer NSIZES, integer, dimension( * ) MVAL, integer, dimension( * ) NVAL,
integer NTYPES, logical, dimension( * ) DOTYPE, integer NRHS, integer, dimension( 4 )
ISEED, real THRESH, complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) BD,
real, dimension( * ) BE, real, dimension( * ) S1, real, dimension( * ) S2, complex,
dimension( ldx, * ) X, integer LDX, complex, dimension( ldx, * ) Y, complex, dimension(
ldx, * ) Z, complex, dimension( ldq, * ) Q, integer LDQ, complex, dimension( ldpt, * ) PT,
integer LDPT, complex, dimension( ldpt, * ) U, complex, dimension( ldpt, * ) VT, complex,
dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer NOUT, integer
INFO)
CCHKBD
Purpose:
CCHKBD checks the singular value decomposition (SVD) routines.
CGEBRD reduces a complex general m by n matrix A to real upper or
lower bidiagonal form by an orthogonal transformation: Q' * A * P = B
(or A = Q * B * P'). The matrix B is upper bidiagonal if m >= n
and lower bidiagonal if m < n.
CUNGBR generates the orthogonal matrices Q and P' from CGEBRD.
Note that Q and P are not necessarily square.
CBDSQR computes the singular value decomposition of the bidiagonal
matrix B as B = U S V'. It is called three times to compute
1) B = U S1 V', where S1 is the diagonal matrix of singular
values and the columns of the matrices U and V are the left
and right singular vectors, respectively, of B.
2) Same as 1), but the singular values are stored in S2 and the
singular vectors are not computed.
3) A = (UQ) S (P'V'), the SVD of the original matrix A.
In addition, CBDSQR has an option to apply the left orthogonal matrix
U to a matrix X, useful in least squares applications.
For each pair of matrix dimensions (M,N) and each selected matrix
type, an M by N matrix A and an M by NRHS matrix X are generated.
The problem dimensions are as follows
A: M x N
Q: M x min(M,N) (but M x M if NRHS > 0)
P: min(M,N) x N
B: min(M,N) x min(M,N)
U, V: min(M,N) x min(M,N)
S1, S2 diagonal, order min(M,N)
X: M x NRHS
For each generated matrix, 14 tests are performed:
Test CGEBRD and CUNGBR
(1) | A - Q B PT | / ( |A| max(M,N) ulp ), PT = P'
(2) | I - Q' Q | / ( M ulp )
(3) | I - PT PT' | / ( N ulp )
Test CBDSQR on bidiagonal matrix B
(4) | B - U S1 VT | / ( |B| min(M,N) ulp ), VT = V'
(5) | Y - U Z | / ( |Y| max(min(M,N),k) ulp ), where Y = Q' X
and Z = U' Y.
(6) | I - U' U | / ( min(M,N) ulp )
(7) | I - VT VT' | / ( min(M,N) ulp )
(8) S1 contains min(M,N) nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(9) 0 if the true singular values of B are within THRESH of
those in S1. 2*THRESH if they are not. (Tested using
SSVDCH)
(10) | S1 - S2 | / ( |S1| ulp ), where S2 is computed without
computing U and V.
Test CBDSQR on matrix A
(11) | A - (QU) S (VT PT) | / ( |A| max(M,N) ulp )
(12) | X - (QU) Z | / ( |X| max(M,k) ulp )
(13) | I - (QU)'(QU) | / ( M ulp )
(14) | I - (VT PT) (PT'VT') | / ( N ulp )
The possible matrix types are
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (3), but multiplied by SQRT( overflow threshold )
(7) Same as (3), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U D V, where U and V are orthogonal and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U D V, where U and V are orthogonal and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U D V, where U and V are orthogonal and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Rectangular matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
Special case:
(16) A bidiagonal matrix with random entries chosen from a
logarithmic distribution on [ulp^2,ulp^(-2)] (I.e., each
entry is e^x, where x is chosen uniformly on
[ 2 log(ulp), -2 log(ulp) ] .) For *this* type:
(a) CGEBRD is not called to reduce it to bidiagonal form.
(b) the bidiagonal is min(M,N) x min(M,N); if M<N, the
matrix will be lower bidiagonal, otherwise upper.
(c) only tests 5--8 and 14 are performed.
A subset of the full set of matrix types may be selected through
the logical array DOTYPE.
Parameters:
NSIZES
NSIZES is INTEGER
The number of values of M and N contained in the vectors
MVAL and NVAL. The matrix sizes are used in pairs (M,N).
MVAL
MVAL is INTEGER array, dimension (NM)
The values of the matrix row dimension M.
NVAL
NVAL is INTEGER array, dimension (NM)
The values of the matrix column dimension N.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CCHKBD
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrices are in A and B.
This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix
of type j will be generated. If NTYPES is smaller than the
maximum number of types defined (PARAMETER MAXTYP), then
types NTYPES+1 through MAXTYP will not be generated. If
NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through
DOTYPE(NTYPES) will be ignored.
NRHS
NRHS is INTEGER
The number of columns in the "right-hand side" matrices X, Y,
and Z, used in testing CBDSQR. If NRHS = 0, then the
operations on the right-hand side will not be tested.
NRHS must be at least 0.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The values of ISEED are changed on exit, and can be
used in the next call to CCHKBD to continue the same random
number sequence.
THRESH
THRESH is REAL
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have
every test ratio printed, use THRESH = 0. Note that the
expected value of the test ratios is O(1), so THRESH should
be a reasonably small multiple of 1, e.g., 10 or 100.
A
A is COMPLEX array, dimension (LDA,NMAX)
where NMAX is the maximum value of N in NVAL.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,MMAX),
where MMAX is the maximum value of M in MVAL.
BD
BD is REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
BE
BE is REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
S1
S1 is REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
S2
S2 is REAL array, dimension
(max(min(MVAL(j),NVAL(j))))
X
X is COMPLEX array, dimension (LDX,NRHS)
LDX
LDX is INTEGER
The leading dimension of the arrays X, Y, and Z.
LDX >= max(1,MMAX).
Y
Y is COMPLEX array, dimension (LDX,NRHS)
Z
Z is COMPLEX array, dimension (LDX,NRHS)
Q
Q is COMPLEX array, dimension (LDQ,MMAX)
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,MMAX).
PT
PT is COMPLEX array, dimension (LDPT,NMAX)
LDPT
LDPT is INTEGER
The leading dimension of the arrays PT, U, and V.
LDPT >= max(1, max(min(MVAL(j),NVAL(j)))).
U
U is COMPLEX array, dimension
(LDPT,max(min(MVAL(j),NVAL(j))))
VT
VT is COMPLEX array, dimension
(LDPT,max(min(MVAL(j),NVAL(j))))
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
3(M+N) and M(M + max(M,N,k) + 1) + N*min(M,N) for all
pairs (M,N)=(MM(j),NN(j))
RWORK
RWORK is REAL array, dimension
(5*max(min(M,N)))
NOUT
NOUT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
INFO
INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some MM(j) < 0
-3: Some NN(j) < 0
-4: NTYPES < 0
-6: NRHS < 0
-8: THRESH < 0
-11: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ).
-17: LDB < 1 or LDB < MMAX.
-21: LDQ < 1 or LDQ < MMAX.
-23: LDP < 1 or LDP < MNMAX.
-27: LWORK too small.
If CLATMR, CLATMS, CGEBRD, CUNGBR, or CBDSQR,
returns an error code, the
absolute value of it is returned.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
MMAX Largest value in NN.
NMAX Largest value in NN.
MNMIN min(MM(j), NN(j)) (the dimension of the bidiagonal
matrix.)
MNMAX The maximum value of MNMIN for j=1,...,NSIZES.
NFAIL The number of tests which have exceeded THRESH
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
RTOVFL, RTUNFL Square roots of the previous 2 values.
ULP, ULPINV Finest relative precision and its inverse.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkbk (integer NIN, integer NOUT)
CCHKBK
Purpose:
CCHKBK tests CGEBAK, a routine for backward transformation of
the computed right or left eigenvectors if the orginal matrix
was preprocessed by balance subroutine CGEBAL.
Parameters:
NIN
NIN is INTEGER
The logical unit number for input. NIN > 0.
NOUT
NOUT is INTEGER
The logical unit number for output. NOUT > 0.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkbl (integer NIN, integer NOUT)
CCHKBL
Purpose:
CCHKBL tests CGEBAL, a routine for balancing a general complex
matrix and isolating some of its eigenvalues.
Parameters:
NIN
NIN is INTEGER
The logical unit number for input. NIN > 0.
NOUT
NOUT is INTEGER
The logical unit number for output. NOUT > 0.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkec (real THRESH, logical TSTERR, integer NIN, integer NOUT)
CCHKEC
Purpose:
CCHKEC tests eigen- condition estimation routines
CTRSYL, CTREXC, CTRSNA, CTRSEN
In all cases, the routine runs through a fixed set of numerical
examples, subjects them to various tests, and compares the test
results to a threshold THRESH. In addition, CTRSNA and CTRSEN are
tested by reading in precomputed examples from a file (on input unit
NIN). Output is written to output unit NOUT.
Parameters:
THRESH
THRESH is REAL
Threshold for residual tests. A computed test ratio passes
the threshold if it is less than THRESH.
TSTERR
TSTERR is LOGICAL
Flag that indicates whether error exits are to be tested.
NIN
NIN is INTEGER
The logical unit number for input.
NOUT
NOUT is INTEGER
The logical unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
program cchkee ()
CCHKEE
Purpose:
CCHKEE tests the COMPLEX LAPACK subroutines for the matrix
eigenvalue problem. The test paths in this version are
NEP (Nonsymmetric Eigenvalue Problem):
Test CGEHRD, CUNGHR, CHSEQR, CTREVC, CHSEIN, and CUNMHR
SEP (Hermitian Eigenvalue Problem):
Test CHETRD, CUNGTR, CSTEQR, CSTERF, CSTEIN, CSTEDC,
and drivers CHEEV(X), CHBEV(X), CHPEV(X),
CHEEVD, CHBEVD, CHPEVD
SVD (Singular Value Decomposition):
Test CGEBRD, CUNGBR, and CBDSQR
and the drivers CGESVD, CGESDD
CEV (Nonsymmetric Eigenvalue/eigenvector Driver):
Test CGEEV
CES (Nonsymmetric Schur form Driver):
Test CGEES
CVX (Nonsymmetric Eigenvalue/eigenvector Expert Driver):
Test CGEEVX
CSX (Nonsymmetric Schur form Expert Driver):
Test CGEESX
CGG (Generalized Nonsymmetric Eigenvalue Problem):
Test CGGHD3, CGGBAL, CGGBAK, CHGEQZ, and CTGEVC
CGS (Generalized Nonsymmetric Schur form Driver):
Test CGGES
CGV (Generalized Nonsymmetric Eigenvalue/eigenvector Driver):
Test CGGEV
CGX (Generalized Nonsymmetric Schur form Expert Driver):
Test CGGESX
CXV (Generalized Nonsymmetric Eigenvalue/eigenvector Expert Driver):
Test CGGEVX
CSG (Hermitian Generalized Eigenvalue Problem):
Test CHEGST, CHEGV, CHEGVD, CHEGVX, CHPGST, CHPGV, CHPGVD,
CHPGVX, CHBGST, CHBGV, CHBGVD, and CHBGVX
CHB (Hermitian Band Eigenvalue Problem):
Test CHBTRD
CBB (Band Singular Value Decomposition):
Test CGBBRD
CEC (Eigencondition estimation):
Test CTRSYL, CTREXC, CTRSNA, and CTRSEN
CBL (Balancing a general matrix)
Test CGEBAL
CBK (Back transformation on a balanced matrix)
Test CGEBAK
CGL (Balancing a matrix pair)
Test CGGBAL
CGK (Back transformation on a matrix pair)
Test CGGBAK
GLM (Generalized Linear Regression Model):
Tests CGGGLM
GQR (Generalized QR and RQ factorizations):
Tests CGGQRF and CGGRQF
GSV (Generalized Singular Value Decomposition):
Tests CGGSVD, CGGSVP, CTGSJA, CLAGS2, CLAPLL, and CLAPMT
CSD (CS decomposition):
Tests CUNCSD
LSE (Constrained Linear Least Squares):
Tests CGGLSE
Each test path has a different set of inputs, but the data sets for
the driver routines xEV, xES, xVX, and xSX can be concatenated in a
single input file. The first line of input should contain one of the
3-character path names in columns 1-3. The number of remaining lines
depends on what is found on the first line.
The number of matrix types used in testing is often controllable from
the input file. The number of matrix types for each path, and the
test routine that describes them, is as follows:
Path name(s) Types Test routine
CHS or NEP 21 CCHKHS
CST or SEP 21 CCHKST (routines)
18 CDRVST (drivers)
CBD or SVD 16 CCHKBD (routines)
5 CDRVBD (drivers)
CEV 21 CDRVEV
CES 21 CDRVES
CVX 21 CDRVVX
CSX 21 CDRVSX
CGG 26 CCHKGG (routines)
CGS 26 CDRGES
CGX 5 CDRGSX
CGV 26 CDRGEV
CXV 2 CDRGVX
CSG 21 CDRVSG
CHB 15 CCHKHB
CBB 15 CCHKBB
CEC - CCHKEC
CBL - CCHKBL
CBK - CCHKBK
CGL - CCHKGL
CGK - CCHKGK
GLM 8 CCKGLM
GQR 8 CCKGQR
GSV 8 CCKGSV
CSD 3 CCKCSD
LSE 8 CCKLSE
-----------------------------------------------------------------------
NEP input file:
line 2: NN, INTEGER
Number of values of N.
line 3: NVAL, INTEGER array, dimension (NN)
The values for the matrix dimension N.
line 4: NPARMS, INTEGER
Number of values of the parameters NB, NBMIN, NX, NS, and
MAXB.
line 5: NBVAL, INTEGER array, dimension (NPARMS)
The values for the blocksize NB.
line 6: NBMIN, INTEGER array, dimension (NPARMS)
The values for the minimum blocksize NBMIN.
line 7: NXVAL, INTEGER array, dimension (NPARMS)
The values for the crossover point NX.
line 8: INMIN, INTEGER array, dimension (NPARMS)
LAHQR vs TTQRE crossover point, >= 11
line 9: INWIN, INTEGER array, dimension (NPARMS)
recommended deflation window size
line 10: INIBL, INTEGER array, dimension (NPARMS)
nibble crossover point
line 11: ISHFTS, INTEGER array, dimension (NPARMS)
number of simultaneous shifts)
line 12: IACC22, INTEGER array, dimension (NPARMS)
select structured matrix multiply: 0, 1 or 2)
line 13: THRESH
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold. To have all of the test
ratios printed, use THRESH = 0.0 .
line 14: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 14 was 2:
line 15: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 15-EOF: The remaining lines occur in sets of 1 or 2 and allow
the user to specify the matrix types. Each line contains
a 3-character path name in columns 1-3, and the number
of matrix types must be the first nonblank item in columns
4-80. If the number of matrix types is at least 1 but is
less than the maximum number of possible types, a second
line will be read to get the numbers of the matrix types to
be used. For example,
NEP 21
requests all of the matrix types for the nonsymmetric
eigenvalue problem, while
NEP 4
9 10 11 12
requests only matrices of type 9, 10, 11, and 12.
The valid 3-character path names are 'NEP' or 'CHS' for the
nonsymmetric eigenvalue routines.
-----------------------------------------------------------------------
SEP or CSG input file:
line 2: NN, INTEGER
Number of values of N.
line 3: NVAL, INTEGER array, dimension (NN)
The values for the matrix dimension N.
line 4: NPARMS, INTEGER
Number of values of the parameters NB, NBMIN, and NX.
line 5: NBVAL, INTEGER array, dimension (NPARMS)
The values for the blocksize NB.
line 6: NBMIN, INTEGER array, dimension (NPARMS)
The values for the minimum blocksize NBMIN.
line 7: NXVAL, INTEGER array, dimension (NPARMS)
The values for the crossover point NX.
line 8: THRESH
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 9: TSTCHK, LOGICAL
Flag indicating whether or not to test the LAPACK routines.
line 10: TSTDRV, LOGICAL
Flag indicating whether or not to test the driver routines.
line 11: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 12: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 12 was 2:
line 13: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 13-EOF: Lines specifying matrix types, as for NEP.
The valid 3-character path names are 'SEP' or 'CST' for the
Hermitian eigenvalue routines and driver routines, and
'CSG' for the routines for the Hermitian generalized
eigenvalue problem.
-----------------------------------------------------------------------
SVD input file:
line 2: NN, INTEGER
Number of values of M and N.
line 3: MVAL, INTEGER array, dimension (NN)
The values for the matrix row dimension M.
line 4: NVAL, INTEGER array, dimension (NN)
The values for the matrix column dimension N.
line 5: NPARMS, INTEGER
Number of values of the parameter NB, NBMIN, NX, and NRHS.
line 6: NBVAL, INTEGER array, dimension (NPARMS)
The values for the blocksize NB.
line 7: NBMIN, INTEGER array, dimension (NPARMS)
The values for the minimum blocksize NBMIN.
line 8: NXVAL, INTEGER array, dimension (NPARMS)
The values for the crossover point NX.
line 9: NSVAL, INTEGER array, dimension (NPARMS)
The values for the number of right hand sides NRHS.
line 10: THRESH
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 11: TSTCHK, LOGICAL
Flag indicating whether or not to test the LAPACK routines.
line 12: TSTDRV, LOGICAL
Flag indicating whether or not to test the driver routines.
line 13: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 14: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 14 was 2:
line 15: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 15-EOF: Lines specifying matrix types, as for NEP.
The 3-character path names are 'SVD' or 'CBD' for both the
SVD routines and the SVD driver routines.
-----------------------------------------------------------------------
CEV and CES data files:
line 1: 'CEV' or 'CES' in columns 1 to 3.
line 2: NSIZES, INTEGER
Number of sizes of matrices to use. Should be at least 0
and at most 20. If NSIZES = 0, no testing is done
(although the remaining 3 lines are still read).
line 3: NN, INTEGER array, dimension(NSIZES)
Dimensions of matrices to be tested.
line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs
These integer parameters determine how blocking is done
(see ILAENV for details)
NB : block size
NBMIN : minimum block size
NX : minimum dimension for blocking
NS : number of shifts in xHSEQR
NBCOL : minimum column dimension for blocking
line 5: THRESH, REAL
The test threshold against which computed residuals are
compared. Should generally be in the range from 10. to 20.
If it is 0., all test case data will be printed.
line 6: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 6 was 2:
line 7: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 8 and following: Lines specifying matrix types, as for NEP.
The 3-character path name is 'CEV' to test CGEEV, or
'CES' to test CGEES.
-----------------------------------------------------------------------
The CVX data has two parts. The first part is identical to CEV,
and the second part consists of test matrices with precomputed
solutions.
line 1: 'CVX' in columns 1-3.
line 2: NSIZES, INTEGER
If NSIZES = 0, no testing of randomly generated examples
is done, but any precomputed examples are tested.
line 3: NN, INTEGER array, dimension(NSIZES)
line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs
line 5: THRESH, REAL
line 6: NEWSD, INTEGER
If line 6 was 2:
line 7: INTEGER array, dimension (4)
lines 8 and following: The first line contains 'CVX' in columns 1-3
followed by the number of matrix types, possibly with
a second line to specify certain matrix types.
If the number of matrix types = 0, no testing of randomly
generated examples is done, but any precomputed examples
are tested.
remaining lines : Each matrix is stored on 1+N+N**2 lines, where N is
its dimension. The first line contains the dimension N and
ISRT (two integers). ISRT indicates whether the last N lines
are sorted by increasing real part of the eigenvalue
(ISRT=0) or by increasing imaginary part (ISRT=1). The next
N**2 lines contain the matrix rowwise, one entry per line.
The last N lines correspond to each eigenvalue. Each of
these last N lines contains 4 real values: the real part of
the eigenvalues, the imaginary part of the eigenvalue, the
reciprocal condition number of the eigenvalues, and the
reciprocal condition number of the vector eigenvector. The
end of data is indicated by dimension N=0. Even if no data
is to be tested, there must be at least one line containing
N=0.
-----------------------------------------------------------------------
The CSX data is like CVX. The first part is identical to CEV, and the
second part consists of test matrices with precomputed solutions.
line 1: 'CSX' in columns 1-3.
line 2: NSIZES, INTEGER
If NSIZES = 0, no testing of randomly generated examples
is done, but any precomputed examples are tested.
line 3: NN, INTEGER array, dimension(NSIZES)
line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs
line 5: THRESH, REAL
line 6: NEWSD, INTEGER
If line 6 was 2:
line 7: INTEGER array, dimension (4)
lines 8 and following: The first line contains 'CSX' in columns 1-3
followed by the number of matrix types, possibly with
a second line to specify certain matrix types.
If the number of matrix types = 0, no testing of randomly
generated examples is done, but any precomputed examples
are tested.
remaining lines : Each matrix is stored on 3+N**2 lines, where N is
its dimension. The first line contains the dimension N, the
dimension M of an invariant subspace, and ISRT. The second
line contains M integers, identifying the eigenvalues in the
invariant subspace (by their position in a list of
eigenvalues ordered by increasing real part (if ISRT=0) or
by increasing imaginary part (if ISRT=1)). The next N**2
lines contain the matrix rowwise. The last line contains the
reciprocal condition number for the average of the selected
eigenvalues, and the reciprocal condition number for the
corresponding right invariant subspace. The end of data in
indicated by a line containing N=0, M=0, and ISRT = 0. Even
if no data is to be tested, there must be at least one line
containing N=0, M=0 and ISRT=0.
-----------------------------------------------------------------------
CGG input file:
line 2: NN, INTEGER
Number of values of N.
line 3: NVAL, INTEGER array, dimension (NN)
The values for the matrix dimension N.
line 4: NPARMS, INTEGER
Number of values of the parameters NB, NBMIN, NBCOL, NS, and
MAXB.
line 5: NBVAL, INTEGER array, dimension (NPARMS)
The values for the blocksize NB.
line 6: NBMIN, INTEGER array, dimension (NPARMS)
The values for NBMIN, the minimum row dimension for blocks.
line 7: NSVAL, INTEGER array, dimension (NPARMS)
The values for the number of shifts.
line 8: MXBVAL, INTEGER array, dimension (NPARMS)
The values for MAXB, used in determining minimum blocksize.
line 9: IACC22, INTEGER array, dimension (NPARMS)
select structured matrix multiply: 1 or 2)
line 10: NBCOL, INTEGER array, dimension (NPARMS)
The values for NBCOL, the minimum column dimension for
blocks.
line 11: THRESH
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 12: TSTCHK, LOGICAL
Flag indicating whether or not to test the LAPACK routines.
line 13: TSTDRV, LOGICAL
Flag indicating whether or not to test the driver routines.
line 14: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 15: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 15 was 2:
line 16: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 17-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'CGG' for the generalized
eigenvalue problem routines and driver routines.
-----------------------------------------------------------------------
CGS and CGV input files:
line 1: 'CGS' or 'CGV' in columns 1 to 3.
line 2: NN, INTEGER
Number of values of N.
line 3: NVAL, INTEGER array, dimension(NN)
Dimensions of matrices to be tested.
line 4: NB, NBMIN, NX, NS, NBCOL, INTEGERs
These integer parameters determine how blocking is done
(see ILAENV for details)
NB : block size
NBMIN : minimum block size
NX : minimum dimension for blocking
NS : number of shifts in xHGEQR
NBCOL : minimum column dimension for blocking
line 5: THRESH, REAL
The test threshold against which computed residuals are
compared. Should generally be in the range from 10. to 20.
If it is 0., all test case data will be printed.
line 6: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits.
line 7: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 17 was 2:
line 7: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 7-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'CGS' for the generalized
eigenvalue problem routines and driver routines.
-----------------------------------------------------------------------
CGX input file:
line 1: 'CGX' in columns 1 to 3.
line 2: N, INTEGER
Value of N.
line 3: NB, NBMIN, NX, NS, NBCOL, INTEGERs
These integer parameters determine how blocking is done
(see ILAENV for details)
NB : block size
NBMIN : minimum block size
NX : minimum dimension for blocking
NS : number of shifts in xHGEQR
NBCOL : minimum column dimension for blocking
line 4: THRESH, REAL
The test threshold against which computed residuals are
compared. Should generally be in the range from 10. to 20.
Information will be printed about each test for which the
test ratio is greater than or equal to the threshold.
line 5: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 6: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 6 was 2:
line 7: INTEGER array, dimension (4)
Four integer values for the random number seed.
If line 2 was 0:
line 7-EOF: Precomputed examples are tested.
remaining lines : Each example is stored on 3+2*N*N lines, where N is
its dimension. The first line contains the dimension (a
single integer). The next line contains an integer k such
that only the last k eigenvalues will be selected and appear
in the leading diagonal blocks of $A$ and $B$. The next N*N
lines contain the matrix A, one element per line. The next N*N
lines contain the matrix B. The last line contains the
reciprocal of the eigenvalue cluster condition number and the
reciprocal of the deflating subspace (associated with the
selected eigencluster) condition number. The end of data is
indicated by dimension N=0. Even if no data is to be tested,
there must be at least one line containing N=0.
-----------------------------------------------------------------------
CXV input files:
line 1: 'CXV' in columns 1 to 3.
line 2: N, INTEGER
Value of N.
line 3: NB, NBMIN, NX, NS, NBCOL, INTEGERs
These integer parameters determine how blocking is done
(see ILAENV for details)
NB : block size
NBMIN : minimum block size
NX : minimum dimension for blocking
NS : number of shifts in xHGEQR
NBCOL : minimum column dimension for blocking
line 4: THRESH, REAL
The test threshold against which computed residuals are
compared. Should generally be in the range from 10. to 20.
Information will be printed about each test for which the
test ratio is greater than or equal to the threshold.
line 5: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 6: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 6 was 2:
line 7: INTEGER array, dimension (4)
Four integer values for the random number seed.
If line 2 was 0:
line 7-EOF: Precomputed examples are tested.
remaining lines : Each example is stored on 3+2*N*N lines, where N is
its dimension. The first line contains the dimension (a
single integer). The next N*N lines contain the matrix A, one
element per line. The next N*N lines contain the matrix B.
The next line contains the reciprocals of the eigenvalue
condition numbers. The last line contains the reciprocals of
the eigenvector condition numbers. The end of data is
indicated by dimension N=0. Even if no data is to be tested,
there must be at least one line containing N=0.
-----------------------------------------------------------------------
CHB input file:
line 2: NN, INTEGER
Number of values of N.
line 3: NVAL, INTEGER array, dimension (NN)
The values for the matrix dimension N.
line 4: NK, INTEGER
Number of values of K.
line 5: KVAL, INTEGER array, dimension (NK)
The values for the matrix dimension K.
line 6: THRESH
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 7: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 7 was 2:
line 8: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 8-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'CHB'.
-----------------------------------------------------------------------
CBB input file:
line 2: NN, INTEGER
Number of values of M and N.
line 3: MVAL, INTEGER array, dimension (NN)
The values for the matrix row dimension M.
line 4: NVAL, INTEGER array, dimension (NN)
The values for the matrix column dimension N.
line 4: NK, INTEGER
Number of values of K.
line 5: KVAL, INTEGER array, dimension (NK)
The values for the matrix bandwidth K.
line 6: NPARMS, INTEGER
Number of values of the parameter NRHS
line 7: NSVAL, INTEGER array, dimension (NPARMS)
The values for the number of right hand sides NRHS.
line 8: THRESH
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 9: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 9 was 2:
line 10: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 10-EOF: Lines specifying matrix types, as for SVD.
The 3-character path name is 'CBB'.
-----------------------------------------------------------------------
CEC input file:
line 2: THRESH, REAL
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
lines 3-EOF:
Input for testing the eigencondition routines consists of a set of
specially constructed test cases and their solutions. The data
format is not intended to be modified by the user.
-----------------------------------------------------------------------
CBL and CBK input files:
line 1: 'CBL' in columns 1-3 to test CGEBAL, or 'CBK' in
columns 1-3 to test CGEBAK.
The remaining lines consist of specially constructed test cases.
-----------------------------------------------------------------------
CGL and CGK input files:
line 1: 'CGL' in columns 1-3 to test CGGBAL, or 'CGK' in
columns 1-3 to test CGGBAK.
The remaining lines consist of specially constructed test cases.
-----------------------------------------------------------------------
GLM data file:
line 1: 'GLM' in columns 1 to 3.
line 2: NN, INTEGER
Number of values of M, P, and N.
line 3: MVAL, INTEGER array, dimension(NN)
Values of M (row dimension).
line 4: PVAL, INTEGER array, dimension(NN)
Values of P (row dimension).
line 5: NVAL, INTEGER array, dimension(NN)
Values of N (column dimension), note M <= N <= M+P.
line 6: THRESH, REAL
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 7: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 8: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 8 was 2:
line 9: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 9-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'GLM' for the generalized
linear regression model routines.
-----------------------------------------------------------------------
GQR data file:
line 1: 'GQR' in columns 1 to 3.
line 2: NN, INTEGER
Number of values of M, P, and N.
line 3: MVAL, INTEGER array, dimension(NN)
Values of M.
line 4: PVAL, INTEGER array, dimension(NN)
Values of P.
line 5: NVAL, INTEGER array, dimension(NN)
Values of N.
line 6: THRESH, REAL
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 7: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 8: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 8 was 2:
line 9: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 9-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'GQR' for the generalized
QR and RQ routines.
-----------------------------------------------------------------------
GSV data file:
line 1: 'GSV' in columns 1 to 3.
line 2: NN, INTEGER
Number of values of M, P, and N.
line 3: MVAL, INTEGER array, dimension(NN)
Values of M (row dimension).
line 4: PVAL, INTEGER array, dimension(NN)
Values of P (row dimension).
line 5: NVAL, INTEGER array, dimension(NN)
Values of N (column dimension).
line 6: THRESH, REAL
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 7: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 8: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 8 was 2:
line 9: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 9-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'GSV' for the generalized
SVD routines.
-----------------------------------------------------------------------
CSD data file:
line 1: 'CSD' in columns 1 to 3.
line 2: NM, INTEGER
Number of values of M, P, and N.
line 3: MVAL, INTEGER array, dimension(NM)
Values of M (row and column dimension of orthogonal matrix).
line 4: PVAL, INTEGER array, dimension(NM)
Values of P (row dimension of top-left block).
line 5: NVAL, INTEGER array, dimension(NM)
Values of N (column dimension of top-left block).
line 6: THRESH, REAL
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 7: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 8: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 8 was 2:
line 9: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 9-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'CSD' for the CSD routine.
-----------------------------------------------------------------------
LSE data file:
line 1: 'LSE' in columns 1 to 3.
line 2: NN, INTEGER
Number of values of M, P, and N.
line 3: MVAL, INTEGER array, dimension(NN)
Values of M.
line 4: PVAL, INTEGER array, dimension(NN)
Values of P.
line 5: NVAL, INTEGER array, dimension(NN)
Values of N, note P <= N <= P+M.
line 6: THRESH, REAL
Threshold value for the test ratios. Information will be
printed about each test for which the test ratio is greater
than or equal to the threshold.
line 7: TSTERR, LOGICAL
Flag indicating whether or not to test the error exits for
the LAPACK routines and driver routines.
line 8: NEWSD, INTEGER
A code indicating how to set the random number seed.
= 0: Set the seed to a default value before each run
= 1: Initialize the seed to a default value only before the
first run
= 2: Like 1, but use the seed values on the next line
If line 8 was 2:
line 9: INTEGER array, dimension (4)
Four integer values for the random number seed.
lines 9-EOF: Lines specifying matrix types, as for NEP.
The 3-character path name is 'GSV' for the generalized
SVD routines.
-----------------------------------------------------------------------
NMAX is currently set to 132 and must be at least 12 for some of the
precomputed examples, and LWORK = NMAX*(5*NMAX+20) in the parameter
statements below. For SVD, we assume NRHS may be as big as N. The
parameter NEED is set to 14 to allow for 14 N-by-N matrices for CGG.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine cchkgg (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, logical TSTDIF, real
THRSHN, integer NOUNIT, complex, dimension( lda, * ) A, integer LDA, complex, dimension(
lda, * ) B, complex, dimension( lda, * ) H, complex, dimension( lda, * ) T, complex,
dimension( lda, * ) S1, complex, dimension( lda, * ) S2, complex, dimension( lda, * ) P1,
complex, dimension( lda, * ) P2, complex, dimension( ldu, * ) U, integer LDU, complex,
dimension( ldu, * ) V, complex, dimension( ldu, * ) Q, complex, dimension( ldu, * ) Z,
complex, dimension( * ) ALPHA1, complex, dimension( * ) BETA1, complex, dimension( * )
ALPHA3, complex, dimension( * ) BETA3, complex, dimension( ldu, * ) EVECTL, complex,
dimension( ldu, * ) EVECTR, complex, dimension( * ) WORK, integer LWORK, real, dimension(
* ) RWORK, logical, dimension( * ) LLWORK, real, dimension( 15 ) RESULT, integer INFO)
CCHKGG
Purpose:
CCHKGG checks the nonsymmetric generalized eigenvalue problem
routines.
H H H
CGGHRD factors A and B as U H V and U T V , where means conjugate
transpose, H is hessenberg, T is triangular and U and V are unitary.
H H
CHGEQZ factors H and T as Q S Z and Q P Z , where P and S are upper
triangular and Q and Z are unitary. It also computes the generalized
eigenvalues (alpha(1),beta(1)),...,(alpha(n),beta(n)), where
alpha(j)=S(j,j) and beta(j)=P(j,j) -- thus, w(j) = alpha(j)/beta(j)
is a root of the generalized eigenvalue problem
det( A - w(j) B ) = 0
and m(j) = beta(j)/alpha(j) is a root of the essentially equivalent
problem
det( m(j) A - B ) = 0
CTGEVC computes the matrix L of left eigenvectors and the matrix R
of right eigenvectors for the matrix pair ( S, P ). In the
description below, l and r are left and right eigenvectors
corresponding to the generalized eigenvalues (alpha,beta).
When CCHKGG is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 13
tests will be performed. The first twelve "test ratios" should be
small -- O(1). They will be compared with the threshold THRESH:
H
(1) | A - U H V | / ( |A| n ulp )
H
(2) | B - U T V | / ( |B| n ulp )
H
(3) | I - UU | / ( n ulp )
H
(4) | I - VV | / ( n ulp )
H
(5) | H - Q S Z | / ( |H| n ulp )
H
(6) | T - Q P Z | / ( |T| n ulp )
H
(7) | I - QQ | / ( n ulp )
H
(8) | I - ZZ | / ( n ulp )
(9) max over all left eigenvalue/-vector pairs (beta/alpha,l) of
H
| (beta A - alpha B) l | / ( ulp max( |beta A|, |alpha B| ) )
(10) max over all left eigenvalue/-vector pairs (beta/alpha,l') of
H
| (beta H - alpha T) l' | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors l' are the result of passing Q to
STGEVC and back transforming (JOB='B').
(11) max over all right eigenvalue/-vector pairs (beta/alpha,r) of
| (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )
(12) max over all right eigenvalue/-vector pairs (beta/alpha,r') of
| (beta H - alpha T) r' | / ( ulp max( |beta H|, |alpha T| ) )
where the eigenvectors r' are the result of passing Z to
STGEVC and back transforming (JOB='B').
The last three test ratios will usually be small, but there is no
mathematical requirement that they be so. They are therefore
compared with THRESH only if TSTDIF is .TRUE.
(13) | S(Q,Z computed) - S(Q,Z not computed) | / ( |S| ulp )
(14) | P(Q,Z computed) - P(Q,Z not computed) | / ( |P| ulp )
(15) max( |alpha(Q,Z computed) - alpha(Q,Z not computed)|/|S| ,
|beta(Q,Z computed) - beta(Q,Z not computed)|/|P| ) / ulp
In addition, the normalization of L and R are checked, and compared
with the threshold THRSHN.
Test Matrices
---- --------
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is P*D1, P is a random unitary diagonal
matrix (i.e., with random magnitude 1 entries
on the diagonal), and D1=diag( 0, 1,..., N-1 )
(i.e., a diagonal matrix with D1(1,1)=0,
D1(2,2)=1, ..., D1(N,N)=N-1.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1=P*diag( 0, 0, 1, ..., N-3, 0 ) and
D2=Q*diag( 0, N-3, N-4,..., 1, 0, 0 ), and
P and Q are random unitary diagonal matrices.
t t
(16) U ( J , J ) V where U and V are random unitary matrices.
(17) U ( T1, T2 ) V where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
P*( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
Q*( 0, N-3, N-4,..., 1, 0, 0 )
(18) U ( T1, T2 ) V diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) U ( T1, T2 ) V diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) U ( T1, T2 ) V diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) U ( big*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) U ( small*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) U ( small*T1, small*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) U ( big*T1, big*T2 ) V diag(T1) = P*( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) U ( T1, T2 ) V where T1 and T2 are random upper-triangular
matrices.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CCHKGG does nothing. It must be at least zero.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CCHKGG
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CCHKGG to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
TSTDIF
TSTDIF is LOGICAL
Specifies whether test ratios 13-15 will be computed and
compared with THRESH.
= .FALSE.: Only test ratios 1-12 will be computed and tested.
Ratios 13-15 will be set to zero.
= .TRUE.: All the test ratios 1-15 will be computed and
tested.
THRSHN
THRSHN is REAL
Threshold for reporting eigenvector normalization error.
If the normalization of any eigenvector differs from 1 by
more than THRSHN*ulp, then a special error message will be
printed. (This is handled separately from the other tests,
since only a compiler or programming error should cause an
error message, at least if THRSHN is at least 5--10.)
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, max(NN))
Used to hold the original A matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
LDA
LDA is INTEGER
The leading dimension of A, B, H, T, S1, P1, S2, and P2.
It must be at least 1 and at least max( NN ).
B
B is COMPLEX array, dimension (LDA, max(NN))
Used to hold the original B matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
H
H is COMPLEX array, dimension (LDA, max(NN))
The upper Hessenberg matrix computed from A by CGGHRD.
T
T is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGHRD.
S1
S1 is COMPLEX array, dimension (LDA, max(NN))
The Schur (upper triangular) matrix computed from H by CHGEQZ
when Q and Z are also computed.
S2
S2 is COMPLEX array, dimension (LDA, max(NN))
The Schur (upper triangular) matrix computed from H by CHGEQZ
when Q and Z are not computed.
P1
P1 is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by CHGEQZ
when Q and Z are also computed.
P2
P2 is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from T by CHGEQZ
when Q and Z are not computed.
U
U is COMPLEX array, dimension (LDU, max(NN))
The (left) unitary matrix computed by CGGHRD.
LDU
LDU is INTEGER
The leading dimension of U, V, Q, Z, EVECTL, and EVECTR. It
must be at least 1 and at least max( NN ).
V
V is COMPLEX array, dimension (LDU, max(NN))
The (right) unitary matrix computed by CGGHRD.
Q
Q is COMPLEX array, dimension (LDU, max(NN))
The (left) unitary matrix computed by CHGEQZ.
Z
Z is COMPLEX array, dimension (LDU, max(NN))
The (left) unitary matrix computed by CHGEQZ.
ALPHA1
ALPHA1 is COMPLEX array, dimension (max(NN))
BETA1
BETA1 is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CHGEQZ
when Q, Z, and the full Schur matrices are computed.
ALPHA3
ALPHA3 is COMPLEX array, dimension (max(NN))
BETA3
BETA3 is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CHGEQZ
when neither Q, Z, nor the Schur matrices are computed.
EVECTL
EVECTL is COMPLEX array, dimension (LDU, max(NN))
The (lower triangular) left eigenvector matrix for the
matrices in S1 and P1.
EVECTR
EVECTR is COMPLEX array, dimension (LDU, max(NN))
The (upper triangular) right eigenvector matrix for the
matrices in S1 and P1.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
max( 4*N, 2 * N**2, 1 ), for all N=NN(j).
RWORK
RWORK is REAL array, dimension (2*max(NN))
LLWORK
LLWORK is LOGICAL array, dimension (max(NN))
RESULT
RESULT is REAL array, dimension (15)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code. INFO is the
absolute value of the INFO value returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkgk (integer NIN, integer NOUT)
CCHKGK
Purpose:
CCHKGK tests CGGBAK, a routine for backward balancing of
a matrix pair (A, B).
Parameters:
NIN
NIN is INTEGER
The logical unit number for input. NIN > 0.
NOUT
NOUT is INTEGER
The logical unit number for output. NOUT > 0.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkgl (integer NIN, integer NOUT)
CCHKGL
Purpose:
CCHKGL tests CGGBAL, a routine for balancing a matrix pair (A, B).
Parameters:
NIN
NIN is INTEGER
The logical unit number for input. NIN > 0.
NOUT
NOUT is INTEGER
The logical unit number for output. NOUT > 0.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkhb (integer NSIZES, integer, dimension( * ) NN, integer NWDTHS, integer,
dimension( * ) KK, integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 )
ISEED, real THRESH, integer NOUNIT, complex, dimension( lda, * ) A, integer LDA, real,
dimension( * ) SD, real, dimension( * ) SE, complex, dimension( ldu, * ) U, integer LDU,
complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, real, dimension(
* ) RESULT, integer INFO)
CCHKHB
Purpose:
CCHKHB tests the reduction of a Hermitian band matrix to tridiagonal
from, used with the Hermitian eigenvalue problem.
CHBTRD factors a Hermitian band matrix A as U S U* , where * means
conjugate transpose, S is symmetric tridiagonal, and U is unitary.
CHBTRD can use either just the lower or just the upper triangle
of A; CCHKHB checks both cases.
When CCHKHB is called, a number of matrix "sizes" ("n's"), a number
of bandwidths ("k's"), and a number of matrix "types" are
specified. For each size ("n"), each bandwidth ("k") less than or
equal to "n", and each type of matrix, one matrix will be generated
and used to test the hermitian banded reduction routine. For each
matrix, a number of tests will be performed:
(1) | A - V S V* | / ( |A| n ulp ) computed by CHBTRD with
UPLO='U'
(2) | I - UU* | / ( n ulp )
(3) | A - V S V* | / ( |A| n ulp ) computed by CHBTRD with
UPLO='L'
(4) | I - UU* | / ( n ulp )
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (4), but multiplied by SQRT( overflow threshold )
(7) Same as (4), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U* D U, where U is unitary and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U* D U, where U is unitary and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U* D U, where U is unitary and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Hermitian matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CCHKHB does nothing. It must be at least zero.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NWDTHS
NWDTHS is INTEGER
The number of bandwidths to use. If it is zero,
CCHKHB does nothing. It must be at least zero.
KK
KK is INTEGER array, dimension (NWDTHS)
An array containing the bandwidths to be used for the band
matrices. The values must be at least zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CCHKHB
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CCHKHB to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is COMPLEX array, dimension
(LDA, max(NN))
Used to hold the matrix whose eigenvalues are to be
computed.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 2 (not 1!)
and at least max( KK )+1.
SD
SD is REAL array, dimension (max(NN))
Used to hold the diagonal of the tridiagonal matrix computed
by CHBTRD.
SE
SE is REAL array, dimension (max(NN))
Used to hold the off-diagonal of the tridiagonal matrix
computed by CHBTRD.
U
U is COMPLEX array, dimension (LDU, max(NN))
Used to hold the unitary matrix computed by CHBTRD.
LDU
LDU is INTEGER
The leading dimension of U. It must be at least 1
and at least max( NN ).
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
max( LDA+1, max(NN)+1 )*max(NN).
RWORK
RWORK is REAL array
RESULT
RESULT is REAL array, dimension (4)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
INFO
INFO is INTEGER
If 0, then everything ran OK.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
NTESTT The total number of tests performed so far.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far.
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkhs (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) H, complex,
dimension( lda, * ) T1, complex, dimension( lda, * ) T2, complex, dimension( ldu, * ) U,
integer LDU, complex, dimension( ldu, * ) Z, complex, dimension( ldu, * ) UZ, complex,
dimension( * ) W1, complex, dimension( * ) W3, complex, dimension( ldu, * ) EVECTL,
complex, dimension( ldu, * ) EVECTR, complex, dimension( ldu, * ) EVECTY, complex,
dimension( ldu, * ) EVECTX, complex, dimension( ldu, * ) UU, complex, dimension( * ) TAU,
complex, dimension( * ) WORK, integer NWORK, real, dimension( * ) RWORK, integer,
dimension( * ) IWORK, logical, dimension( * ) SELECT, real, dimension( 14 ) RESULT,
integer INFO)
CCHKHS
Purpose:
CCHKHS checks the nonsymmetric eigenvalue problem routines.
CGEHRD factors A as U H U' , where ' means conjugate
transpose, H is hessenberg, and U is unitary.
CUNGHR generates the unitary matrix U.
CUNMHR multiplies a matrix by the unitary matrix U.
CHSEQR factors H as Z T Z' , where Z is unitary and T
is upper triangular. It also computes the eigenvalues,
w(1), ..., w(n); we define a diagonal matrix W whose
(diagonal) entries are the eigenvalues.
CTREVC computes the left eigenvector matrix L and the
right eigenvector matrix R for the matrix T. The
columns of L are the complex conjugates of the left
eigenvectors of T. The columns of R are the right
eigenvectors of T. L is lower triangular, and R is
upper triangular.
CHSEIN computes the left eigenvector matrix Y and the
right eigenvector matrix X for the matrix H. The
columns of Y are the complex conjugates of the left
eigenvectors of H. The columns of X are the right
eigenvectors of H. Y is lower triangular, and X is
upper triangular.
When CCHKHS is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 14
tests will be performed:
(1) | A - U H U**H | / ( |A| n ulp )
(2) | I - UU**H | / ( n ulp )
(3) | H - Z T Z**H | / ( |H| n ulp )
(4) | I - ZZ**H | / ( n ulp )
(5) | A - UZ H (UZ)**H | / ( |A| n ulp )
(6) | I - UZ (UZ)**H | / ( n ulp )
(7) | T(Z computed) - T(Z not computed) | / ( |T| ulp )
(8) | W(Z computed) - W(Z not computed) | / ( |W| ulp )
(9) | TR - RW | / ( |T| |R| ulp )
(10) | L**H T - W**H L | / ( |T| |L| ulp )
(11) | HX - XW | / ( |H| |X| ulp )
(12) | Y**H H - W**H Y | / ( |H| |Y| ulp )
(13) | AX - XW | / ( |A| |X| ulp )
(14) | Y**H A - W**H Y | / ( |A| |Y| ulp )
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A (transposed) Jordan block, with 1's on the diagonal.
(4) A diagonal matrix with evenly spaced entries
1, ..., ULP and random complex angles.
(ULP = (first number larger than 1) - 1 )
(5) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random complex angles.
(6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random complex angles.
(7) Same as (4), but multiplied by SQRT( overflow threshold )
(8) Same as (4), but multiplied by SQRT( underflow threshold )
(9) A matrix of the form U' T U, where U is unitary and
T has evenly spaced entries 1, ..., ULP with random complex
angles on the diagonal and random O(1) entries in the upper
triangle.
(10) A matrix of the form U' T U, where U is unitary and
T has geometrically spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(11) A matrix of the form U' T U, where U is unitary and
T has "clustered" entries 1, ULP,..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(12) A matrix of the form U' T U, where U is unitary and
T has complex eigenvalues randomly chosen from
ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(13) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(14) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has geometrically spaced entries
1, ..., ULP with random complex angles on the diagonal
and random O(1) entries in the upper triangle.
(15) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(16) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has complex eigenvalues randomly chosen
from ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(17) Same as (16), but multiplied by SQRT( overflow threshold )
(18) Same as (16), but multiplied by SQRT( underflow threshold )
(19) Nonsymmetric matrix with random entries chosen from |z| < 1
(20) Same as (19), but multiplied by SQRT( overflow threshold )
(21) Same as (19), but multiplied by SQRT( underflow threshold )
NSIZES - INTEGER
The number of sizes of matrices to use. If it is zero,
CCHKHS does nothing. It must be at least zero.
Not modified.
NN - INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
Not modified.
NTYPES - INTEGER
The number of elements in DOTYPE. If it is zero, CCHKHS
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
Not modified.
DOTYPE - LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
Not modified.
ISEED - INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CCHKHS to continue the same random number
sequence.
Modified.
THRESH - REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
Not modified.
NOUNIT - INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
Not modified.
A - COMPLEX array, dimension (LDA,max(NN))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually
used.
Modified.
LDA - INTEGER
The leading dimension of A, H, T1 and T2. It must be at
least 1 and at least max( NN ).
Not modified.
H - COMPLEX array, dimension (LDA,max(NN))
The upper hessenberg matrix computed by CGEHRD. On exit,
H contains the Hessenberg form of the matrix in A.
Modified.
T1 - COMPLEX array, dimension (LDA,max(NN))
The Schur (="quasi-triangular") matrix computed by CHSEQR
if Z is computed. On exit, T1 contains the Schur form of
the matrix in A.
Modified.
T2 - COMPLEX array, dimension (LDA,max(NN))
The Schur matrix computed by CHSEQR when Z is not computed.
This should be identical to T1.
Modified.
LDU - INTEGER
The leading dimension of U, Z, UZ and UU. It must be at
least 1 and at least max( NN ).
Not modified.
U - COMPLEX array, dimension (LDU,max(NN))
The unitary matrix computed by CGEHRD.
Modified.
Z - COMPLEX array, dimension (LDU,max(NN))
The unitary matrix computed by CHSEQR.
Modified.
UZ - COMPLEX array, dimension (LDU,max(NN))
The product of U times Z.
Modified.
W1 - COMPLEX array, dimension (max(NN))
The eigenvalues of A, as computed by a full Schur
decomposition H = Z T Z'. On exit, W1 contains the
eigenvalues of the matrix in A.
Modified.
W3 - COMPLEX array, dimension (max(NN))
The eigenvalues of A, as computed by a partial Schur
decomposition (Z not computed, T only computed as much
as is necessary for determining eigenvalues). On exit,
W3 contains the eigenvalues of the matrix in A, possibly
perturbed by CHSEIN.
Modified.
EVECTL - COMPLEX array, dimension (LDU,max(NN))
The conjugate transpose of the (upper triangular) left
eigenvector matrix for the matrix in T1.
Modified.
EVECTR - COMPLEX array, dimension (LDU,max(NN))
The (upper triangular) right eigenvector matrix for the
matrix in T1.
Modified.
EVECTY - COMPLEX array, dimension (LDU,max(NN))
The conjugate transpose of the left eigenvector matrix
for the matrix in H.
Modified.
EVECTX - COMPLEX array, dimension (LDU,max(NN))
The right eigenvector matrix for the matrix in H.
Modified.
UU - COMPLEX array, dimension (LDU,max(NN))
Details of the unitary matrix computed by CGEHRD.
Modified.
TAU - COMPLEX array, dimension (max(NN))
Further details of the unitary matrix computed by CGEHRD.
Modified.
WORK - COMPLEX array, dimension (NWORK)
Workspace.
Modified.
NWORK - INTEGER
The number of entries in WORK. NWORK >= 4*NN(j)*NN(j) + 2.
RWORK - REAL array, dimension (max(NN))
Workspace. Could be equivalenced to IWORK, but not SELECT.
Modified.
IWORK - INTEGER array, dimension (max(NN))
Workspace.
Modified.
SELECT - LOGICAL array, dimension (max(NN))
Workspace. Could be equivalenced to IWORK, but not RWORK.
Modified.
RESULT - REAL array, dimension (14)
The values computed by the fourteen tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
Modified.
INFO - INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-6: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-14: LDU < 1 or LDU < NMAX.
-26: NWORK too small.
If CLATMR, CLATMS, or CLATME returns an error code, the
absolute value of it is returned.
If 1, then CHSEQR could not find all the shifts.
If 2, then the EISPACK code (for small blocks) failed.
If >2, then 30*N iterations were not enough to find an
eigenvalue or to decompose the problem.
Modified.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
MTEST The number of tests defined: care must be taken
that (1) the size of RESULT, (2) the number of
tests actually performed, and (3) MTEST agree.
NTEST The number of tests performed on this matrix
so far. This should be less than MTEST, and
equal to it by the last test. It will be less
if any of the routines being tested indicates
that it could not compute the matrices that
would be tested.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far (computed by SLAFTS).
COND, CONDS,
IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL,
RTULP, RTULPI Square roots of the previous 4 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
KCONDS(j) Selects whether CONDS is to be 1 or
1/sqrt(ulp). (0 means irrelevant.)
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cchkst (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( * ) AP, real, dimension(
* ) SD, real, dimension( * ) SE, real, dimension( * ) D1, real, dimension( * ) D2, real,
dimension( * ) D3, real, dimension( * ) D4, real, dimension( * ) D5, real, dimension( * )
WA1, real, dimension( * ) WA2, real, dimension( * ) WA3, real, dimension( * ) WR, complex,
dimension( ldu, * ) U, integer LDU, complex, dimension( ldu, * ) V, complex, dimension( *
) VP, complex, dimension( * ) TAU, complex, dimension( ldu, * ) Z, complex, dimension( * )
WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * )
IWORK, integer LIWORK, real, dimension( * ) RESULT, integer INFO)
CCHKST
Purpose:
CCHKST checks the Hermitian eigenvalue problem routines.
CHETRD factors A as U S U* , where * means conjugate transpose,
S is real symmetric tridiagonal, and U is unitary.
CHETRD can use either just the lower or just the upper triangle
of A; CCHKST checks both cases.
U is represented as a product of Householder
transformations, whose vectors are stored in the first
n-1 columns of V, and whose scale factors are in TAU.
CHPTRD does the same as CHETRD, except that A and V are stored
in "packed" format.
CUNGTR constructs the matrix U from the contents of V and TAU.
CUPGTR constructs the matrix U from the contents of VP and TAU.
CSTEQR factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal. D2 is the matrix of
eigenvalues computed when Z is not computed.
SSTERF computes D3, the matrix of eigenvalues, by the
PWK method, which does not yield eigenvectors.
CPTEQR factors S as Z4 D4 Z4* , for a
Hermitian positive definite tridiagonal matrix.
D5 is the matrix of eigenvalues computed when Z is not
computed.
SSTEBZ computes selected eigenvalues. WA1, WA2, and
WA3 will denote eigenvalues computed to high
absolute accuracy, with different range options.
WR will denote eigenvalues computed to high relative
accuracy.
CSTEIN computes Y, the eigenvectors of S, given the
eigenvalues.
CSTEDC factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). It may also
update an input unitary matrix, usually the output
from CHETRD/CUNGTR or CHPTRD/CUPGTR ('V' option). It may
also just compute eigenvalues ('N' option).
CSTEMR factors S as Z D1 Z* , where Z is the unitary
matrix of eigenvectors and D1 is a diagonal matrix with
the eigenvalues on the diagonal ('I' option). CSTEMR
uses the Relatively Robust Representation whenever possible.
When CCHKST is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the Hermitian eigenroutines. For each matrix, a number
of tests will be performed:
(1) | A - V S V* | / ( |A| n ulp ) CHETRD( UPLO='U', ... )
(2) | I - UV* | / ( n ulp ) CUNGTR( UPLO='U', ... )
(3) | A - V S V* | / ( |A| n ulp ) CHETRD( UPLO='L', ... )
(4) | I - UV* | / ( n ulp ) CUNGTR( UPLO='L', ... )
(5-8) Same as 1-4, but for CHPTRD and CUPGTR.
(9) | S - Z D Z* | / ( |S| n ulp ) CSTEQR('V',...)
(10) | I - ZZ* | / ( n ulp ) CSTEQR('V',...)
(11) | D1 - D2 | / ( |D1| ulp ) CSTEQR('N',...)
(12) | D1 - D3 | / ( |D1| ulp ) SSTERF
(13) 0 if the true eigenvalues (computed by sturm count)
of S are within THRESH of
those in D1. 2*THRESH if they are not. (Tested using
SSTECH)
For S positive definite,
(14) | S - Z4 D4 Z4* | / ( |S| n ulp ) CPTEQR('V',...)
(15) | I - Z4 Z4* | / ( n ulp ) CPTEQR('V',...)
(16) | D4 - D5 | / ( 100 |D4| ulp ) CPTEQR('N',...)
When S is also diagonally dominant by the factor gamma < 1,
(17) max | D4(i) - WR(i) | / ( |D4(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
SSTEBZ( 'A', 'E', ...)
(18) | WA1 - D3 | / ( |D3| ulp ) SSTEBZ( 'A', 'E', ...)
(19) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
SSTEBZ( 'I', 'E', ...)
(20) | S - Y WA1 Y* | / ( |S| n ulp ) SSTEBZ, CSTEIN
(21) | I - Y Y* | / ( n ulp ) SSTEBZ, CSTEIN
(22) | S - Z D Z* | / ( |S| n ulp ) CSTEDC('I')
(23) | I - ZZ* | / ( n ulp ) CSTEDC('I')
(24) | S - Z D Z* | / ( |S| n ulp ) CSTEDC('V')
(25) | I - ZZ* | / ( n ulp ) CSTEDC('V')
(26) | D1 - D2 | / ( |D1| ulp ) CSTEDC('V') and
CSTEDC('N')
Test 27 is disabled at the moment because CSTEMR does not
guarantee high relatvie accuracy.
(27) max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
CSTEMR('V', 'A')
(28) max | D6(i) - WR(i) | / ( |D6(i)| omega ) ,
i
omega = 2 (2n-1) ULP (1 + 8 gamma**2) / (1 - gamma)**4
CSTEMR('V', 'I')
Tests 29 through 34 are disable at present because CSTEMR
does not handle partial specturm requests.
(29) | S - Z D Z* | / ( |S| n ulp ) CSTEMR('V', 'I')
(30) | I - ZZ* | / ( n ulp ) CSTEMR('V', 'I')
(31) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
CSTEMR('N', 'I') vs. CSTEMR('V', 'I')
(32) | S - Z D Z* | / ( |S| n ulp ) CSTEMR('V', 'V')
(33) | I - ZZ* | / ( n ulp ) CSTEMR('V', 'V')
(34) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
CSTEMR('N', 'V') vs. CSTEMR('V', 'V')
(35) | S - Z D Z* | / ( |S| n ulp ) CSTEMR('V', 'A')
(36) | I - ZZ* | / ( n ulp ) CSTEMR('V', 'A')
(37) ( max { min | WA2(i)-WA3(j) | } +
i j
max { min | WA3(i)-WA2(j) | } ) / ( |D3| ulp )
i j
CSTEMR('N', 'A') vs. CSTEMR('V', 'A')
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (4), but multiplied by SQRT( overflow threshold )
(7) Same as (4), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U* D U, where U is unitary and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U* D U, where U is unitary and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U* D U, where U is unitary and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Hermitian matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
(16) Same as (8), but diagonal elements are all positive.
(17) Same as (9), but diagonal elements are all positive.
(18) Same as (10), but diagonal elements are all positive.
(19) Same as (16), but multiplied by SQRT( overflow threshold )
(20) Same as (16), but multiplied by SQRT( underflow threshold )
(21) A diagonally dominant tridiagonal matrix with geometrically
spaced diagonal entries 1, ..., ULP.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CCHKST does nothing. It must be at least zero.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CCHKST
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CCHKST to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is COMPLEX array of
dimension ( LDA , max(NN) )
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually
used.
LDA
LDA is INTEGER
The leading dimension of A. It must be at
least 1 and at least max( NN ).
AP
AP is COMPLEX array of
dimension( max(NN)*max(NN+1)/2 )
The matrix A stored in packed format.
SD
SD is REAL array of
dimension( max(NN) )
The diagonal of the tridiagonal matrix computed by CHETRD.
On exit, SD and SE contain the tridiagonal form of the
matrix in A.
SE
SE is REAL array of
dimension( max(NN) )
The off-diagonal of the tridiagonal matrix computed by
CHETRD. On exit, SD and SE contain the tridiagonal form of
the matrix in A.
D1
D1 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by CSTEQR simlutaneously
with Z. On exit, the eigenvalues in D1 correspond with the
matrix in A.
D2
D2 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by CSTEQR if Z is not
computed. On exit, the eigenvalues in D2 correspond with
the matrix in A.
D3
D3 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by SSTERF. On exit, the
eigenvalues in D3 correspond with the matrix in A.
D4
D4 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by CPTEQR(V).
ZPTEQR factors S as Z4 D4 Z4*
On exit, the eigenvalues in D4 correspond with the matrix in A.
D5
D5 is REAL array of
dimension( max(NN) )
The eigenvalues of A, as computed by ZPTEQR(N)
when Z is not computed. On exit, the
eigenvalues in D4 correspond with the matrix in A.
WA1
WA1 is REAL array of
dimension( max(NN) )
All eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by SSTEBZ.
WA2
WA2 is REAL array of
dimension( max(NN) )
Selected eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by SSTEBZ.
Choose random values for IL and IU, and ask for the
IL-th through IU-th eigenvalues.
WA3
WA3 is REAL array of
dimension( max(NN) )
Selected eigenvalues of A, computed to high
absolute accuracy, with different range options.
as computed by SSTEBZ.
Determine the values VL and VU of the IL-th and IU-th
eigenvalues and ask for all eigenvalues in this range.
WR
WR is DOUBLE PRECISION array of
dimension( max(NN) )
All eigenvalues of A, computed to high
absolute accuracy, with different options.
as computed by DSTEBZ.
U
U is COMPLEX array of
dimension( LDU, max(NN) ).
The unitary matrix computed by CHETRD + CUNGTR.
LDU
LDU is INTEGER
The leading dimension of U, Z, and V. It must be at least 1
and at least max( NN ).
V
V is COMPLEX array of
dimension( LDU, max(NN) ).
The Housholder vectors computed by CHETRD in reducing A to
tridiagonal form. The vectors computed with UPLO='U' are
in the upper triangle, and the vectors computed with UPLO='L'
are in the lower triangle. (As described in CHETRD, the
sub- and superdiagonal are not set to 1, although the
true Householder vector has a 1 in that position. The
routines that use V, such as CUNGTR, set those entries to
1 before using them, and then restore them later.)
VP
VP is COMPLEX array of
dimension( max(NN)*max(NN+1)/2 )
The matrix V stored in packed format.
TAU
TAU is COMPLEX array of
dimension( max(NN) )
The Householder factors computed by CHETRD in reducing A
to tridiagonal form.
Z
Z is COMPLEX array of
dimension( LDU, max(NN) ).
The unitary matrix of eigenvectors computed by CSTEQR,
CPTEQR, and CSTEIN.
WORK
WORK is COMPLEX array of
dimension( LWORK )
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
1 + 4 * Nmax + 2 * Nmax * lg Nmax + 3 * Nmax**2
where Nmax = max( NN(j), 2 ) and lg = log base 2.
IWORK
IWORK is INTEGER array,
Workspace.
LIWORK
LIWORK is INTEGER
The number of entries in IWORK. This must be at least
6 + 6*Nmax + 5 * Nmax * lg Nmax
where Nmax = max( NN(j), 2 ) and lg = log base 2.
RWORK
RWORK is REAL array
LRWORK
LRWORK is INTEGER
The number of entries in LRWORK (dimension( ??? )
RESULT
RESULT is REAL array, dimension (26)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
INFO
INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-5: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-23: LDU < 1 or LDU < NMAX.
-29: LWORK too small.
If CLATMR, CLATMS, CHETRD, CUNGTR, CSTEQR, SSTERF,
or CUNMC2 returns an error code, the
absolute value of it is returned.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
NTESTT The total number of tests performed so far.
NBLOCK Blocksize as returned by ENVIR.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far.
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cckcsd (integer NM, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL,
integer, dimension( * ) QVAL, integer NMATS, integer, dimension( 4 ) ISEED, real THRESH,
integer MMAX, complex, dimension( * ) X, complex, dimension( * ) XF, complex, dimension( *
) U1, complex, dimension( * ) U2, complex, dimension( * ) V1T, complex, dimension( * )
V2T, real, dimension( * ) THETA, integer, dimension( * ) IWORK, complex, dimension( * )
WORK, real, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
CCKCSD
Purpose:
CCKCSD tests CUNCSD:
the CSD for an M-by-M unitary matrix X partitioned as
[ X11 X12; X21 X22 ]. X11 is P-by-Q.
Parameters:
NM
NM is INTEGER
The number of values of M contained in the vector MVAL.
MVAL
MVAL is INTEGER array, dimension (NM)
The values of the matrix row dimension M.
PVAL
PVAL is INTEGER array, dimension (NM)
The values of the matrix row dimension P.
QVAL
QVAL is INTEGER array, dimension (NM)
The values of the matrix column dimension Q.
NMATS
NMATS is INTEGER
The number of matrix types to be tested for each combination
of matrix dimensions. If NMATS >= NTYPES (the maximum
number of matrix types), then all the different types are
generated for testing. If NMATS < NTYPES, another input line
is read to get the numbers of the matrix types to be used.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator. The array
elements should be between 0 and 4095, otherwise they will be
reduced mod 4096, and ISEED(4) must be odd.
On exit, the next seed in the random number sequence after
all the test matrices have been generated.
THRESH
THRESH is REAL
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have
every test ratio printed, use THRESH = 0.
MMAX
MMAX is INTEGER
The maximum value permitted for M, used in dimensioning the
work arrays.
X
X is COMPLEX array, dimension (MMAX*MMAX)
XF
XF is COMPLEX array, dimension (MMAX*MMAX)
U1
U1 is COMPLEX array, dimension (MMAX*MMAX)
U2
U2 is COMPLEX array, dimension (MMAX*MMAX)
V1T
V1T is COMPLEX array, dimension (MMAX*MMAX)
V2T
V2T is COMPLEX array, dimension (MMAX*MMAX)
THETA
THETA is REAL array, dimension (MMAX)
IWORK
IWORK is INTEGER array, dimension (MMAX)
WORK
WORK is COMPLEX array
RWORK
RWORK is REAL array
NIN
NIN is INTEGER
The unit number for input.
NOUT
NOUT is INTEGER
The unit number for output.
INFO
INFO is INTEGER
= 0 : successful exit
> 0 : If CLAROR returns an error code, the absolute value
of it is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cckglm (integer NN, integer, dimension( * ) NVAL, integer, dimension( * ) MVAL,
integer, dimension( * ) PVAL, integer NMATS, integer, dimension( 4 ) ISEED, real THRESH,
integer NMAX, complex, dimension( * ) A, complex, dimension( * ) AF, complex, dimension( *
) B, complex, dimension( * ) BF, complex, dimension( * ) X, complex, dimension( * ) WORK,
real, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
CCKGLM
Purpose:
CCKGLM tests CGGGLM - subroutine for solving generalized linear
model problem.
Parameters:
NN
NN is INTEGER
The number of values of N, M and P contained in the vectors
NVAL, MVAL and PVAL.
NVAL
NVAL is INTEGER array, dimension (NN)
The values of the matrix row dimension N.
MVAL
MVAL is INTEGER array, dimension (NN)
The values of the matrix column dimension M.
PVAL
PVAL is INTEGER array, dimension (NN)
The values of the matrix column dimension P.
NMATS
NMATS is INTEGER
The number of matrix types to be tested for each combination
of matrix dimensions. If NMATS >= NTYPES (the maximum
number of matrix types), then all the different types are
generated for testing. If NMATS < NTYPES, another input line
is read to get the numbers of the matrix types to be used.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator. The array
elements should be between 0 and 4095, otherwise they will be
reduced mod 4096, and ISEED(4) must be odd.
On exit, the next seed in the random number sequence after
all the test matrices have been generated.
THRESH
THRESH is REAL
The threshold value for the test ratios. A result is
included in the output file if RESID >= THRESH. To have
every test ratio printed, use THRESH = 0.
NMAX
NMAX is INTEGER
The maximum value permitted for M or N, used in dimensioning
the work arrays.
A
A is COMPLEX array, dimension (NMAX*NMAX)
AF
AF is COMPLEX array, dimension (NMAX*NMAX)
B
B is COMPLEX array, dimension (NMAX*NMAX)
BF
BF is COMPLEX array, dimension (NMAX*NMAX)
X
X is COMPLEX array, dimension (4*NMAX)
RWORK
RWORK is REAL array, dimension (NMAX)
WORK
WORK is COMPLEX array, dimension (NMAX*NMAX)
NIN
NIN is INTEGER
The unit number for input.
NOUT
NOUT is INTEGER
The unit number for output.
INFO
INFO is INTEGER
= 0 : successful exit
> 0 : If CLATMS returns an error code, the absolute value
of it is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cckgqr (integer NM, integer, dimension( * ) MVAL, integer NP, integer, dimension( *
) PVAL, integer NN, integer, dimension( * ) NVAL, integer NMATS, integer, dimension( 4 )
ISEED, real THRESH, integer NMAX, complex, dimension( * ) A, complex, dimension( * ) AF,
complex, dimension( * ) AQ, complex, dimension( * ) AR, complex, dimension( * ) TAUA,
complex, dimension( * ) B, complex, dimension( * ) BF, complex, dimension( * ) BZ,
complex, dimension( * ) BT, complex, dimension( * ) BWK, complex, dimension( * ) TAUB,
complex, dimension( * ) WORK, real, dimension( * ) RWORK, integer NIN, integer NOUT,
integer INFO)
CCKGQR
Purpose:
CCKGQR tests
CGGQRF: GQR factorization for N-by-M matrix A and N-by-P matrix B,
CGGRQF: GRQ factorization for M-by-N matrix A and P-by-N matrix B.
Parameters:
NM
NM is INTEGER
The number of values of M contained in the vector MVAL.
MVAL
MVAL is INTEGER array, dimension (NM)
The values of the matrix row(column) dimension M.
NP
NP is INTEGER
The number of values of P contained in the vector PVAL.
PVAL
PVAL is INTEGER array, dimension (NP)
The values of the matrix row(column) dimension P.
NN
NN is INTEGER
The number of values of N contained in the vector NVAL.
NVAL
NVAL is INTEGER array, dimension (NN)
The values of the matrix column(row) dimension N.
NMATS
NMATS is INTEGER
The number of matrix types to be tested for each combination
of matrix dimensions. If NMATS >= NTYPES (the maximum
number of matrix types), then all the different types are
generated for testing. If NMATS < NTYPES, another input line
is read to get the numbers of the matrix types to be used.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator. The array
elements should be between 0 and 4095, otherwise they will be
reduced mod 4096, and ISEED(4) must be odd.
On exit, the next seed in the random number sequence after
all the test matrices have been generated.
THRESH
THRESH is REAL
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have
every test ratio printed, use THRESH = 0.
NMAX
NMAX is INTEGER
The maximum value permitted for M or N, used in dimensioning
the work arrays.
A
A is COMPLEX array, dimension (NMAX*NMAX)
AF
AF is COMPLEX array, dimension (NMAX*NMAX)
AQ
AQ is COMPLEX array, dimension (NMAX*NMAX)
AR
AR is COMPLEX array, dimension (NMAX*NMAX)
TAUA
TAUA is COMPLEX array, dimension (NMAX)
B
B is COMPLEX array, dimension (NMAX*NMAX)
BF
BF is COMPLEX array, dimension (NMAX*NMAX)
BZ
BZ is COMPLEX array, dimension (NMAX*NMAX)
BT
BT is COMPLEX array, dimension (NMAX*NMAX)
BWK
BWK is COMPLEX array, dimension (NMAX*NMAX)
TAUB
TAUB is COMPLEX array, dimension (NMAX)
WORK
WORK is COMPLEX array, dimension (NMAX*NMAX)
RWORK
RWORK is REAL array, dimension (NMAX)
NIN
NIN is INTEGER
The unit number for input.
NOUT
NOUT is INTEGER
The unit number for output.
INFO
INFO is INTEGER
= 0 : successful exit
> 0 : If CLATMS returns an error code, the absolute value
of it is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cckgsv (integer NM, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL,
integer, dimension( * ) NVAL, integer NMATS, integer, dimension( 4 ) ISEED, real THRESH,
integer NMAX, complex, dimension( * ) A, complex, dimension( * ) AF, complex, dimension( *
) B, complex, dimension( * ) BF, complex, dimension( * ) U, complex, dimension( * ) V,
complex, dimension( * ) Q, real, dimension( * ) ALPHA, real, dimension( * ) BETA, complex,
dimension( * ) R, integer, dimension( * ) IWORK, complex, dimension( * ) WORK, real,
dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
CCKGSV
Purpose:
CCKGSV tests CGGSVD:
the GSVD for M-by-N matrix A and P-by-N matrix B.
Parameters:
NM
NM is INTEGER
The number of values of M contained in the vector MVAL.
MVAL
MVAL is INTEGER array, dimension (NM)
The values of the matrix row dimension M.
PVAL
PVAL is INTEGER array, dimension (NP)
The values of the matrix row dimension P.
NVAL
NVAL is INTEGER array, dimension (NN)
The values of the matrix column dimension N.
NMATS
NMATS is INTEGER
The number of matrix types to be tested for each combination
of matrix dimensions. If NMATS >= NTYPES (the maximum
number of matrix types), then all the different types are
generated for testing. If NMATS < NTYPES, another input line
is read to get the numbers of the matrix types to be used.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator. The array
elements should be between 0 and 4095, otherwise they will be
reduced mod 4096, and ISEED(4) must be odd.
On exit, the next seed in the random number sequence after
all the test matrices have been generated.
THRESH
THRESH is REAL
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have
every test ratio printed, use THRESH = 0.
NMAX
NMAX is INTEGER
The maximum value permitted for M or N, used in dimensioning
the work arrays.
A
A is COMPLEX array, dimension (NMAX*NMAX)
AF
AF is COMPLEX array, dimension (NMAX*NMAX)
B
B is COMPLEX array, dimension (NMAX*NMAX)
BF
BF is COMPLEX array, dimension (NMAX*NMAX)
U
U is COMPLEX array, dimension (NMAX*NMAX)
V
V is COMPLEX array, dimension (NMAX*NMAX)
Q
Q is COMPLEX array, dimension (NMAX*NMAX)
ALPHA
ALPHA is REAL array, dimension (NMAX)
BETA
BETA is REAL array, dimension (NMAX)
R
R is COMPLEX array, dimension (NMAX*NMAX)
IWORK
IWORK is INTEGER array, dimension (NMAX)
WORK
WORK is COMPLEX array, dimension (NMAX*NMAX)
RWORK
RWORK is REAL array, dimension (NMAX)
NIN
NIN is INTEGER
The unit number for input.
NOUT
NOUT is INTEGER
The unit number for output.
INFO
INFO is INTEGER
= 0 : successful exit
> 0 : If CLATMS returns an error code, the absolute value
of it is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine ccklse (integer NN, integer, dimension( * ) MVAL, integer, dimension( * ) PVAL,
integer, dimension( * ) NVAL, integer NMATS, integer, dimension( 4 ) ISEED, real THRESH,
integer NMAX, complex, dimension( * ) A, complex, dimension( * ) AF, complex, dimension( *
) B, complex, dimension( * ) BF, complex, dimension( * ) X, complex, dimension( * ) WORK,
real, dimension( * ) RWORK, integer NIN, integer NOUT, integer INFO)
CCKLSE
Purpose:
CCKLSE tests CGGLSE - a subroutine for solving linear equality
constrained least square problem (LSE).
Parameters:
NN
NN is INTEGER
The number of values of (M,P,N) contained in the vectors
(MVAL, PVAL, NVAL).
MVAL
MVAL is INTEGER array, dimension (NN)
The values of the matrix row(column) dimension M.
PVAL
PVAL is INTEGER array, dimension (NN)
The values of the matrix row(column) dimension P.
NVAL
NVAL is INTEGER array, dimension (NN)
The values of the matrix column(row) dimension N.
NMATS
NMATS is INTEGER
The number of matrix types to be tested for each combination
of matrix dimensions. If NMATS >= NTYPES (the maximum
number of matrix types), then all the different types are
generated for testing. If NMATS < NTYPES, another input line
is read to get the numbers of the matrix types to be used.
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator. The array
elements should be between 0 and 4095, otherwise they will be
reduced mod 4096, and ISEED(4) must be odd.
On exit, the next seed in the random number sequence after
all the test matrices have been generated.
THRESH
THRESH is REAL
The threshold value for the test ratios. A result is
included in the output file if RESULT >= THRESH. To have
every test ratio printed, use THRESH = 0.
NMAX
NMAX is INTEGER
The maximum value permitted for M or N, used in dimensioning
the work arrays.
A
A is COMPLEX array, dimension (NMAX*NMAX)
AF
AF is COMPLEX array, dimension (NMAX*NMAX)
B
B is COMPLEX array, dimension (NMAX*NMAX)
BF
BF is COMPLEX array, dimension (NMAX*NMAX)
X
X is COMPLEX array, dimension (5*NMAX)
WORK
WORK is COMPLEX array, dimension (NMAX*NMAX)
RWORK
RWORK is REAL array, dimension (NMAX)
NIN
NIN is INTEGER
The unit number for input.
NOUT
NOUT is INTEGER
The unit number for output.
INFO
INFO is INTEGER
= 0 : successful exit
> 0 : If CLATMS returns an error code, the absolute value
of it is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine ccsdts (integer M, integer P, integer Q, complex, dimension( ldx, * ) X, complex,
dimension( ldx, * ) XF, integer LDX, complex, dimension( ldu1, * ) U1, integer LDU1,
complex, dimension( ldu2, * ) U2, integer LDU2, complex, dimension( ldv1t, * ) V1T,
integer LDV1T, complex, dimension( ldv2t, * ) V2T, integer LDV2T, real, dimension( * )
THETA, integer, dimension( * ) IWORK, complex, dimension( lwork ) WORK, integer LWORK,
real, dimension( * ) RWORK, real, dimension( 15 ) RESULT)
CCSDTS
Purpose:
CCSDTS tests CUNCSD, which, given an M-by-M partitioned unitary
matrix X,
Q M-Q
X = [ X11 X12 ] P ,
[ X21 X22 ] M-P
computes the CSD
[ U1 ]**T * [ X11 X12 ] * [ V1 ]
[ U2 ] [ X21 X22 ] [ V2 ]
[ I 0 0 | 0 0 0 ]
[ 0 C 0 | 0 -S 0 ]
[ 0 0 0 | 0 0 -I ]
= [---------------------] = [ D11 D12 ] .
[ 0 0 0 | I 0 0 ] [ D21 D22 ]
[ 0 S 0 | 0 C 0 ]
[ 0 0 I | 0 0 0 ]
and also SORCSD2BY1, which, given
Q
[ X11 ] P ,
[ X21 ] M-P
computes the 2-by-1 CSD
[ I 0 0 ]
[ 0 C 0 ]
[ 0 0 0 ]
[ U1 ]**T * [ X11 ] * V1 = [----------] = [ D11 ] ,
[ U2 ] [ X21 ] [ 0 0 0 ] [ D21 ]
[ 0 S 0 ]
[ 0 0 I ]
Parameters:
M
M is INTEGER
The number of rows of the matrix X. M >= 0.
P
P is INTEGER
The number of rows of the matrix X11. P >= 0.
Q
Q is INTEGER
The number of columns of the matrix X11. Q >= 0.
X
X is COMPLEX array, dimension (LDX,M)
The M-by-M matrix X.
XF
XF is COMPLEX array, dimension (LDX,M)
Details of the CSD of X, as returned by CUNCSD;
see CUNCSD for further details.
LDX
LDX is INTEGER
The leading dimension of the arrays X and XF.
LDX >= max( 1,M ).
U1
U1 is COMPLEX array, dimension(LDU1,P)
The P-by-P unitary matrix U1.
LDU1
LDU1 is INTEGER
The leading dimension of the array U1. LDU >= max(1,P).
U2
U2 is COMPLEX array, dimension(LDU2,M-P)
The (M-P)-by-(M-P) unitary matrix U2.
LDU2
LDU2 is INTEGER
The leading dimension of the array U2. LDU >= max(1,M-P).
V1T
V1T is COMPLEX array, dimension(LDV1T,Q)
The Q-by-Q unitary matrix V1T.
LDV1T
LDV1T is INTEGER
The leading dimension of the array V1T. LDV1T >=
max(1,Q).
V2T
V2T is COMPLEX array, dimension(LDV2T,M-Q)
The (M-Q)-by-(M-Q) unitary matrix V2T.
LDV2T
LDV2T is INTEGER
The leading dimension of the array V2T. LDV2T >=
max(1,M-Q).
THETA
THETA is REAL array, dimension MIN(P,M-P,Q,M-Q)
The CS values of X; the essentially diagonal matrices C and
S are constructed from THETA; see subroutine CUNCSD for
details.
IWORK
IWORK is INTEGER array, dimension (M)
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK
RWORK
RWORK is REAL array
RESULT
RESULT is REAL array, dimension (15)
The test ratios:
First, the 2-by-2 CSD:
RESULT(1) = norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 )
RESULT(2) = norm( U1'*X12*V2 - D12 ) / ( MAX(1,P,M-Q)*EPS2 )
RESULT(3) = norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 )
RESULT(4) = norm( U2'*X22*V2 - D22 ) / ( MAX(1,M-P,M-Q)*EPS2 )
RESULT(5) = norm( I - U1'*U1 ) / ( MAX(1,P)*ULP )
RESULT(6) = norm( I - U2'*U2 ) / ( MAX(1,M-P)*ULP )
RESULT(7) = norm( I - V1T'*V1T ) / ( MAX(1,Q)*ULP )
RESULT(8) = norm( I - V2T'*V2T ) / ( MAX(1,M-Q)*ULP )
RESULT(9) = 0 if THETA is in increasing order and
all angles are in [0,pi/2];
= ULPINV otherwise.
Then, the 2-by-1 CSD:
RESULT(10) = norm( U1'*X11*V1 - D11 ) / ( MAX(1,P,Q)*EPS2 )
RESULT(11) = norm( U2'*X21*V1 - D21 ) / ( MAX(1,M-P,Q)*EPS2 )
RESULT(12) = norm( I - U1'*U1 ) / ( MAX(1,P)*ULP )
RESULT(13) = norm( I - U2'*U2 ) / ( MAX(1,M-P)*ULP )
RESULT(14) = norm( I - V1T'*V1T ) / ( MAX(1,Q)*ULP )
RESULT(15) = 0 if THETA is in increasing order and
all angles are in [0,pi/2];
= ULPINV otherwise.
( EPS2 = MAX( norm( I - X'*X ) / M, ULP ). )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine cdrges (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) B, complex,
dimension( lda, * ) S, complex, dimension( lda, * ) T, complex, dimension( ldq, * ) Q,
integer LDQ, complex, dimension( ldq, * ) Z, complex, dimension( * ) ALPHA, complex,
dimension( * ) BETA, complex, dimension( * ) WORK, integer LWORK, real, dimension( * )
RWORK, real, dimension( 13 ) RESULT, logical, dimension( * ) BWORK, integer INFO)
CDRGES
Purpose:
CDRGES checks the nonsymmetric generalized eigenvalue (Schur form)
problem driver CGGES.
CGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate
transpose, S and T are upper triangular (i.e., in generalized Schur
form), and Q and Z are unitary. It also computes the generalized
eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus,
w(j) = alpha(j)/beta(j) is a root of the characteristic equation
det( A - w(j) B ) = 0
Optionally it also reorder the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms.
When CDRGES is called, a number of matrix "sizes" ("N's") and a
number of matrix "TYPES" are specified. For each size ("N")
and each TYPE of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following 13 tests
will be performed and compared with the threshold THRESH except
the tests (5), (11) and (13).
(1) | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
(2) | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
(3) | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
(4) | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
(5) if A is in Schur form (i.e. triangular form) (no sorting of
eigenvalues)
(6) if eigenvalues = diagonal elements of the Schur form (S, T),
i.e., test the maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(no sorting of eigenvalues)
(7) | (A,B) - Q (S,T) Z' | / ( |(A,B)| n ulp )
(with sorting of eigenvalues).
(8) | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
(9) | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
(10) if A is in Schur form (i.e. quasi-triangular form)
(with sorting of eigenvalues).
(11) if eigenvalues = diagonal elements of the Schur form (S, T),
i.e. test the maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(with sorting of eigenvalues).
(12) if sorting worked and SDIM is the number of eigenvalues
which were CELECTed.
Test Matrices
=============
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
SDRGES does nothing. NSIZES >= 0.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. NN >= 0.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, SDRGES
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A on input.
This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to SDRGES to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error is
scaled to be O(1), so THRESH should be a reasonably small
multiple of 1, e.g., 10 or 100. In particular, it should
not depend on the precision (single vs. double) or the size
of the matrix. THRESH >= 0.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original A matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
LDA
LDA is INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ).
B
B is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original B matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
S
S is COMPLEX array, dimension (LDA, max(NN))
The Schur form matrix computed from A by CGGES. On exit, S
contains the Schur form matrix corresponding to the matrix
in A.
T
T is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGES.
Q
Q is COMPLEX array, dimension (LDQ, max(NN))
The (left) orthogonal matrix computed by CGGES.
LDQ
LDQ is INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ).
Z
Z is COMPLEX array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by CGGES.
ALPHA
ALPHA is COMPLEX array, dimension (max(NN))
BETA
BETA is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CGGES.
ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A
and B.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 3*N*N.
RWORK
RWORK is REAL array, dimension ( 8*N )
Real workspace.
RESULT
RESULT is REAL array, dimension (15)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
BWORK
BWORK is LOGICAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code. INFO is the
absolute value of the INFO value returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrges3 (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) B, complex,
dimension( lda, * ) S, complex, dimension( lda, * ) T, complex, dimension( ldq, * ) Q,
integer LDQ, complex, dimension( ldq, * ) Z, complex, dimension( * ) ALPHA, complex,
dimension( * ) BETA, complex, dimension( * ) WORK, integer LWORK, real, dimension( * )
RWORK, real, dimension( 13 ) RESULT, logical, dimension( * ) BWORK, integer INFO)
CDRGES3
Purpose:
CDRGES3 checks the nonsymmetric generalized eigenvalue (Schur form)
problem driver CGGES3.
CGGES3 factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate
transpose, S and T are upper triangular (i.e., in generalized Schur
form), and Q and Z are unitary. It also computes the generalized
eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus,
w(j) = alpha(j)/beta(j) is a root of the characteristic equation
det( A - w(j) B ) = 0
Optionally it also reorder the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms.
When CDRGES3 is called, a number of matrix "sizes" ("N's") and a
number of matrix "TYPES" are specified. For each size ("N")
and each TYPE of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following 13 tests
will be performed and compared with the threshold THRESH except
the tests (5), (11) and (13).
(1) | A - Q S Z' | / ( |A| n ulp ) (no sorting of eigenvalues)
(2) | B - Q T Z' | / ( |B| n ulp ) (no sorting of eigenvalues)
(3) | I - QQ' | / ( n ulp ) (no sorting of eigenvalues)
(4) | I - ZZ' | / ( n ulp ) (no sorting of eigenvalues)
(5) if A is in Schur form (i.e. triangular form) (no sorting of
eigenvalues)
(6) if eigenvalues = diagonal elements of the Schur form (S, T),
i.e., test the maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(no sorting of eigenvalues)
(7) | (A,B) - Q (S,T) Z' | / ( |(A,B)| n ulp )
(with sorting of eigenvalues).
(8) | I - QQ' | / ( n ulp ) (with sorting of eigenvalues).
(9) | I - ZZ' | / ( n ulp ) (with sorting of eigenvalues).
(10) if A is in Schur form (i.e. quasi-triangular form)
(with sorting of eigenvalues).
(11) if eigenvalues = diagonal elements of the Schur form (S, T),
i.e. test the maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(with sorting of eigenvalues).
(12) if sorting worked and SDIM is the number of eigenvalues
which were CELECTed.
Test Matrices
=============
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
SDRGES3 does nothing. NSIZES >= 0.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. NN >= 0.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, SDRGES3
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A on input.
This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to SDRGES3 to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error is
scaled to be O(1), so THRESH should be a reasonably small
multiple of 1, e.g., 10 or 100. In particular, it should
not depend on the precision (single vs. double) or the size
of the matrix. THRESH >= 0.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original A matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
LDA
LDA is INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ).
B
B is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original B matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
S
S is COMPLEX array, dimension (LDA, max(NN))
The Schur form matrix computed from A by CGGES3. On exit, S
contains the Schur form matrix corresponding to the matrix
in A.
T
T is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGES3.
Q
Q is COMPLEX array, dimension (LDQ, max(NN))
The (left) orthogonal matrix computed by CGGES3.
LDQ
LDQ is INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ).
Z
Z is COMPLEX array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by CGGES3.
ALPHA
ALPHA is COMPLEX array, dimension (max(NN))
BETA
BETA is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CGGES3.
ALPHA(k) / BETA(k) is the k-th generalized eigenvalue of A
and B.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 3*N*N.
RWORK
RWORK is REAL array, dimension ( 8*N )
Real workspace.
RESULT
RESULT is REAL array, dimension (15)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
BWORK
BWORK is LOGICAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code. INFO is the
absolute value of the INFO value returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
February 2015
subroutine cdrgev (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) B, complex,
dimension( lda, * ) S, complex, dimension( lda, * ) T, complex, dimension( ldq, * ) Q,
integer LDQ, complex, dimension( ldq, * ) Z, complex, dimension( ldqe, * ) QE, integer
LDQE, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( * )
ALPHA1, complex, dimension( * ) BETA1, complex, dimension( * ) WORK, integer LWORK, real,
dimension( * ) RWORK, real, dimension( * ) RESULT, integer INFO)
CDRGEV
Purpose:
CDRGEV checks the nonsymmetric generalized eigenvalue problem driver
routine CGGEV.
CGGEV computes for a pair of n-by-n nonsymmetric matrices (A,B) the
generalized eigenvalues and, optionally, the left and right
eigenvectors.
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that A - w*B is singular. It is
usually represented as the pair (alpha,beta), as there is reasonable
interpretation for beta=0, and even for both being zero.
A right generalized eigenvector corresponding to a generalized
eigenvalue w for a pair of matrices (A,B) is a vector r such that
(A - wB) * r = 0. A left generalized eigenvector is a vector l such
that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.
When CDRGEV is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following tests
will be performed and compared with the threshold THRESH.
Results from CGGEV:
(1) max over all left eigenvalue/-vector pairs (alpha/beta,l) of
| VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) )
where VL**H is the conjugate-transpose of VL.
(2) | |VL(i)| - 1 | / ulp and whether largest component real
VL(i) denotes the i-th column of VL.
(3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of
| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )
(4) | |VR(i)| - 1 | / ulp and whether largest component real
VR(i) denotes the i-th column of VR.
(5) W(full) = W(partial)
W(full) denotes the eigenvalues computed when both l and r
are also computed, and W(partial) denotes the eigenvalues
computed when only W, only W and r, or only W and l are
computed.
(6) VL(full) = VL(partial)
VL(full) denotes the left eigenvectors computed when both l
and r are computed, and VL(partial) denotes the result
when only l is computed.
(7) VR(full) = VR(partial)
VR(full) denotes the right eigenvectors computed when both l
and r are also computed, and VR(partial) denotes the result
when only l is computed.
Test Matrices
---- --------
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CDRGES does nothing. NSIZES >= 0.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. NN >= 0.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CDRGEV
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRGES to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error is
scaled to be O(1), so THRESH should be a reasonably small
multiple of 1, e.g., 10 or 100. In particular, it should
not depend on the precision (single vs. double) or the size
of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IERR not equal to 0.)
A
A is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original A matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
LDA
LDA is INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ).
B
B is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original B matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
S
S is COMPLEX array, dimension (LDA, max(NN))
The Schur form matrix computed from A by CGGEV. On exit, S
contains the Schur form matrix corresponding to the matrix
in A.
T
T is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGEV.
Q
Q is COMPLEX array, dimension (LDQ, max(NN))
The (left) eigenvectors matrix computed by CGGEV.
LDQ
LDQ is INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ).
Z
Z is COMPLEX array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by CGGEV.
QE
QE is COMPLEX array, dimension( LDQ, max(NN) )
QE holds the computed right or left eigenvectors.
LDQE
LDQE is INTEGER
The leading dimension of QE. LDQE >= max(1,max(NN)).
ALPHA
ALPHA is COMPLEX array, dimension (max(NN))
BETA
BETA is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CGGEV.
( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
generalized eigenvalue of A and B.
ALPHA1
ALPHA1 is COMPLEX array, dimension (max(NN))
BETA1
BETA1 is COMPLEX array, dimension (max(NN))
Like ALPHAR, ALPHAI, BETA, these arrays contain the
eigenvalues of A and B, but those computed when CGGEV only
computes a partial eigendecomposition, i.e. not the
eigenvalues and left and right eigenvectors.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. LWORK >= N*(N+1)
RWORK
RWORK is REAL array, dimension (8*N)
Real workspace.
RESULT
RESULT is REAL array, dimension (2)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code. INFO is the
absolute value of the INFO value returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine cdrgev3 (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) B, complex,
dimension( lda, * ) S, complex, dimension( lda, * ) T, complex, dimension( ldq, * ) Q,
integer LDQ, complex, dimension( ldq, * ) Z, complex, dimension( ldqe, * ) QE, integer
LDQE, complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( * )
ALPHA1, complex, dimension( * ) BETA1, complex, dimension( * ) WORK, integer LWORK, real,
dimension( * ) RWORK, real, dimension( * ) RESULT, integer INFO)
CDRGEV3
Purpose:
CDRGEV3 checks the nonsymmetric generalized eigenvalue problem driver
routine CGGEV3.
CGGEV3 computes for a pair of n-by-n nonsymmetric matrices (A,B) the
generalized eigenvalues and, optionally, the left and right
eigenvectors.
A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
or a ratio alpha/beta = w, such that A - w*B is singular. It is
usually represented as the pair (alpha,beta), as there is reasonable
interpretation for beta=0, and even for both being zero.
A right generalized eigenvector corresponding to a generalized
eigenvalue w for a pair of matrices (A,B) is a vector r such that
(A - wB) * r = 0. A left generalized eigenvector is a vector l such
that l**H * (A - wB) = 0, where l**H is the conjugate-transpose of l.
When CDRGEV3 is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, a pair of matrices (A, B) will be generated
and used for testing. For each matrix pair, the following tests
will be performed and compared with the threshold THRESH.
Results from CGGEV3:
(1) max over all left eigenvalue/-vector pairs (alpha/beta,l) of
| VL**H * (beta A - alpha B) |/( ulp max(|beta A|, |alpha B|) )
where VL**H is the conjugate-transpose of VL.
(2) | |VL(i)| - 1 | / ulp and whether largest component real
VL(i) denotes the i-th column of VL.
(3) max over all left eigenvalue/-vector pairs (alpha/beta,r) of
| (beta A - alpha B) * VR | / ( ulp max(|beta A|, |alpha B|) )
(4) | |VR(i)| - 1 | / ulp and whether largest component real
VR(i) denotes the i-th column of VR.
(5) W(full) = W(partial)
W(full) denotes the eigenvalues computed when both l and r
are also computed, and W(partial) denotes the eigenvalues
computed when only W, only W and r, or only W and l are
computed.
(6) VL(full) = VL(partial)
VL(full) denotes the left eigenvectors computed when both l
and r are computed, and VL(partial) denotes the result
when only l is computed.
(7) VR(full) = VR(partial)
VR(full) denotes the right eigenvectors computed when both l
and r are also computed, and VR(partial) denotes the result
when only l is computed.
Test Matrices
---- --------
The sizes of the test matrices are specified by an array
NN(1:NSIZES); the value of each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES ); if
DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) ( 0, 0 ) (a pair of zero matrices)
(2) ( I, 0 ) (an identity and a zero matrix)
(3) ( 0, I ) (an identity and a zero matrix)
(4) ( I, I ) (a pair of identity matrices)
t t
(5) ( J , J ) (a pair of transposed Jordan blocks)
t ( I 0 )
(6) ( X, Y ) where X = ( J 0 ) and Y = ( t )
( 0 I ) ( 0 J )
and I is a k x k identity and J a (k+1)x(k+1)
Jordan block; k=(N-1)/2
(7) ( D, I ) where D is diag( 0, 1,..., N-1 ) (a diagonal
matrix with those diagonal entries.)
(8) ( I, D )
(9) ( big*D, small*I ) where "big" is near overflow and small=1/big
(10) ( small*D, big*I )
(11) ( big*I, small*D )
(12) ( small*I, big*D )
(13) ( big*D, big*I )
(14) ( small*D, small*I )
(15) ( D1, D2 ) where D1 is diag( 0, 0, 1, ..., N-3, 0 ) and
D2 is diag( 0, N-3, N-4,..., 1, 0, 0 )
t t
(16) Q ( J , J ) Z where Q and Z are random orthogonal matrices.
(17) Q ( T1, T2 ) Z where T1 and T2 are upper triangular matrices
with random O(1) entries above the diagonal
and diagonal entries diag(T1) =
( 0, 0, 1, ..., N-3, 0 ) and diag(T2) =
( 0, N-3, N-4,..., 1, 0, 0 )
(18) Q ( T1, T2 ) Z diag(T1) = ( 0, 0, 1, 1, s, ..., s, 0 )
diag(T2) = ( 0, 1, 0, 1,..., 1, 0 )
s = machine precision.
(19) Q ( T1, T2 ) Z diag(T1)=( 0,0,1,1, 1-d, ..., 1-(N-5)*d=s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0 )
N-5
(20) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, 1, a, ..., a =s, 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
(21) Q ( T1, T2 ) Z diag(T1)=( 0, 0, 1, r1, r2, ..., r(N-4), 0 )
diag(T2) = ( 0, 1, 0, 1, ..., 1, 0, 0 )
where r1,..., r(N-4) are random.
(22) Q ( big*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(23) Q ( small*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(24) Q ( small*T1, small*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(25) Q ( big*T1, big*T2 ) Z diag(T1) = ( 0, 0, 1, ..., N-3, 0 )
diag(T2) = ( 0, 1, ..., 1, 0, 0 )
(26) Q ( T1, T2 ) Z where T1 and T2 are random upper-triangular
matrices.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CDRGEV3 does nothing. NSIZES >= 0.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. NN >= 0.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CDRGEV3
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRGEV3 to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error is
scaled to be O(1), so THRESH should be a reasonably small
multiple of 1, e.g., 10 or 100. In particular, it should
not depend on the precision (single vs. double) or the size
of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IERR not equal to 0.)
A
A is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original A matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
LDA
LDA is INTEGER
The leading dimension of A, B, S, and T.
It must be at least 1 and at least max( NN ).
B
B is COMPLEX array, dimension(LDA, max(NN))
Used to hold the original B matrix. Used as input only
if NTYPES=MAXTYP+1, DOTYPE(1:MAXTYP)=.FALSE., and
DOTYPE(MAXTYP+1)=.TRUE.
S
S is COMPLEX array, dimension (LDA, max(NN))
The Schur form matrix computed from A by CGGEV3. On exit, S
contains the Schur form matrix corresponding to the matrix
in A.
T
T is COMPLEX array, dimension (LDA, max(NN))
The upper triangular matrix computed from B by CGGEV3.
Q
Q is COMPLEX array, dimension (LDQ, max(NN))
The (left) eigenvectors matrix computed by CGGEV3.
LDQ
LDQ is INTEGER
The leading dimension of Q and Z. It must
be at least 1 and at least max( NN ).
Z
Z is COMPLEX array, dimension( LDQ, max(NN) )
The (right) orthogonal matrix computed by CGGEV3.
QE
QE is COMPLEX array, dimension( LDQ, max(NN) )
QE holds the computed right or left eigenvectors.
LDQE
LDQE is INTEGER
The leading dimension of QE. LDQE >= max(1,max(NN)).
ALPHA
ALPHA is COMPLEX array, dimension (max(NN))
BETA
BETA is COMPLEX array, dimension (max(NN))
The generalized eigenvalues of (A,B) computed by CGGEV3.
( ALPHAR(k)+ALPHAI(k)*i ) / BETA(k) is the k-th
generalized eigenvalue of A and B.
ALPHA1
ALPHA1 is COMPLEX array, dimension (max(NN))
BETA1
BETA1 is COMPLEX array, dimension (max(NN))
Like ALPHAR, ALPHAI, BETA, these arrays contain the
eigenvalues of A and B, but those computed when CGGEV3 only
computes a partial eigendecomposition, i.e. not the
eigenvalues and left and right eigenvectors.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. LWORK >= N*(N+1)
RWORK
RWORK is REAL array, dimension (8*N)
Real workspace.
RESULT
RESULT is REAL array, dimension (2)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code. INFO is the
absolute value of the INFO value returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
January 2015
subroutine cdrgsx (integer NSIZE, integer NCMAX, real THRESH, integer NIN, integer NOUT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) B, complex,
dimension( lda, * ) AI, complex, dimension( lda, * ) BI, complex, dimension( lda, * ) Z,
complex, dimension( lda, * ) Q, complex, dimension( * ) ALPHA, complex, dimension( * )
BETA, complex, dimension( ldc, * ) C, integer LDC, real, dimension( * ) S, complex,
dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * )
IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)
CDRGSX
Purpose:
CDRGSX checks the nonsymmetric generalized eigenvalue (Schur form)
problem expert driver CGGESX.
CGGES factors A and B as Q*S*Z' and Q*T*Z' , where ' means conjugate
transpose, S and T are upper triangular (i.e., in generalized Schur
form), and Q and Z are unitary. It also computes the generalized
eigenvalues (alpha(j),beta(j)), j=1,...,n. Thus,
w(j) = alpha(j)/beta(j) is a root of the characteristic equation
det( A - w(j) B ) = 0
Optionally it also reorders the eigenvalues so that a selected
cluster of eigenvalues appears in the leading diagonal block of the
Schur forms; computes a reciprocal condition number for the average
of the selected eigenvalues; and computes a reciprocal condition
number for the right and left deflating subspaces corresponding to
the selected eigenvalues.
When CDRGSX is called with NSIZE > 0, five (5) types of built-in
matrix pairs are used to test the routine CGGESX.
When CDRGSX is called with NSIZE = 0, it reads in test matrix data
to test CGGESX.
(need more details on what kind of read-in data are needed).
For each matrix pair, the following tests will be performed and
compared with the threshold THRESH except for the tests (7) and (9):
(1) | A - Q S Z' | / ( |A| n ulp )
(2) | B - Q T Z' | / ( |B| n ulp )
(3) | I - QQ' | / ( n ulp )
(4) | I - ZZ' | / ( n ulp )
(5) if A is in Schur form (i.e. triangular form)
(6) maximum over j of D(j) where:
|alpha(j) - S(j,j)| |beta(j) - T(j,j)|
D(j) = ------------------------ + -----------------------
max(|alpha(j)|,|S(j,j)|) max(|beta(j)|,|T(j,j)|)
(7) if sorting worked and SDIM is the number of eigenvalues
which were selected.
(8) the estimated value DIF does not differ from the true values of
Difu and Difl more than a factor 10*THRESH. If the estimate DIF
equals zero the corresponding true values of Difu and Difl
should be less than EPS*norm(A, B). If the true value of Difu
and Difl equal zero, the estimate DIF should be less than
EPS*norm(A, B).
(9) If INFO = N+3 is returned by CGGESX, the reordering "failed"
and we check that DIF = PL = PR = 0 and that the true value of
Difu and Difl is < EPS*norm(A, B). We count the events when
INFO=N+3.
For read-in test matrices, the same tests are run except that the
exact value for DIF (and PL) is input data. Additionally, there is
one more test run for read-in test matrices:
(10) the estimated value PL does not differ from the true value of
PLTRU more than a factor THRESH. If the estimate PL equals
zero the corresponding true value of PLTRU should be less than
EPS*norm(A, B). If the true value of PLTRU equal zero, the
estimate PL should be less than EPS*norm(A, B).
Note that for the built-in tests, a total of 10*NSIZE*(NSIZE-1)
matrix pairs are generated and tested. NSIZE should be kept small.
SVD (routine CGESVD) is used for computing the true value of DIF_u
and DIF_l when testing the built-in test problems.
Built-in Test Matrices
======================
All built-in test matrices are the 2 by 2 block of triangular
matrices
A = [ A11 A12 ] and B = [ B11 B12 ]
[ A22 ] [ B22 ]
where for different type of A11 and A22 are given as the following.
A12 and B12 are chosen so that the generalized Sylvester equation
A11*R - L*A22 = -A12
B11*R - L*B22 = -B12
have prescribed solution R and L.
Type 1: A11 = J_m(1,-1) and A_22 = J_k(1-a,1).
B11 = I_m, B22 = I_k
where J_k(a,b) is the k-by-k Jordan block with ``a'' on
diagonal and ``b'' on superdiagonal.
Type 2: A11 = (a_ij) = ( 2(.5-sin(i)) ) and
B11 = (b_ij) = ( 2(.5-sin(ij)) ) for i=1,...,m, j=i,...,m
A22 = (a_ij) = ( 2(.5-sin(i+j)) ) and
B22 = (b_ij) = ( 2(.5-sin(ij)) ) for i=m+1,...,k, j=i,...,k
Type 3: A11, A22 and B11, B22 are chosen as for Type 2, but each
second diagonal block in A_11 and each third diagonal block
in A_22 are made as 2 by 2 blocks.
Type 4: A11 = ( 20(.5 - sin(ij)) ) and B22 = ( 2(.5 - sin(i+j)) )
for i=1,...,m, j=1,...,m and
A22 = ( 20(.5 - sin(i+j)) ) and B22 = ( 2(.5 - sin(ij)) )
for i=m+1,...,k, j=m+1,...,k
Type 5: (A,B) and have potentially close or common eigenvalues and
very large departure from block diagonality A_11 is chosen
as the m x m leading submatrix of A_1:
| 1 b |
| -b 1 |
| 1+d b |
| -b 1+d |
A_1 = | d 1 |
| -1 d |
| -d 1 |
| -1 -d |
| 1 |
and A_22 is chosen as the k x k leading submatrix of A_2:
| -1 b |
| -b -1 |
| 1-d b |
| -b 1-d |
A_2 = | d 1+b |
| -1-b d |
| -d 1+b |
| -1+b -d |
| 1-d |
and matrix B are chosen as identity matrices (see SLATM5).
Parameters:
NSIZE
NSIZE is INTEGER
The maximum size of the matrices to use. NSIZE >= 0.
If NSIZE = 0, no built-in tests matrices are used, but
read-in test matrices are used to test SGGESX.
NCMAX
NCMAX is INTEGER
Maximum allowable NMAX for generating Kroneker matrix
in call to CLAKF2
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. THRESH >= 0.
NIN
NIN is INTEGER
The FORTRAN unit number for reading in the data file of
problems to solve.
NOUT
NOUT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, NSIZE)
Used to store the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used.
LDA
LDA is INTEGER
The leading dimension of A, B, AI, BI, Z and Q,
LDA >= max( 1, NSIZE ). For the read-in test,
LDA >= max( 1, N ), N is the size of the test matrices.
B
B is COMPLEX array, dimension (LDA, NSIZE)
Used to store the matrix whose eigenvalues are to be
computed. On exit, B contains the last matrix actually used.
AI
AI is COMPLEX array, dimension (LDA, NSIZE)
Copy of A, modified by CGGESX.
BI
BI is COMPLEX array, dimension (LDA, NSIZE)
Copy of B, modified by CGGESX.
Z
Z is COMPLEX array, dimension (LDA, NSIZE)
Z holds the left Schur vectors computed by CGGESX.
Q
Q is COMPLEX array, dimension (LDA, NSIZE)
Q holds the right Schur vectors computed by CGGESX.
ALPHA
ALPHA is COMPLEX array, dimension (NSIZE)
BETA
BETA is COMPLEX array, dimension (NSIZE)
On exit, ALPHA/BETA are the eigenvalues.
C
C is COMPLEX array, dimension (LDC, LDC)
Store the matrix generated by subroutine CLAKF2, this is the
matrix formed by Kronecker products used for estimating
DIF.
LDC
LDC is INTEGER
The leading dimension of C. LDC >= max(1, LDA*LDA/2 ).
S
S is REAL array, dimension (LDC)
Singular values of C
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= 3*NSIZE*NSIZE/2
RWORK
RWORK is REAL array,
dimension (5*NSIZE*NSIZE/2 - 4)
IWORK
IWORK is INTEGER array, dimension (LIWORK)
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK. LIWORK >= NSIZE + 2.
BWORK
BWORK is LOGICAL array, dimension (NSIZE)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrgvx (integer NSIZE, real THRESH, integer NIN, integer NOUT, complex, dimension(
lda, * ) A, integer LDA, complex, dimension( lda, * ) B, complex, dimension( lda, * ) AI,
complex, dimension( lda, * ) BI, complex, dimension( * ) ALPHA, complex, dimension( * )
BETA, complex, dimension( lda, * ) VL, complex, dimension( lda, * ) VR, integer ILO,
integer IHI, real, dimension( * ) LSCALE, real, dimension( * ) RSCALE, real, dimension( *
) S, real, dimension( * ) STRU, real, dimension( * ) DIF, real, dimension( * ) DIFTRU,
complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer,
dimension( * ) IWORK, integer LIWORK, real, dimension( 4 ) RESULT, logical, dimension( * )
BWORK, integer INFO)
CDRGVX
Purpose:
CDRGVX checks the nonsymmetric generalized eigenvalue problem
expert driver CGGEVX.
CGGEVX computes the generalized eigenvalues, (optionally) the left
and/or right eigenvectors, (optionally) computes a balancing
transformation to improve the conditioning, and (optionally)
reciprocal condition numbers for the eigenvalues and eigenvectors.
When CDRGVX is called with NSIZE > 0, two types of test matrix pairs
are generated by the subroutine SLATM6 and test the driver CGGEVX.
The test matrices have the known exact condition numbers for
eigenvalues. For the condition numbers of the eigenvectors
corresponding the first and last eigenvalues are also know
``exactly'' (see CLATM6).
For each matrix pair, the following tests will be performed and
compared with the threshold THRESH.
(1) max over all left eigenvalue/-vector pairs (beta/alpha,l) of
| l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) )
where l**H is the conjugate tranpose of l.
(2) max over all right eigenvalue/-vector pairs (beta/alpha,r) of
| (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )
(3) The condition number S(i) of eigenvalues computed by CGGEVX
differs less than a factor THRESH from the exact S(i) (see
CLATM6).
(4) DIF(i) computed by CTGSNA differs less than a factor 10*THRESH
from the exact value (for the 1st and 5th vectors only).
Test Matrices
=============
Two kinds of test matrix pairs
(A, B) = inverse(YH) * (Da, Db) * inverse(X)
are used in the tests:
1: Da = 1+a 0 0 0 0 Db = 1 0 0 0 0
0 2+a 0 0 0 0 1 0 0 0
0 0 3+a 0 0 0 0 1 0 0
0 0 0 4+a 0 0 0 0 1 0
0 0 0 0 5+a , 0 0 0 0 1 , and
2: Da = 1 -1 0 0 0 Db = 1 0 0 0 0
1 1 0 0 0 0 1 0 0 0
0 0 1 0 0 0 0 1 0 0
0 0 0 1+a 1+b 0 0 0 1 0
0 0 0 -1-b 1+a , 0 0 0 0 1 .
In both cases the same inverse(YH) and inverse(X) are used to compute
(A, B), giving the exact eigenvectors to (A,B) as (YH, X):
YH: = 1 0 -y y -y X = 1 0 -x -x x
0 1 -y y -y 0 1 x -x -x
0 0 1 0 0 0 0 1 0 0
0 0 0 1 0 0 0 0 1 0
0 0 0 0 1, 0 0 0 0 1 , where
a, b, x and y will have all values independently of each other from
{ sqrt(sqrt(ULP)), 0.1, 1, 10, 1/sqrt(sqrt(ULP)) }.
Parameters:
NSIZE
NSIZE is INTEGER
The number of sizes of matrices to use. NSIZE must be at
least zero. If it is zero, no randomly generated matrices
are tested, but any test matrices read from NIN will be
tested. If it is not zero, then N = 5.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NIN
NIN is INTEGER
The FORTRAN unit number for reading in the data file of
problems to solve.
NOUT
NOUT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, NSIZE)
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used.
LDA
LDA is INTEGER
The leading dimension of A, B, AI, BI, Ao, and Bo.
It must be at least 1 and at least NSIZE.
B
B is COMPLEX array, dimension (LDA, NSIZE)
Used to hold the matrix whose eigenvalues are to be
computed. On exit, B contains the last matrix actually used.
AI
AI is COMPLEX array, dimension (LDA, NSIZE)
Copy of A, modified by CGGEVX.
BI
BI is COMPLEX array, dimension (LDA, NSIZE)
Copy of B, modified by CGGEVX.
ALPHA
ALPHA is COMPLEX array, dimension (NSIZE)
BETA
BETA is COMPLEX array, dimension (NSIZE)
On exit, ALPHA/BETA are the eigenvalues.
VL
VL is COMPLEX array, dimension (LDA, NSIZE)
VL holds the left eigenvectors computed by CGGEVX.
VR
VR is COMPLEX array, dimension (LDA, NSIZE)
VR holds the right eigenvectors computed by CGGEVX.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
LSCALE
LSCALE is REAL array, dimension (N)
RSCALE
RSCALE is REAL array, dimension (N)
S
S is REAL array, dimension (N)
STRU
STRU is REAL array, dimension (N)
DIF
DIF is REAL array, dimension (N)
DIFTRU
DIFTRU is REAL array, dimension (N)
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
Leading dimension of WORK. LWORK >= 2*N*N + 2*N
RWORK
RWORK is REAL array, dimension (6*N)
IWORK
IWORK is INTEGER array, dimension (LIWORK)
LIWORK
LIWORK is INTEGER
Leading dimension of IWORK. LIWORK >= N+2.
RESULT
RESULT is REAL array, dimension (4)
BWORK
BWORK is LOGICAL array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: A routine returned an error code.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrvbd (integer NSIZES, integer, dimension( * ) MM, integer, dimension( * ) NN,
integer NTYPES, logical, dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real
THRESH, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldu, * ) U,
integer LDU, complex, dimension( ldvt, * ) VT, integer LDVT, complex, dimension( lda, * )
ASAV, complex, dimension( ldu, * ) USAV, complex, dimension( ldvt, * ) VTSAV, real,
dimension( * ) S, real, dimension( * ) SSAV, real, dimension( * ) E, complex, dimension( *
) WORK, integer LWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer
NOUNIT, integer INFO)
CDRVBD
Purpose:
CDRVBD checks the singular value decomposition (SVD) driver CGESVD
and CGESDD.
CGESVD and CGESDD factors A = U diag(S) VT, where U and VT are
unitary and diag(S) is diagonal with the entries of the array S on
its diagonal. The entries of S are the singular values, nonnegative
and stored in decreasing order. U and VT can be optionally not
computed, overwritten on A, or computed partially.
A is M by N. Let MNMIN = min( M, N ). S has dimension MNMIN.
U can be M by M or M by MNMIN. VT can be N by N or MNMIN by N.
When CDRVBD is called, a number of matrix "sizes" (M's and N's)
and a number of matrix "types" are specified. For each size (M,N)
and each type of matrix, and for the minimal workspace as well as
workspace adequate to permit blocking, an M x N matrix "A" will be
generated and used to test the SVD routines. For each matrix, A will
be factored as A = U diag(S) VT and the following 12 tests computed:
Test for CGESVD:
(1) | A - U diag(S) VT | / ( |A| max(M,N) ulp )
(2) | I - U'U | / ( M ulp )
(3) | I - VT VT' | / ( N ulp )
(4) S contains MNMIN nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(5) | U - Upartial | / ( M ulp ) where Upartial is a partially
computed U.
(6) | VT - VTpartial | / ( N ulp ) where VTpartial is a partially
computed VT.
(7) | S - Spartial | / ( MNMIN ulp |S| ) where Spartial is the
vector of singular values from the partial SVD
Test for CGESDD:
(1) | A - U diag(S) VT | / ( |A| max(M,N) ulp )
(2) | I - U'U | / ( M ulp )
(3) | I - VT VT' | / ( N ulp )
(4) S contains MNMIN nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(5) | U - Upartial | / ( M ulp ) where Upartial is a partially
computed U.
(6) | VT - VTpartial | / ( N ulp ) where VTpartial is a partially
computed VT.
(7) | S - Spartial | / ( MNMIN ulp |S| ) where Spartial is the
vector of singular values from the partial SVD
Test for CGESVJ:
(1) | A - U diag(S) VT | / ( |A| max(M,N) ulp )
(2) | I - U'U | / ( M ulp )
(3) | I - VT VT' | / ( N ulp )
(4) S contains MNMIN nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
Test for CGEJSV:
(1) | A - U diag(S) VT | / ( |A| max(M,N) ulp )
(2) | I - U'U | / ( M ulp )
(3) | I - VT VT' | / ( N ulp )
(4) S contains MNMIN nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
Test for CGESVDX( 'V', 'V', 'A' )/CGESVDX( 'N', 'N', 'A' )
(1) | A - U diag(S) VT | / ( |A| max(M,N) ulp )
(2) | I - U'U | / ( M ulp )
(3) | I - VT VT' | / ( N ulp )
(4) S contains MNMIN nonnegative values in decreasing order.
(Return 0 if true, 1/ULP if false.)
(5) | U - Upartial | / ( M ulp ) where Upartial is a partially
computed U.
(6) | VT - VTpartial | / ( N ulp ) where VTpartial is a partially
computed VT.
(7) | S - Spartial | / ( MNMIN ulp |S| ) where Spartial is the
vector of singular values from the partial SVD
Test for CGESVDX( 'V', 'V', 'I' )
(8) | U' A VT''' - diag(S) | / ( |A| max(M,N) ulp )
(9) | I - U'U | / ( M ulp )
(10) | I - VT VT' | / ( N ulp )
Test for CGESVDX( 'V', 'V', 'V' )
(11) | U' A VT''' - diag(S) | / ( |A| max(M,N) ulp )
(12) | I - U'U | / ( M ulp )
(13) | I - VT VT' | / ( N ulp )
The "sizes" are specified by the arrays MM(1:NSIZES) and
NN(1:NSIZES); the value of each element pair (MM(j),NN(j))
specifies one size. The "types" are specified by a logical array
DOTYPE( 1:NTYPES ); if DOTYPE(j) is .TRUE., then matrix type "j"
will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A matrix of the form U D V, where U and V are unitary and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(4) Same as (3), but multiplied by the underflow-threshold / ULP.
(5) Same as (3), but multiplied by the overflow-threshold * ULP.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CDRVBD does nothing. It must be at least zero.
MM
MM is INTEGER array, dimension (NSIZES)
An array containing the matrix "heights" to be used. For
each j=1,...,NSIZES, if MM(j) is zero, then MM(j) and NN(j)
will be ignored. The MM(j) values must be at least zero.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the matrix "widths" to be used. For
each j=1,...,NSIZES, if NN(j) is zero, then MM(j) and NN(j)
will be ignored. The NN(j) values must be at least zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CDRVBD
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrices are in A and B.
This is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size (m,n), a matrix
of type j will be generated. If NTYPES is smaller than the
maximum number of types defined (PARAMETER MAXTYP), then
types NTYPES+1 through MAXTYP will not be generated. If
NTYPES is larger than MAXTYP, DOTYPE(MAXTYP+1) through
DOTYPE(NTYPES) will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVBD to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
A
A is COMPLEX array, dimension (LDA,max(NN))
Used to hold the matrix whose singular values are to be
computed. On exit, A contains the last matrix actually
used.
LDA
LDA is INTEGER
The leading dimension of A. It must be at
least 1 and at least max( MM ).
U
U is COMPLEX array, dimension (LDU,max(MM))
Used to hold the computed matrix of right singular vectors.
On exit, U contains the last such vectors actually computed.
LDU
LDU is INTEGER
The leading dimension of U. It must be at
least 1 and at least max( MM ).
VT
VT is COMPLEX array, dimension (LDVT,max(NN))
Used to hold the computed matrix of left singular vectors.
On exit, VT contains the last such vectors actually computed.
LDVT
LDVT is INTEGER
The leading dimension of VT. It must be at
least 1 and at least max( NN ).
ASAV
ASAV is COMPLEX array, dimension (LDA,max(NN))
Used to hold a different copy of the matrix whose singular
values are to be computed. On exit, A contains the last
matrix actually used.
USAV
USAV is COMPLEX array, dimension (LDU,max(MM))
Used to hold a different copy of the computed matrix of
right singular vectors. On exit, USAV contains the last such
vectors actually computed.
VTSAV
VTSAV is COMPLEX array, dimension (LDVT,max(NN))
Used to hold a different copy of the computed matrix of
left singular vectors. On exit, VTSAV contains the last such
vectors actually computed.
S
S is REAL array, dimension (max(min(MM,NN)))
Contains the computed singular values.
SSAV
SSAV is REAL array, dimension (max(min(MM,NN)))
Contains another copy of the computed singular values.
E
E is REAL array, dimension (max(min(MM,NN)))
Workspace for CGESVD.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
MAX(3*MIN(M,N)+MAX(M,N)**2,5*MIN(M,N),3*MAX(M,N)) for all
pairs (M,N)=(MM(j),NN(j))
RWORK
RWORK is REAL array,
dimension ( 5*max(max(MM,NN)) )
IWORK
IWORK is INTEGER array, dimension at least 8*min(M,N)
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
INFO
INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some MM(j) < 0
-3: Some NN(j) < 0
-4: NTYPES < 0
-7: THRESH < 0
-10: LDA < 1 or LDA < MMAX, where MMAX is max( MM(j) ).
-12: LDU < 1 or LDU < MMAX.
-14: LDVT < 1 or LDVT < NMAX, where NMAX is max( NN(j) ).
-29: LWORK too small.
If CLATMS, or CGESVD returns an error code, the
absolute value of it is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine cdrves (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) H, complex,
dimension( lda, * ) HT, complex, dimension( * ) W, complex, dimension( * ) WT, complex,
dimension( ldvs, * ) VS, integer LDVS, real, dimension( 13 ) RESULT, complex, dimension( *
) WORK, integer NWORK, real, dimension( * ) RWORK, integer, dimension( * ) IWORK, logical,
dimension( * ) BWORK, integer INFO)
CDRVES
Purpose:
CDRVES checks the nonsymmetric eigenvalue (Schur form) problem
driver CGEES.
When CDRVES is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 13
tests will be performed:
(1) 0 if T is in Schur form, 1/ulp otherwise
(no sorting of eigenvalues)
(2) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (no sorting of eigenvalues).
(3) | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).
(4) 0 if W are eigenvalues of T
1/ulp otherwise
(no sorting of eigenvalues)
(5) 0 if T(with VS) = T(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(6) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(7) 0 if T is in Schur form, 1/ulp otherwise
(with sorting of eigenvalues)
(8) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (with sorting of eigenvalues).
(9) | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).
(10) 0 if W are eigenvalues of T
1/ulp otherwise
(with sorting of eigenvalues)
(11) 0 if T(with VS) = T(without VS),
1/ulp otherwise
(with sorting of eigenvalues)
(12) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(with sorting of eigenvalues)
(13) if sorting worked and SDIM is the number of
eigenvalues which were SELECTed
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A (transposed) Jordan block, with 1's on the diagonal.
(4) A diagonal matrix with evenly spaced entries
1, ..., ULP and random complex angles.
(ULP = (first number larger than 1) - 1 )
(5) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random complex angles.
(6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random complex angles.
(7) Same as (4), but multiplied by a constant near
the overflow threshold
(8) Same as (4), but multiplied by a constant near
the underflow threshold
(9) A matrix of the form U' T U, where U is unitary and
T has evenly spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(10) A matrix of the form U' T U, where U is unitary and
T has geometrically spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(11) A matrix of the form U' T U, where U is orthogonal and
T has "clustered" entries 1, ULP,..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(12) A matrix of the form U' T U, where U is unitary and
T has complex eigenvalues randomly chosen from
ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(13) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(14) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has geometrically spaced entries
1, ..., ULP with random complex angles on the diagonal
and random O(1) entries in the upper triangle.
(15) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(16) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has complex eigenvalues randomly chosen
from ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(17) Same as (16), but multiplied by a constant
near the overflow threshold
(18) Same as (16), but multiplied by a constant
near the underflow threshold
(19) Nonsymmetric matrix with random entries chosen from (-1,1).
If N is at least 4, all entries in first two rows and last
row, and first column and last two columns are zero.
(20) Same as (19), but multiplied by a constant
near the overflow threshold
(21) Same as (19), but multiplied by a constant
near the underflow threshold
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CDRVES does nothing. It must be at least zero.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CDRVES
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVES to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, max(NN))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used.
LDA
LDA is INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least max( NN ).
H
H is COMPLEX array, dimension (LDA, max(NN))
Another copy of the test matrix A, modified by CGEES.
HT
HT is COMPLEX array, dimension (LDA, max(NN))
Yet another copy of the test matrix A, modified by CGEES.
W
W is COMPLEX array, dimension (max(NN))
The computed eigenvalues of A.
WT
WT is COMPLEX array, dimension (max(NN))
Like W, this array contains the eigenvalues of A,
but those computed when CGEES only computes a partial
eigendecomposition, i.e. not Schur vectors
VS
VS is COMPLEX array, dimension (LDVS, max(NN))
VS holds the computed Schur vectors.
LDVS
LDVS is INTEGER
Leading dimension of VS. Must be at least max(1,max(NN)).
RESULT
RESULT is REAL array, dimension (13)
The values computed by the 13 tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
WORK
WORK is COMPLEX array, dimension (NWORK)
NWORK
NWORK is INTEGER
The number of entries in WORK. This must be at least
5*NN(j)+2*NN(j)**2 for all j.
RWORK
RWORK is REAL array, dimension (max(NN))
IWORK
IWORK is INTEGER array, dimension (max(NN))
BWORK
BWORK is LOGICAL array, dimension (max(NN))
INFO
INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-6: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-15: LDVS < 1 or LDVS < NMAX, where NMAX is max( NN(j) ).
-18: NWORK too small.
If CLATMR, CLATMS, CLATME or CGEES returns an error code,
the absolute value of it is returned.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NMAX Largest value in NN.
NERRS The number of tests which have exceeded THRESH
COND, CONDS,
IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTULP, RTULPI Square roots of the previous 4 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
KCONDS(j) Select whether CONDS is to be 1 or
1/sqrt(ulp). (0 means irrelevant.)
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrvev (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) H, complex,
dimension( * ) W, complex, dimension( * ) W1, complex, dimension( ldvl, * ) VL, integer
LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( ldlre, * ) LRE,
integer LDLRE, real, dimension( 7 ) RESULT, complex, dimension( * ) WORK, integer NWORK,
real, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer INFO)
CDRVEV
Purpose:
CDRVEV checks the nonsymmetric eigenvalue problem driver CGEEV.
When CDRVEV is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 7
tests will be performed:
(1) | A * VR - VR * W | / ( n |A| ulp )
Here VR is the matrix of unit right eigenvectors.
W is a diagonal matrix with diagonal entries W(j).
(2) | A**H * VL - VL * W**H | / ( n |A| ulp )
Here VL is the matrix of unit left eigenvectors, A**H is the
conjugate-transpose of A, and W is as above.
(3) | |VR(i)| - 1 | / ulp and whether largest component real
VR(i) denotes the i-th column of VR.
(4) | |VL(i)| - 1 | / ulp and whether largest component real
VL(i) denotes the i-th column of VL.
(5) W(full) = W(partial)
W(full) denotes the eigenvalues computed when both VR and VL
are also computed, and W(partial) denotes the eigenvalues
computed when only W, only W and VR, or only W and VL are
computed.
(6) VR(full) = VR(partial)
VR(full) denotes the right eigenvectors computed when both VR
and VL are computed, and VR(partial) denotes the result
when only VR is computed.
(7) VL(full) = VL(partial)
VL(full) denotes the left eigenvectors computed when both VR
and VL are also computed, and VL(partial) denotes the result
when only VL is computed.
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A (transposed) Jordan block, with 1's on the diagonal.
(4) A diagonal matrix with evenly spaced entries
1, ..., ULP and random complex angles.
(ULP = (first number larger than 1) - 1 )
(5) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random complex angles.
(6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random complex angles.
(7) Same as (4), but multiplied by a constant near
the overflow threshold
(8) Same as (4), but multiplied by a constant near
the underflow threshold
(9) A matrix of the form U' T U, where U is unitary and
T has evenly spaced entries 1, ..., ULP with random complex
angles on the diagonal and random O(1) entries in the upper
triangle.
(10) A matrix of the form U' T U, where U is unitary and
T has geometrically spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(11) A matrix of the form U' T U, where U is unitary and
T has "clustered" entries 1, ULP,..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(12) A matrix of the form U' T U, where U is unitary and
T has complex eigenvalues randomly chosen from
ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(13) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(14) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has geometrically spaced entries
1, ..., ULP with random complex angles on the diagonal
and random O(1) entries in the upper triangle.
(15) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(16) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has complex eigenvalues randomly chosen
from ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(17) Same as (16), but multiplied by a constant
near the overflow threshold
(18) Same as (16), but multiplied by a constant
near the underflow threshold
(19) Nonsymmetric matrix with random entries chosen from |z| < 1
If N is at least 4, all entries in first two rows and last
row, and first column and last two columns are zero.
(20) Same as (19), but multiplied by a constant
near the overflow threshold
(21) Same as (19), but multiplied by a constant
near the underflow threshold
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. If it is zero,
CDRVEV does nothing. It must be at least zero.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. If it is zero, CDRVEV
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVEV to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, max(NN))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used.
LDA
LDA is INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least max(NN).
H
H is COMPLEX array, dimension (LDA, max(NN))
Another copy of the test matrix A, modified by CGEEV.
W
W is COMPLEX array, dimension (max(NN))
The eigenvalues of A. On exit, W are the eigenvalues of
the matrix in A.
W1
W1 is COMPLEX array, dimension (max(NN))
Like W, this array contains the eigenvalues of A,
but those computed when CGEEV only computes a partial
eigendecomposition, i.e. not the eigenvalues and left
and right eigenvectors.
VL
VL is COMPLEX array, dimension (LDVL, max(NN))
VL holds the computed left eigenvectors.
LDVL
LDVL is INTEGER
Leading dimension of VL. Must be at least max(1,max(NN)).
VR
VR is COMPLEX array, dimension (LDVR, max(NN))
VR holds the computed right eigenvectors.
LDVR
LDVR is INTEGER
Leading dimension of VR. Must be at least max(1,max(NN)).
LRE
LRE is COMPLEX array, dimension (LDLRE, max(NN))
LRE holds the computed right or left eigenvectors.
LDLRE
LDLRE is INTEGER
Leading dimension of LRE. Must be at least max(1,max(NN)).
RESULT
RESULT is REAL array, dimension (7)
The values computed by the seven tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
WORK
WORK is COMPLEX array, dimension (NWORK)
NWORK
NWORK is INTEGER
The number of entries in WORK. This must be at least
5*NN(j)+2*NN(j)**2 for all j.
RWORK
RWORK is REAL array, dimension (2*max(NN))
IWORK
IWORK is INTEGER array, dimension (max(NN))
INFO
INFO is INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-6: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-14: LDVL < 1 or LDVL < NMAX, where NMAX is max( NN(j) ).
-16: LDVR < 1 or LDVR < NMAX, where NMAX is max( NN(j) ).
-18: LDLRE < 1 or LDLRE < NMAX, where NMAX is max( NN(j) ).
-21: NWORK too small.
If CLATMR, CLATMS, CLATME or CGEEV returns an error code,
the absolute value of it is returned.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NMAX Largest value in NN.
NERRS The number of tests which have exceeded THRESH
COND, CONDS,
IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTULP, RTULPI Square roots of the previous 4 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
KCONDS(j) Selectw whether CONDS is to be 1 or
1/sqrt(ulp). (0 means irrelevant.)
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrvsg (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB,
real, dimension( * ) D, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension(
lda, * ) AB, complex, dimension( ldb, * ) BB, complex, dimension( * ) AP, complex,
dimension( * ) BP, complex, dimension( * ) WORK, integer NWORK, real, dimension( * )
RWORK, integer LRWORK, integer, dimension( * ) IWORK, integer LIWORK, real, dimension( * )
RESULT, integer INFO)
CDRVSG
Purpose:
CDRVSG checks the complex Hermitian generalized eigenproblem
drivers.
CHEGV computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite generalized
eigenproblem.
CHEGVD computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite generalized
eigenproblem using a divide and conquer algorithm.
CHEGVX computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite generalized
eigenproblem.
CHPGV computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite generalized
eigenproblem in packed storage.
CHPGVD computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite generalized
eigenproblem in packed storage using a divide and
conquer algorithm.
CHPGVX computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite generalized
eigenproblem in packed storage.
CHBGV computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite banded
generalized eigenproblem.
CHBGVD computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite banded
generalized eigenproblem using a divide and conquer
algorithm.
CHBGVX computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian-definite banded
generalized eigenproblem.
When CDRVSG is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix A of the given type will be
generated; a random well-conditioned matrix B is also generated
and the pair (A,B) is used to test the drivers.
For each pair (A,B), the following tests are performed:
(1) CHEGV with ITYPE = 1 and UPLO ='U':
| A Z - B Z D | / ( |A| |Z| n ulp )
(2) as (1) but calling CHPGV
(3) as (1) but calling CHBGV
(4) as (1) but with UPLO = 'L'
(5) as (4) but calling CHPGV
(6) as (4) but calling CHBGV
(7) CHEGV with ITYPE = 2 and UPLO ='U':
| A B Z - Z D | / ( |A| |Z| n ulp )
(8) as (7) but calling CHPGV
(9) as (7) but with UPLO = 'L'
(10) as (9) but calling CHPGV
(11) CHEGV with ITYPE = 3 and UPLO ='U':
| B A Z - Z D | / ( |A| |Z| n ulp )
(12) as (11) but calling CHPGV
(13) as (11) but with UPLO = 'L'
(14) as (13) but calling CHPGV
CHEGVD, CHPGVD and CHBGVD performed the same 14 tests.
CHEGVX, CHPGVX and CHBGVX performed the above 14 tests with
the parameter RANGE = 'A', 'N' and 'I', respectively.
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
This type is used for the matrix A which has half-bandwidth KA.
B is generated as a well-conditioned positive definite matrix
with half-bandwidth KB (<= KA).
Currently, the list of possible types for A is:
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (4), but multiplied by SQRT( overflow threshold )
(7) Same as (4), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U* D U, where U is unitary and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U* D U, where U is unitary and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U* D U, where U is unitary and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Hermitian matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
(16) Same as (8), but with KA = 1 and KB = 1
(17) Same as (8), but with KA = 2 and KB = 1
(18) Same as (8), but with KA = 2 and KB = 2
(19) Same as (8), but with KA = 3 and KB = 1
(20) Same as (8), but with KA = 3 and KB = 2
(21) Same as (8), but with KA = 3 and KB = 3
NSIZES INTEGER
The number of sizes of matrices to use. If it is zero,
CDRVSG does nothing. It must be at least zero.
Not modified.
NN INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
Not modified.
NTYPES INTEGER
The number of elements in DOTYPE. If it is zero, CDRVSG
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
Not modified.
DOTYPE LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
Not modified.
ISEED INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVSG to continue the same random number
sequence.
Modified.
THRESH REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
Not modified.
NOUNIT INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
Not modified.
A COMPLEX array, dimension (LDA , max(NN))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually
used.
Modified.
LDA INTEGER
The leading dimension of A. It must be at
least 1 and at least max( NN ).
Not modified.
B COMPLEX array, dimension (LDB , max(NN))
Used to hold the Hermitian positive definite matrix for
the generailzed problem.
On exit, B contains the last matrix actually
used.
Modified.
LDB INTEGER
The leading dimension of B. It must be at
least 1 and at least max( NN ).
Not modified.
D REAL array, dimension (max(NN))
The eigenvalues of A. On exit, the eigenvalues in D
correspond with the matrix in A.
Modified.
Z COMPLEX array, dimension (LDZ, max(NN))
The matrix of eigenvectors.
Modified.
LDZ INTEGER
The leading dimension of ZZ. It must be at least 1 and
at least max( NN ).
Not modified.
AB COMPLEX array, dimension (LDA, max(NN))
Workspace.
Modified.
BB COMPLEX array, dimension (LDB, max(NN))
Workspace.
Modified.
AP COMPLEX array, dimension (max(NN)**2)
Workspace.
Modified.
BP COMPLEX array, dimension (max(NN)**2)
Workspace.
Modified.
WORK COMPLEX array, dimension (NWORK)
Workspace.
Modified.
NWORK INTEGER
The number of entries in WORK. This must be at least
2*N + N**2 where N = max( NN(j), 2 ).
Not modified.
RWORK REAL array, dimension (LRWORK)
Workspace.
Modified.
LRWORK INTEGER
The number of entries in RWORK. This must be at least
max( 7*N, 1 + 4*N + 2*N*lg(N) + 3*N**2 ) where
N = max( NN(j) ) and lg( N ) = smallest integer k such
that 2**k >= N .
Not modified.
IWORK INTEGER array, dimension (LIWORK))
Workspace.
Modified.
LIWORK INTEGER
The number of entries in IWORK. This must be at least
2 + 5*max( NN(j) ).
Not modified.
RESULT REAL array, dimension (70)
The values computed by the 70 tests described above.
Modified.
INFO INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-5: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-16: LDZ < 1 or LDZ < NMAX.
-21: NWORK too small.
-23: LRWORK too small.
-25: LIWORK too small.
If CLATMR, CLATMS, CHEGV, CHPGV, CHBGV, CHEGVD, CHPGVD,
CHPGVD, CHEGVX, CHPGVX, CHBGVX returns an error code,
the absolute value of it is returned.
Modified.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests that have been run
on this matrix.
NTESTT The total number of tests for this call.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far (computed by SLAFTS).
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrvst (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NOUNIT,
complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D1, real, dimension( * )
D2, real, dimension( * ) D3, real, dimension( * ) WA1, real, dimension( * ) WA2, real,
dimension( * ) WA3, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldu,
* ) V, complex, dimension( * ) TAU, complex, dimension( ldu, * ) Z, complex, dimension( *
) WORK, integer LWORK, real, dimension( * ) RWORK, integer LRWORK, integer, dimension( * )
IWORK, integer LIWORK, real, dimension( * ) RESULT, integer INFO)
CDRVST
Purpose:
CDRVST checks the Hermitian eigenvalue problem drivers.
CHEEVD computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix,
using a divide-and-conquer algorithm.
CHEEVX computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix.
CHEEVR computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix
using the Relatively Robust Representation where it can.
CHPEVD computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix in packed
storage, using a divide-and-conquer algorithm.
CHPEVX computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix in packed
storage.
CHBEVD computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian band matrix,
using a divide-and-conquer algorithm.
CHBEVX computes selected eigenvalues and, optionally,
eigenvectors of a complex Hermitian band matrix.
CHEEV computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix.
CHPEV computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian matrix in packed
storage.
CHBEV computes all eigenvalues and, optionally,
eigenvectors of a complex Hermitian band matrix.
When CDRVST is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the appropriate drivers. For each matrix and each
driver routine called, the following tests will be performed:
(1) | A - Z D Z' | / ( |A| n ulp )
(2) | I - Z Z' | / ( n ulp )
(3) | D1 - D2 | / ( |D1| ulp )
where Z is the matrix of eigenvectors returned when the
eigenvector option is given and D1 and D2 are the eigenvalues
returned with and without the eigenvector option.
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A diagonal matrix with evenly spaced entries
1, ..., ULP and random signs.
(ULP = (first number larger than 1) - 1 )
(4) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random signs.
(5) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random signs.
(6) Same as (4), but multiplied by SQRT( overflow threshold )
(7) Same as (4), but multiplied by SQRT( underflow threshold )
(8) A matrix of the form U* D U, where U is unitary and
D has evenly spaced entries 1, ..., ULP with random signs
on the diagonal.
(9) A matrix of the form U* D U, where U is unitary and
D has geometrically spaced entries 1, ..., ULP with random
signs on the diagonal.
(10) A matrix of the form U* D U, where U is unitary and
D has "clustered" entries 1, ULP,..., ULP with random
signs on the diagonal.
(11) Same as (8), but multiplied by SQRT( overflow threshold )
(12) Same as (8), but multiplied by SQRT( underflow threshold )
(13) Symmetric matrix with random entries chosen from (-1,1).
(14) Same as (13), but multiplied by SQRT( overflow threshold )
(15) Same as (13), but multiplied by SQRT( underflow threshold )
(16) A band matrix with half bandwidth randomly chosen between
0 and N-1, with evenly spaced eigenvalues 1, ..., ULP
with random signs.
(17) Same as (16), but multiplied by SQRT( overflow threshold )
(18) Same as (16), but multiplied by SQRT( underflow threshold )
NSIZES INTEGER
The number of sizes of matrices to use. If it is zero,
CDRVST does nothing. It must be at least zero.
Not modified.
NN INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
Not modified.
NTYPES INTEGER
The number of elements in DOTYPE. If it is zero, CDRVST
does nothing. It must be at least zero. If it is MAXTYP+1
and NSIZES is 1, then an additional type, MAXTYP+1 is
defined, which is to use whatever matrix is in A. This
is only useful if DOTYPE(1:MAXTYP) is .FALSE. and
DOTYPE(MAXTYP+1) is .TRUE. .
Not modified.
DOTYPE LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
Not modified.
ISEED INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVST to continue the same random number
sequence.
Modified.
THRESH REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
Not modified.
NOUNIT INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns IINFO not equal to 0.)
Not modified.
A COMPLEX array, dimension (LDA , max(NN))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually
used.
Modified.
LDA INTEGER
The leading dimension of A. It must be at
least 1 and at least max( NN ).
Not modified.
D1 REAL array, dimension (max(NN))
The eigenvalues of A, as computed by CSTEQR simlutaneously
with Z. On exit, the eigenvalues in D1 correspond with the
matrix in A.
Modified.
D2 REAL array, dimension (max(NN))
The eigenvalues of A, as computed by CSTEQR if Z is not
computed. On exit, the eigenvalues in D2 correspond with
the matrix in A.
Modified.
D3 REAL array, dimension (max(NN))
The eigenvalues of A, as computed by SSTERF. On exit, the
eigenvalues in D3 correspond with the matrix in A.
Modified.
WA1 REAL array, dimension
WA2 REAL array, dimension
WA3 REAL array, dimension
U COMPLEX array, dimension (LDU, max(NN))
The unitary matrix computed by CHETRD + CUNGC3.
Modified.
LDU INTEGER
The leading dimension of U, Z, and V. It must be at
least 1 and at least max( NN ).
Not modified.
V COMPLEX array, dimension (LDU, max(NN))
The Housholder vectors computed by CHETRD in reducing A to
tridiagonal form.
Modified.
TAU COMPLEX array, dimension (max(NN))
The Householder factors computed by CHETRD in reducing A
to tridiagonal form.
Modified.
Z COMPLEX array, dimension (LDU, max(NN))
The unitary matrix of eigenvectors computed by CHEEVD,
CHEEVX, CHPEVD, CHPEVX, CHBEVD, and CHBEVX.
Modified.
WORK - COMPLEX array of dimension ( LWORK )
Workspace.
Modified.
LWORK - INTEGER
The number of entries in WORK. This must be at least
2*max( NN(j), 2 )**2.
Not modified.
RWORK REAL array, dimension (3*max(NN))
Workspace.
Modified.
LRWORK - INTEGER
The number of entries in RWORK.
IWORK INTEGER array, dimension (6*max(NN))
Workspace.
Modified.
LIWORK - INTEGER
The number of entries in IWORK.
RESULT REAL array, dimension (??)
The values computed by the tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
Modified.
INFO INTEGER
If 0, then everything ran OK.
-1: NSIZES < 0
-2: Some NN(j) < 0
-3: NTYPES < 0
-5: THRESH < 0
-9: LDA < 1 or LDA < NMAX, where NMAX is max( NN(j) ).
-16: LDU < 1 or LDU < NMAX.
-21: LWORK too small.
If SLATMR, SLATMS, CHETRD, SORGC3, CSTEQR, SSTERF,
or SORMC2 returns an error code, the
absolute value of it is returned.
Modified.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NTEST The number of tests performed, or which can
be performed so far, for the current matrix.
NTESTT The total number of tests performed so far.
NMAX Largest value in NN.
NMATS The number of matrices generated so far.
NERRS The number of tests which have exceeded THRESH
so far (computed by SLAFTS).
COND, IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTOVFL, RTUNFL Square roots of the previous 2 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrvsx (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NIUNIT, integer
NOUNIT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) H,
complex, dimension( lda, * ) HT, complex, dimension( * ) W, complex, dimension( * ) WT,
complex, dimension( * ) WTMP, complex, dimension( ldvs, * ) VS, integer LDVS, complex,
dimension( ldvs, * ) VS1, real, dimension( 17 ) RESULT, complex, dimension( * ) WORK,
integer LWORK, real, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)
CDRVSX
Purpose:
CDRVSX checks the nonsymmetric eigenvalue (Schur form) problem
expert driver CGEESX.
CDRVSX uses both test matrices generated randomly depending on
data supplied in the calling sequence, as well as on data
read from an input file and including precomputed condition
numbers to which it compares the ones it computes.
When CDRVSX is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified. For each size ("n")
and each type of matrix, one matrix will be generated and used
to test the nonsymmetric eigenroutines. For each matrix, 15
tests will be performed:
(1) 0 if T is in Schur form, 1/ulp otherwise
(no sorting of eigenvalues)
(2) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (no sorting of eigenvalues).
(3) | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).
(4) 0 if W are eigenvalues of T
1/ulp otherwise
(no sorting of eigenvalues)
(5) 0 if T(with VS) = T(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(6) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(7) 0 if T is in Schur form, 1/ulp otherwise
(with sorting of eigenvalues)
(8) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (with sorting of eigenvalues).
(9) | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).
(10) 0 if W are eigenvalues of T
1/ulp otherwise
If workspace sufficient, also compare W with and
without reciprocal condition numbers
(with sorting of eigenvalues)
(11) 0 if T(with VS) = T(without VS),
1/ulp otherwise
If workspace sufficient, also compare T with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(12) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
If workspace sufficient, also compare VS with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(13) if sorting worked and SDIM is the number of
eigenvalues which were SELECTed
If workspace sufficient, also compare SDIM with and
without reciprocal condition numbers
(14) if RCONDE the same no matter if VS and/or RCONDV computed
(15) if RCONDV the same no matter if VS and/or RCONDE computed
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A (transposed) Jordan block, with 1's on the diagonal.
(4) A diagonal matrix with evenly spaced entries
1, ..., ULP and random complex angles.
(ULP = (first number larger than 1) - 1 )
(5) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random complex angles.
(6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random complex angles.
(7) Same as (4), but multiplied by a constant near
the overflow threshold
(8) Same as (4), but multiplied by a constant near
the underflow threshold
(9) A matrix of the form U' T U, where U is unitary and
T has evenly spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(10) A matrix of the form U' T U, where U is unitary and
T has geometrically spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(11) A matrix of the form U' T U, where U is orthogonal and
T has "clustered" entries 1, ULP,..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(12) A matrix of the form U' T U, where U is unitary and
T has complex eigenvalues randomly chosen from
ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(13) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(14) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has geometrically spaced entries
1, ..., ULP with random complex angles on the diagonal
and random O(1) entries in the upper triangle.
(15) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(16) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has complex eigenvalues randomly chosen
from ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(17) Same as (16), but multiplied by a constant
near the overflow threshold
(18) Same as (16), but multiplied by a constant
near the underflow threshold
(19) Nonsymmetric matrix with random entries chosen from (-1,1).
If N is at least 4, all entries in first two rows and last
row, and first column and last two columns are zero.
(20) Same as (19), but multiplied by a constant
near the overflow threshold
(21) Same as (19), but multiplied by a constant
near the underflow threshold
In addition, an input file will be read from logical unit number
NIUNIT. The file contains matrices along with precomputed
eigenvalues and reciprocal condition numbers for the eigenvalue
average and right invariant subspace. For these matrices, in
addition to tests (1) to (15) we will compute the following two
tests:
(16) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal average eigenvalue condition number
computed by CGEESX and RCDEIN (the precomputed true value)
is supplied as input. cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.
(17) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right invariant subspace condition
number computed by CGEESX and RCDVIN (the precomputed true
value) is supplied as input. cond(RCONDV) is the condition
number of RCONDV, and takes errors in computing RCONDV into
account, so that the resulting quantity should be O(ULP).
cond(RCONDV) is essentially given by norm(A)/RCONDE.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. NSIZES must be at
least zero. If it is zero, no randomly generated matrices
are tested, but any test matrices read from NIUNIT will be
tested.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. NTYPES must be at least
zero. If it is zero, no randomly generated test matrices
are tested, but and test matrices read from NIUNIT will be
tested. If it is MAXTYP+1 and NSIZES is 1, then an
additional type, MAXTYP+1 is defined, which is to use
whatever matrix is in A. This is only useful if
DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVSX to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NIUNIT
NIUNIT is INTEGER
The FORTRAN unit number for reading in the data file of
problems to solve.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, max(NN))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used.
LDA
LDA is INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least max( NN ).
H
H is COMPLEX array, dimension (LDA, max(NN))
Another copy of the test matrix A, modified by CGEESX.
HT
HT is COMPLEX array, dimension (LDA, max(NN))
Yet another copy of the test matrix A, modified by CGEESX.
W
W is COMPLEX array, dimension (max(NN))
The computed eigenvalues of A.
WT
WT is COMPLEX array, dimension (max(NN))
Like W, this array contains the eigenvalues of A,
but those computed when CGEESX only computes a partial
eigendecomposition, i.e. not Schur vectors
WTMP
WTMP is COMPLEX array, dimension (max(NN))
More temporary storage for eigenvalues.
VS
VS is COMPLEX array, dimension (LDVS, max(NN))
VS holds the computed Schur vectors.
LDVS
LDVS is INTEGER
Leading dimension of VS. Must be at least max(1,max(NN)).
VS1
VS1 is COMPLEX array, dimension (LDVS, max(NN))
VS1 holds another copy of the computed Schur vectors.
RESULT
RESULT is REAL array, dimension (17)
The values computed by the 17 tests described above.
The values are currently limited to 1/ulp, to avoid overflow.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
max(1,2*NN(j)**2) for all j.
RWORK
RWORK is REAL array, dimension (max(NN))
BWORK
BWORK is LOGICAL array, dimension (max(NN))
INFO
INFO is INTEGER
If 0, successful exit.
<0, input parameter -INFO is incorrect
>0, CLATMR, CLATMS, CLATME or CGET24 returned an error
code and INFO is its absolute value
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NMAX Largest value in NN.
NERRS The number of tests which have exceeded THRESH
COND, CONDS,
IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTULP, RTULPI Square roots of the previous 4 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
KCONDS(j) Selectw whether CONDS is to be 1 or
1/sqrt(ulp). (0 means irrelevant.)
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cdrvvx (integer NSIZES, integer, dimension( * ) NN, integer NTYPES, logical,
dimension( * ) DOTYPE, integer, dimension( 4 ) ISEED, real THRESH, integer NIUNIT, integer
NOUNIT, complex, dimension( lda, * ) A, integer LDA, complex, dimension( lda, * ) H,
complex, dimension( * ) W, complex, dimension( * ) W1, complex, dimension( ldvl, * ) VL,
integer LDVL, complex, dimension( ldvr, * ) VR, integer LDVR, complex, dimension( ldlre, *
) LRE, integer LDLRE, real, dimension( * ) RCONDV, real, dimension( * ) RCNDV1, real,
dimension( * ) RCDVIN, real, dimension( * ) RCONDE, real, dimension( * ) RCNDE1, real,
dimension( * ) RCDEIN, real, dimension( * ) SCALE, real, dimension( * ) SCALE1, real,
dimension( 11 ) RESULT, complex, dimension( * ) WORK, integer NWORK, real, dimension( * )
RWORK, integer INFO)
CDRVVX
Purpose:
CDRVVX checks the nonsymmetric eigenvalue problem expert driver
CGEEVX.
CDRVVX uses both test matrices generated randomly depending on
data supplied in the calling sequence, as well as on data
read from an input file and including precomputed condition
numbers to which it compares the ones it computes.
When CDRVVX is called, a number of matrix "sizes" ("n's") and a
number of matrix "types" are specified in the calling sequence.
For each size ("n") and each type of matrix, one matrix will be
generated and used to test the nonsymmetric eigenroutines. For
each matrix, 9 tests will be performed:
(1) | A * VR - VR * W | / ( n |A| ulp )
Here VR is the matrix of unit right eigenvectors.
W is a diagonal matrix with diagonal entries W(j).
(2) | A**H * VL - VL * W**H | / ( n |A| ulp )
Here VL is the matrix of unit left eigenvectors, A**H is the
conjugate transpose of A, and W is as above.
(3) | |VR(i)| - 1 | / ulp and largest component real
VR(i) denotes the i-th column of VR.
(4) | |VL(i)| - 1 | / ulp and largest component real
VL(i) denotes the i-th column of VL.
(5) W(full) = W(partial)
W(full) denotes the eigenvalues computed when VR, VL, RCONDV
and RCONDE are also computed, and W(partial) denotes the
eigenvalues computed when only some of VR, VL, RCONDV, and
RCONDE are computed.
(6) VR(full) = VR(partial)
VR(full) denotes the right eigenvectors computed when VL, RCONDV
and RCONDE are computed, and VR(partial) denotes the result
when only some of VL and RCONDV are computed.
(7) VL(full) = VL(partial)
VL(full) denotes the left eigenvectors computed when VR, RCONDV
and RCONDE are computed, and VL(partial) denotes the result
when only some of VR and RCONDV are computed.
(8) 0 if SCALE, ILO, IHI, ABNRM (full) =
SCALE, ILO, IHI, ABNRM (partial)
1/ulp otherwise
SCALE, ILO, IHI and ABNRM describe how the matrix is balanced.
(full) is when VR, VL, RCONDE and RCONDV are also computed, and
(partial) is when some are not computed.
(9) RCONDV(full) = RCONDV(partial)
RCONDV(full) denotes the reciprocal condition numbers of the
right eigenvectors computed when VR, VL and RCONDE are also
computed. RCONDV(partial) denotes the reciprocal condition
numbers when only some of VR, VL and RCONDE are computed.
The "sizes" are specified by an array NN(1:NSIZES); the value of
each element NN(j) specifies one size.
The "types" are specified by a logical array DOTYPE( 1:NTYPES );
if DOTYPE(j) is .TRUE., then matrix type "j" will be generated.
Currently, the list of possible types is:
(1) The zero matrix.
(2) The identity matrix.
(3) A (transposed) Jordan block, with 1's on the diagonal.
(4) A diagonal matrix with evenly spaced entries
1, ..., ULP and random complex angles.
(ULP = (first number larger than 1) - 1 )
(5) A diagonal matrix with geometrically spaced entries
1, ..., ULP and random complex angles.
(6) A diagonal matrix with "clustered" entries 1, ULP, ..., ULP
and random complex angles.
(7) Same as (4), but multiplied by a constant near
the overflow threshold
(8) Same as (4), but multiplied by a constant near
the underflow threshold
(9) A matrix of the form U' T U, where U is unitary and
T has evenly spaced entries 1, ..., ULP with random complex
angles on the diagonal and random O(1) entries in the upper
triangle.
(10) A matrix of the form U' T U, where U is unitary and
T has geometrically spaced entries 1, ..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(11) A matrix of the form U' T U, where U is unitary and
T has "clustered" entries 1, ULP,..., ULP with random
complex angles on the diagonal and random O(1) entries in
the upper triangle.
(12) A matrix of the form U' T U, where U is unitary and
T has complex eigenvalues randomly chosen from
ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(13) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has evenly spaced entries 1, ..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(14) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has geometrically spaced entries
1, ..., ULP with random complex angles on the diagonal
and random O(1) entries in the upper triangle.
(15) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has "clustered" entries 1, ULP,..., ULP
with random complex angles on the diagonal and random O(1)
entries in the upper triangle.
(16) A matrix of the form X' T X, where X has condition
SQRT( ULP ) and T has complex eigenvalues randomly chosen
from ULP < |z| < 1 and random O(1) entries in the upper
triangle.
(17) Same as (16), but multiplied by a constant
near the overflow threshold
(18) Same as (16), but multiplied by a constant
near the underflow threshold
(19) Nonsymmetric matrix with random entries chosen from |z| < 1
If N is at least 4, all entries in first two rows and last
row, and first column and last two columns are zero.
(20) Same as (19), but multiplied by a constant
near the overflow threshold
(21) Same as (19), but multiplied by a constant
near the underflow threshold
In addition, an input file will be read from logical unit number
NIUNIT. The file contains matrices along with precomputed
eigenvalues and reciprocal condition numbers for the eigenvalues
and right eigenvectors. For these matrices, in addition to tests
(1) to (9) we will compute the following two tests:
(10) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right eigenvector condition number
computed by CGEEVX and RCDVIN (the precomputed true value)
is supplied as input. cond(RCONDV) is the condition number of
RCONDV, and takes errors in computing RCONDV into account, so
that the resulting quantity should be O(ULP). cond(RCONDV) is
essentially given by norm(A)/RCONDE.
(11) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal eigenvalue condition number
computed by CGEEVX and RCDEIN (the precomputed true value)
is supplied as input. cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.
Parameters:
NSIZES
NSIZES is INTEGER
The number of sizes of matrices to use. NSIZES must be at
least zero. If it is zero, no randomly generated matrices
are tested, but any test matrices read from NIUNIT will be
tested.
NN
NN is INTEGER array, dimension (NSIZES)
An array containing the sizes to be used for the matrices.
Zero values will be skipped. The values must be at least
zero.
NTYPES
NTYPES is INTEGER
The number of elements in DOTYPE. NTYPES must be at least
zero. If it is zero, no randomly generated test matrices
are tested, but and test matrices read from NIUNIT will be
tested. If it is MAXTYP+1 and NSIZES is 1, then an
additional type, MAXTYP+1 is defined, which is to use
whatever matrix is in A. This is only useful if
DOTYPE(1:MAXTYP) is .FALSE. and DOTYPE(MAXTYP+1) is .TRUE. .
DOTYPE
DOTYPE is LOGICAL array, dimension (NTYPES)
If DOTYPE(j) is .TRUE., then for each size in NN a
matrix of that size and of type j will be generated.
If NTYPES is smaller than the maximum number of types
defined (PARAMETER MAXTYP), then types NTYPES+1 through
MAXTYP will not be generated. If NTYPES is larger
than MAXTYP, DOTYPE(MAXTYP+1) through DOTYPE(NTYPES)
will be ignored.
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The array elements should be between 0 and 4095;
if not they will be reduced mod 4096. Also, ISEED(4) must
be odd. The random number generator uses a linear
congruential sequence limited to small integers, and so
should produce machine independent random numbers. The
values of ISEED are changed on exit, and can be used in the
next call to CDRVVX to continue the same random number
sequence.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NIUNIT
NIUNIT is INTEGER
The FORTRAN unit number for reading in the data file of
problems to solve.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
A
A is COMPLEX array, dimension (LDA, max(NN,12))
Used to hold the matrix whose eigenvalues are to be
computed. On exit, A contains the last matrix actually used.
LDA
LDA is INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least max( NN, 12 ). (12 is the
dimension of the largest matrix on the precomputed
input file.)
H
H is COMPLEX array, dimension (LDA, max(NN,12))
Another copy of the test matrix A, modified by CGEEVX.
W
W is COMPLEX array, dimension (max(NN,12))
Contains the eigenvalues of A.
W1
W1 is COMPLEX array, dimension (max(NN,12))
Like W, this array contains the eigenvalues of A,
but those computed when CGEEVX only computes a partial
eigendecomposition, i.e. not the eigenvalues and left
and right eigenvectors.
VL
VL is COMPLEX array, dimension (LDVL, max(NN,12))
VL holds the computed left eigenvectors.
LDVL
LDVL is INTEGER
Leading dimension of VL. Must be at least max(1,max(NN,12)).
VR
VR is COMPLEX array, dimension (LDVR, max(NN,12))
VR holds the computed right eigenvectors.
LDVR
LDVR is INTEGER
Leading dimension of VR. Must be at least max(1,max(NN,12)).
LRE
LRE is COMPLEX array, dimension (LDLRE, max(NN,12))
LRE holds the computed right or left eigenvectors.
LDLRE
LDLRE is INTEGER
Leading dimension of LRE. Must be at least max(1,max(NN,12))
RCONDV
RCONDV is REAL array, dimension (N)
RCONDV holds the computed reciprocal condition numbers
for eigenvectors.
RCNDV1
RCNDV1 is REAL array, dimension (N)
RCNDV1 holds more computed reciprocal condition numbers
for eigenvectors.
RCDVIN
RCDVIN is REAL array, dimension (N)
When COMP = .TRUE. RCDVIN holds the precomputed reciprocal
condition numbers for eigenvectors to be compared with
RCONDV.
RCONDE
RCONDE is REAL array, dimension (N)
RCONDE holds the computed reciprocal condition numbers
for eigenvalues.
RCNDE1
RCNDE1 is REAL array, dimension (N)
RCNDE1 holds more computed reciprocal condition numbers
for eigenvalues.
RCDEIN
RCDEIN is REAL array, dimension (N)
When COMP = .TRUE. RCDEIN holds the precomputed reciprocal
condition numbers for eigenvalues to be compared with
RCONDE.
SCALE
SCALE is REAL array, dimension (N)
Holds information describing balancing of matrix.
SCALE1
SCALE1 is REAL array, dimension (N)
Holds information describing balancing of matrix.
RESULT
RESULT is REAL array, dimension (11)
The values computed by the seven tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
WORK
WORK is COMPLEX array, dimension (NWORK)
NWORK
NWORK is INTEGER
The number of entries in WORK. This must be at least
max(6*12+2*12**2,6*NN(j)+2*NN(j)**2) =
max( 360 ,6*NN(j)+2*NN(j)**2) for all j.
RWORK
RWORK is REAL array, dimension (2*max(NN,12))
INFO
INFO is INTEGER
If 0, then successful exit.
If <0, then input parameter -INFO is incorrect.
If >0, CLATMR, CLATMS, CLATME or CGET23 returned an error
code, and INFO is its absolute value.
-----------------------------------------------------------------------
Some Local Variables and Parameters:
---- ----- --------- --- ----------
ZERO, ONE Real 0 and 1.
MAXTYP The number of types defined.
NMAX Largest value in NN or 12.
NERRS The number of tests which have exceeded THRESH
COND, CONDS,
IMODE Values to be passed to the matrix generators.
ANORM Norm of A; passed to matrix generators.
OVFL, UNFL Overflow and underflow thresholds.
ULP, ULPINV Finest relative precision and its inverse.
RTULP, RTULPI Square roots of the previous 4 values.
The following four arrays decode JTYPE:
KTYPE(j) The general type (1-10) for type "j".
KMODE(j) The MODE value to be passed to the matrix
generator for type "j".
KMAGN(j) The order of magnitude ( O(1),
O(overflow^(1/2) ), O(underflow^(1/2) )
KCONDS(j) Selectw whether CONDS is to be 1 or
1/sqrt(ulp). (0 means irrelevant.)
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cerrbd (character*3 PATH, integer NUNIT)
CERRBD
Purpose:
CERRBD tests the error exits for CGEBRD, CUNGBR, CUNMBR, and CBDSQR.
Parameters:
PATH
PATH is CHARACTER*3
The LAPACK path name for the routines to be tested.
NUNIT
NUNIT is INTEGER
The unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cerrec (character*3 PATH, integer NUNIT)
CERREC
Purpose:
CERREC tests the error exits for the routines for eigen- condition
estimation for REAL matrices:
CTRSYL, CTREXC, CTRSNA and CTRSEN.
Parameters:
PATH
PATH is CHARACTER*3
The LAPACK path name for the routines to be tested.
NUNIT
NUNIT is INTEGER
The unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cerred (character*3 PATH, integer NUNIT)
CERRED
Purpose:
CERRED tests the error exits for the eigenvalue driver routines for
REAL matrices:
PATH driver description
---- ------ -----------
CEV CGEEV find eigenvalues/eigenvectors for nonsymmetric A
CES CGEES find eigenvalues/Schur form for nonsymmetric A
CVX CGEEVX CGEEV + balancing and condition estimation
CSX CGEESX CGEES + balancing and condition estimation
CBD CGESVD compute SVD of an M-by-N matrix A
CGESDD compute SVD of an M-by-N matrix A(by divide and
conquer)
CGEJSV compute SVD of an M-by-N matrix A where M >= N
CGESVDX compute SVD of an M-by-N matrix A(by bisection
and inverse iteration)
Parameters:
PATH
PATH is CHARACTER*3
The LAPACK path name for the routines to be tested.
NUNIT
NUNIT is INTEGER
The unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine cerrgg (character*3 PATH, integer NUNIT)
CERRGG
Purpose:
CERRGG tests the error exits for CGGES, CGGESX, CGGEV, CGGEVX,
CGGES3, CGGEV3, CGGGLM, CGGHRD, CGGLSE, CGGQRF, CGGRQF,
CGGSVD3, CGGSVP3, CHGEQZ, CTGEVC, CTGEXC, CTGSEN, CTGSJA,
CTGSNA, CTGSYL, and CUNCSD.
Parameters:
PATH
PATH is CHARACTER*3
The LAPACK path name for the routines to be tested.
NUNIT
NUNIT is INTEGER
The unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2015
subroutine cerrhs (character*3 PATH, integer NUNIT)
CERRHS
Purpose:
CERRHS tests the error exits for CGEBAK, CGEBAL, CGEHRD, CUNGHR,
CUNMHR, CHSEQR, CHSEIN, and CTREVC.
Parameters:
PATH
PATH is CHARACTER*3
The LAPACK path name for the routines to be tested.
NUNIT
NUNIT is INTEGER
The unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cerrst (character*3 PATH, integer NUNIT)
CERRST
Purpose:
CERRST tests the error exits for CHETRD, CUNGTR, CUNMTR, CHPTRD,
CUNGTR, CUPMTR, CSTEQR, CSTEIN, CPTEQR, CHBTRD,
CHEEV, CHEEVX, CHEEVD, CHBEV, CHBEVX, CHBEVD,
CHPEV, CHPEVX, CHPEVD, and CSTEDC.
Parameters:
PATH
PATH is CHARACTER*3
The LAPACK path name for the routines to be tested.
NUNIT
NUNIT is INTEGER
The unit number for output.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget02 (character TRANS, integer M, integer N, integer NRHS, complex, dimension(
lda, * ) A, integer LDA, complex, dimension( ldx, * ) X, integer LDX, complex, dimension(
ldb, * ) B, integer LDB, real, dimension( * ) RWORK, real RESID)
CGET02
Purpose:
CGET02 computes the residual for a solution of a system of linear
equations A*x = b or A'*x = b:
RESID = norm(B - A*X) / ( norm(A) * norm(X) * EPS ),
where EPS is the machine epsilon.
Parameters:
TRANS
TRANS is CHARACTER*1
Specifies the form of the system of equations:
= 'N': A *x = b
= 'T': A^T*x = b, where A^T is the transpose of A
= 'C': A^H*x = b, where A^H is the conjugate transpose of A
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of columns of B, the matrix of right hand sides.
NRHS >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The original M x N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
X
X is COMPLEX array, dimension (LDX,NRHS)
The computed solution vectors for the system of linear
equations.
LDX
LDX is INTEGER
The leading dimension of the array X. If TRANS = 'N',
LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M).
B
B is COMPLEX array, dimension (LDB,NRHS)
On entry, the right hand side vectors for the system of
linear equations.
On exit, B is overwritten with the difference B - A*X.
LDB
LDB is INTEGER
The leading dimension of the array B. IF TRANS = 'N',
LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N).
RWORK
RWORK is REAL array, dimension (M)
RESID
RESID is REAL
The maximum over the number of right hand sides of
norm(B - A*X) / ( norm(A) * norm(X) * EPS ).
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget10 (integer M, integer N, complex, dimension( lda, * ) A, integer LDA, complex,
dimension( ldb, * ) B, integer LDB, complex, dimension( * ) WORK, real, dimension( * )
RWORK, real RESULT)
CGET10
Purpose:
CGET10 compares two matrices A and B and computes the ratio
RESULT = norm( A - B ) / ( norm(A) * M * EPS )
Parameters:
M
M is INTEGER
The number of rows of the matrices A and B.
N
N is INTEGER
The number of columns of the matrices A and B.
A
A is COMPLEX array, dimension (LDA,N)
The m by n matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX array, dimension (LDB,N)
The m by n matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
WORK
WORK is COMPLEX array, dimension (M)
RWORK
RWORK is COMPLEX array, dimension (M)
RESULT
RESULT is REAL
RESULT = norm( A - B ) / ( norm(A) * M * EPS )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget22 (character TRANSA, character TRANSE, character TRANSW, integer N, complex,
dimension( lda, * ) A, integer LDA, complex, dimension( lde, * ) E, integer LDE, complex,
dimension( * ) W, complex, dimension( * ) WORK, real, dimension( * ) RWORK, real,
dimension( 2 ) RESULT)
CGET22
Purpose:
CGET22 does an eigenvector check.
The basic test is:
RESULT(1) = | A E - E W | / ( |A| |E| ulp )
using the 1-norm. It also tests the normalization of E:
RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
j
where E(j) is the j-th eigenvector, and m-norm is the max-norm of a
vector. The max-norm of a complex n-vector x in this case is the
maximum of |re(x(i)| + |im(x(i)| over i = 1, ..., n.
Parameters:
TRANSA
TRANSA is CHARACTER*1
Specifies whether or not A is transposed.
= 'N': No transpose
= 'T': Transpose
= 'C': Conjugate transpose
TRANSE
TRANSE is CHARACTER*1
Specifies whether or not E is transposed.
= 'N': No transpose, eigenvectors are in columns of E
= 'T': Transpose, eigenvectors are in rows of E
= 'C': Conjugate transpose, eigenvectors are in rows of E
TRANSW
TRANSW is CHARACTER*1
Specifies whether or not W is transposed.
= 'N': No transpose
= 'T': Transpose, same as TRANSW = 'N'
= 'C': Conjugate transpose, use -WI(j) instead of WI(j)
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The matrix whose eigenvectors are in E.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is COMPLEX array, dimension (LDE,N)
The matrix of eigenvectors. If TRANSE = 'N', the eigenvectors
are stored in the columns of E, if TRANSE = 'T' or 'C', the
eigenvectors are stored in the rows of E.
LDE
LDE is INTEGER
The leading dimension of the array E. LDE >= max(1,N).
W
W is COMPLEX array, dimension (N)
The eigenvalues of A.
WORK
WORK is COMPLEX array, dimension (N*N)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
RESULT(1) = | A E - E W | / ( |A| |E| ulp )
RESULT(2) = max | m-norm(E(j)) - 1 | / ( n ulp )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget23 (logical COMP, integer ISRT, character BALANC, integer JTYPE, real THRESH,
integer, dimension( 4 ) ISEED, integer NOUNIT, integer N, complex, dimension( lda, * ) A,
integer LDA, complex, dimension( lda, * ) H, complex, dimension( * ) W, complex,
dimension( * ) W1, complex, dimension( ldvl, * ) VL, integer LDVL, complex, dimension(
ldvr, * ) VR, integer LDVR, complex, dimension( ldlre, * ) LRE, integer LDLRE, real,
dimension( * ) RCONDV, real, dimension( * ) RCNDV1, real, dimension( * ) RCDVIN, real,
dimension( * ) RCONDE, real, dimension( * ) RCNDE1, real, dimension( * ) RCDEIN, real,
dimension( * ) SCALE, real, dimension( * ) SCALE1, real, dimension( 11 ) RESULT, complex,
dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, integer INFO)
CGET23
Purpose:
CGET23 checks the nonsymmetric eigenvalue problem driver CGEEVX.
If COMP = .FALSE., the first 8 of the following tests will be
performed on the input matrix A, and also test 9 if LWORK is
sufficiently large.
if COMP is .TRUE. all 11 tests will be performed.
(1) | A * VR - VR * W | / ( n |A| ulp )
Here VR is the matrix of unit right eigenvectors.
W is a diagonal matrix with diagonal entries W(j).
(2) | A**H * VL - VL * W**H | / ( n |A| ulp )
Here VL is the matrix of unit left eigenvectors, A**H is the
conjugate transpose of A, and W is as above.
(3) | |VR(i)| - 1 | / ulp and largest component real
VR(i) denotes the i-th column of VR.
(4) | |VL(i)| - 1 | / ulp and largest component real
VL(i) denotes the i-th column of VL.
(5) 0 if W(full) = W(partial), 1/ulp otherwise
W(full) denotes the eigenvalues computed when VR, VL, RCONDV
and RCONDE are also computed, and W(partial) denotes the
eigenvalues computed when only some of VR, VL, RCONDV, and
RCONDE are computed.
(6) 0 if VR(full) = VR(partial), 1/ulp otherwise
VR(full) denotes the right eigenvectors computed when VL, RCONDV
and RCONDE are computed, and VR(partial) denotes the result
when only some of VL and RCONDV are computed.
(7) 0 if VL(full) = VL(partial), 1/ulp otherwise
VL(full) denotes the left eigenvectors computed when VR, RCONDV
and RCONDE are computed, and VL(partial) denotes the result
when only some of VR and RCONDV are computed.
(8) 0 if SCALE, ILO, IHI, ABNRM (full) =
SCALE, ILO, IHI, ABNRM (partial)
1/ulp otherwise
SCALE, ILO, IHI and ABNRM describe how the matrix is balanced.
(full) is when VR, VL, RCONDE and RCONDV are also computed, and
(partial) is when some are not computed.
(9) 0 if RCONDV(full) = RCONDV(partial), 1/ulp otherwise
RCONDV(full) denotes the reciprocal condition numbers of the
right eigenvectors computed when VR, VL and RCONDE are also
computed. RCONDV(partial) denotes the reciprocal condition
numbers when only some of VR, VL and RCONDE are computed.
(10) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right eigenvector condition number
computed by CGEEVX and RCDVIN (the precomputed true value)
is supplied as input. cond(RCONDV) is the condition number of
RCONDV, and takes errors in computing RCONDV into account, so
that the resulting quantity should be O(ULP). cond(RCONDV) is
essentially given by norm(A)/RCONDE.
(11) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal eigenvalue condition number
computed by CGEEVX and RCDEIN (the precomputed true value)
is supplied as input. cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.
Parameters:
COMP
COMP is LOGICAL
COMP describes which input tests to perform:
= .FALSE. if the computed condition numbers are not to
be tested against RCDVIN and RCDEIN
= .TRUE. if they are to be compared
ISRT
ISRT is INTEGER
If COMP = .TRUE., ISRT indicates in how the eigenvalues
corresponding to values in RCDVIN and RCDEIN are ordered:
= 0 means the eigenvalues are sorted by
increasing real part
= 1 means the eigenvalues are sorted by
increasing imaginary part
If COMP = .FALSE., ISRT is not referenced.
BALANC
BALANC is CHARACTER
Describes the balancing option to be tested.
= 'N' for no permuting or diagonal scaling
= 'P' for permuting but no diagonal scaling
= 'S' for no permuting but diagonal scaling
= 'B' for permuting and diagonal scaling
JTYPE
JTYPE is INTEGER
Type of input matrix. Used to label output if error occurs.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
ISEED
ISEED is INTEGER array, dimension (4)
If COMP = .FALSE., the random number generator seed
used to produce matrix.
If COMP = .TRUE., ISEED(1) = the number of the example.
Used to label output if error occurs.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
N
N is INTEGER
The dimension of A. N must be at least 0.
A
A is COMPLEX array, dimension (LDA,N)
Used to hold the matrix whose eigenvalues are to be
computed.
LDA
LDA is INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least N.
H
H is COMPLEX array, dimension (LDA,N)
Another copy of the test matrix A, modified by CGEEVX.
W
W is COMPLEX array, dimension (N)
Contains the eigenvalues of A.
W1
W1 is COMPLEX array, dimension (N)
Like W, this array contains the eigenvalues of A,
but those computed when CGEEVX only computes a partial
eigendecomposition, i.e. not the eigenvalues and left
and right eigenvectors.
VL
VL is COMPLEX array, dimension (LDVL,N)
VL holds the computed left eigenvectors.
LDVL
LDVL is INTEGER
Leading dimension of VL. Must be at least max(1,N).
VR
VR is COMPLEX array, dimension (LDVR,N)
VR holds the computed right eigenvectors.
LDVR
LDVR is INTEGER
Leading dimension of VR. Must be at least max(1,N).
LRE
LRE is COMPLEX array, dimension (LDLRE,N)
LRE holds the computed right or left eigenvectors.
LDLRE
LDLRE is INTEGER
Leading dimension of LRE. Must be at least max(1,N).
RCONDV
RCONDV is REAL array, dimension (N)
RCONDV holds the computed reciprocal condition numbers
for eigenvectors.
RCNDV1
RCNDV1 is REAL array, dimension (N)
RCNDV1 holds more computed reciprocal condition numbers
for eigenvectors.
RCDVIN
RCDVIN is REAL array, dimension (N)
When COMP = .TRUE. RCDVIN holds the precomputed reciprocal
condition numbers for eigenvectors to be compared with
RCONDV.
RCONDE
RCONDE is REAL array, dimension (N)
RCONDE holds the computed reciprocal condition numbers
for eigenvalues.
RCNDE1
RCNDE1 is REAL array, dimension (N)
RCNDE1 holds more computed reciprocal condition numbers
for eigenvalues.
RCDEIN
RCDEIN is REAL array, dimension (N)
When COMP = .TRUE. RCDEIN holds the precomputed reciprocal
condition numbers for eigenvalues to be compared with
RCONDE.
SCALE
SCALE is REAL array, dimension (N)
Holds information describing balancing of matrix.
SCALE1
SCALE1 is REAL array, dimension (N)
Holds information describing balancing of matrix.
RESULT
RESULT is REAL array, dimension (11)
The values computed by the 11 tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The number of entries in WORK. This must be at least
2*N, and 2*N+N**2 if tests 9, 10 or 11 are to be performed.
RWORK
RWORK is REAL array, dimension (2*N)
INFO
INFO is INTEGER
If 0, successful exit.
If <0, input parameter -INFO had an incorrect value.
If >0, CGEEVX returned an error code, the absolute
value of which is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget24 (logical COMP, integer JTYPE, real THRESH, integer, dimension( 4 ) ISEED,
integer NOUNIT, integer N, complex, dimension( lda, * ) A, integer LDA, complex,
dimension( lda, * ) H, complex, dimension( lda, * ) HT, complex, dimension( * ) W,
complex, dimension( * ) WT, complex, dimension( * ) WTMP, complex, dimension( ldvs, * )
VS, integer LDVS, complex, dimension( ldvs, * ) VS1, real RCDEIN, real RCDVIN, integer
NSLCT, integer, dimension( * ) ISLCT, integer ISRT, real, dimension( 17 ) RESULT, complex,
dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, logical, dimension( * )
BWORK, integer INFO)
CGET24
Purpose:
CGET24 checks the nonsymmetric eigenvalue (Schur form) problem
expert driver CGEESX.
If COMP = .FALSE., the first 13 of the following tests will be
be performed on the input matrix A, and also tests 14 and 15
if LWORK is sufficiently large.
If COMP = .TRUE., all 17 test will be performed.
(1) 0 if T is in Schur form, 1/ulp otherwise
(no sorting of eigenvalues)
(2) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (no sorting of eigenvalues).
(3) | I - VS VS' | / ( n ulp ) (no sorting of eigenvalues).
(4) 0 if W are eigenvalues of T
1/ulp otherwise
(no sorting of eigenvalues)
(5) 0 if T(with VS) = T(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(6) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
(no sorting of eigenvalues)
(7) 0 if T is in Schur form, 1/ulp otherwise
(with sorting of eigenvalues)
(8) | A - VS T VS' | / ( n |A| ulp )
Here VS is the matrix of Schur eigenvectors, and T is in Schur
form (with sorting of eigenvalues).
(9) | I - VS VS' | / ( n ulp ) (with sorting of eigenvalues).
(10) 0 if W are eigenvalues of T
1/ulp otherwise
If workspace sufficient, also compare W with and
without reciprocal condition numbers
(with sorting of eigenvalues)
(11) 0 if T(with VS) = T(without VS),
1/ulp otherwise
If workspace sufficient, also compare T with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(12) 0 if eigenvalues(with VS) = eigenvalues(without VS),
1/ulp otherwise
If workspace sufficient, also compare VS with and without
reciprocal condition numbers
(with sorting of eigenvalues)
(13) if sorting worked and SDIM is the number of
eigenvalues which were SELECTed
If workspace sufficient, also compare SDIM with and
without reciprocal condition numbers
(14) if RCONDE the same no matter if VS and/or RCONDV computed
(15) if RCONDV the same no matter if VS and/or RCONDE computed
(16) |RCONDE - RCDEIN| / cond(RCONDE)
RCONDE is the reciprocal average eigenvalue condition number
computed by CGEESX and RCDEIN (the precomputed true value)
is supplied as input. cond(RCONDE) is the condition number
of RCONDE, and takes errors in computing RCONDE into account,
so that the resulting quantity should be O(ULP). cond(RCONDE)
is essentially given by norm(A)/RCONDV.
(17) |RCONDV - RCDVIN| / cond(RCONDV)
RCONDV is the reciprocal right invariant subspace condition
number computed by CGEESX and RCDVIN (the precomputed true
value) is supplied as input. cond(RCONDV) is the condition
number of RCONDV, and takes errors in computing RCONDV into
account, so that the resulting quantity should be O(ULP).
cond(RCONDV) is essentially given by norm(A)/RCONDE.
Parameters:
COMP
COMP is LOGICAL
COMP describes which input tests to perform:
= .FALSE. if the computed condition numbers are not to
be tested against RCDVIN and RCDEIN
= .TRUE. if they are to be compared
JTYPE
JTYPE is INTEGER
Type of input matrix. Used to label output if error occurs.
ISEED
ISEED is INTEGER array, dimension (4)
If COMP = .FALSE., the random number generator seed
used to produce matrix.
If COMP = .TRUE., ISEED(1) = the number of the example.
Used to label output if error occurs.
THRESH
THRESH is REAL
A test will count as "failed" if the "error", computed as
described above, exceeds THRESH. Note that the error
is scaled to be O(1), so THRESH should be a reasonably
small multiple of 1, e.g., 10 or 100. In particular,
it should not depend on the precision (single vs. double)
or the size of the matrix. It must be at least zero.
NOUNIT
NOUNIT is INTEGER
The FORTRAN unit number for printing out error messages
(e.g., if a routine returns INFO not equal to 0.)
N
N is INTEGER
The dimension of A. N must be at least 0.
A
A is COMPLEX array, dimension (LDA, N)
Used to hold the matrix whose eigenvalues are to be
computed.
LDA
LDA is INTEGER
The leading dimension of A, and H. LDA must be at
least 1 and at least N.
H
H is COMPLEX array, dimension (LDA, N)
Another copy of the test matrix A, modified by CGEESX.
HT
HT is COMPLEX array, dimension (LDA, N)
Yet another copy of the test matrix A, modified by CGEESX.
W
W is COMPLEX array, dimension (N)
The computed eigenvalues of A.
WT
WT is COMPLEX array, dimension (N)
Like W, this array contains the eigenvalues of A,
but those computed when CGEESX only computes a partial
eigendecomposition, i.e. not Schur vectors
WTMP
WTMP is COMPLEX array, dimension (N)
Like W, this array contains the eigenvalues of A,
but sorted by increasing real or imaginary part.
VS
VS is COMPLEX array, dimension (LDVS, N)
VS holds the computed Schur vectors.
LDVS
LDVS is INTEGER
Leading dimension of VS. Must be at least max(1, N).
VS1
VS1 is COMPLEX array, dimension (LDVS, N)
VS1 holds another copy of the computed Schur vectors.
RCDEIN
RCDEIN is REAL
When COMP = .TRUE. RCDEIN holds the precomputed reciprocal
condition number for the average of selected eigenvalues.
RCDVIN
RCDVIN is REAL
When COMP = .TRUE. RCDVIN holds the precomputed reciprocal
condition number for the selected right invariant subspace.
NSLCT
NSLCT is INTEGER
When COMP = .TRUE. the number of selected eigenvalues
corresponding to the precomputed values RCDEIN and RCDVIN.
ISLCT
ISLCT is INTEGER array, dimension (NSLCT)
When COMP = .TRUE. ISLCT selects the eigenvalues of the
input matrix corresponding to the precomputed values RCDEIN
and RCDVIN. For I=1, ... ,NSLCT, if ISLCT(I) = J, then the
eigenvalue with the J-th largest real or imaginary part is
selected. The real part is used if ISRT = 0, and the
imaginary part if ISRT = 1.
Not referenced if COMP = .FALSE.
ISRT
ISRT is INTEGER
When COMP = .TRUE., ISRT describes how ISLCT is used to
choose a subset of the spectrum.
Not referenced if COMP = .FALSE.
RESULT
RESULT is REAL array, dimension (17)
The values computed by the 17 tests described above.
The values are currently limited to 1/ulp, to avoid
overflow.
WORK
WORK is COMPLEX array, dimension (2*N*N)
LWORK
LWORK is INTEGER
The number of entries in WORK to be passed to CGEESX. This
must be at least 2*N, and N*(N+1)/2 if tests 14--16 are to
be performed.
RWORK
RWORK is REAL array, dimension (N)
BWORK
BWORK is LOGICAL array, dimension (N)
INFO
INFO is INTEGER
If 0, successful exit.
If <0, input parameter -INFO had an incorrect value.
If >0, CGEESX returned an error code, the absolute
value of which is returned.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget35 (real RMAX, integer LMAX, integer NINFO, integer KNT, integer NIN)
CGET35
Purpose:
CGET35 tests CTRSYL, a routine for solving the Sylvester matrix
equation
op(A)*X + ISGN*X*op(B) = scale*C,
A and B are assumed to be in Schur canonical form, op() represents an
optional transpose, and ISGN can be -1 or +1. Scale is an output
less than or equal to 1, chosen to avoid overflow in X.
The test code verifies that the following residual is order 1:
norm(op(A)*X + ISGN*X*op(B) - scale*C) /
(EPS*max(norm(A),norm(B))*norm(X))
Parameters:
RMAX
RMAX is REAL
Value of the largest test ratio.
LMAX
LMAX is INTEGER
Example number where largest test ratio achieved.
NINFO
NINFO is INTEGER
Number of examples where INFO is nonzero.
KNT
KNT is INTEGER
Total number of examples tested.
NIN
NIN is INTEGER
Input logical unit number.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget36 (real RMAX, integer LMAX, integer NINFO, integer KNT, integer NIN)
CGET36
Purpose:
CGET36 tests CTREXC, a routine for reordering diagonal entries of a
matrix in complex Schur form. Thus, CLAEXC computes a unitary matrix
Q such that
Q' * T1 * Q = T2
and where one of the diagonal blocks of T1 (the one at row IFST) has
been moved to position ILST.
The test code verifies that the residual Q'*T1*Q-T2 is small, that T2
is in Schur form, and that the final position of the IFST block is
ILST.
The test matrices are read from a file with logical unit number NIN.
Parameters:
RMAX
RMAX is REAL
Value of the largest test ratio.
LMAX
LMAX is INTEGER
Example number where largest test ratio achieved.
NINFO
NINFO is INTEGER
Number of examples where INFO is nonzero.
KNT
KNT is INTEGER
Total number of examples tested.
NIN
NIN is INTEGER
Input logical unit number.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget37 (real, dimension( 3 ) RMAX, integer, dimension( 3 ) LMAX, integer,
dimension( 3 ) NINFO, integer KNT, integer NIN)
CGET37
Purpose:
CGET37 tests CTRSNA, a routine for estimating condition numbers of
eigenvalues and/or right eigenvectors of a matrix.
The test matrices are read from a file with logical unit number NIN.
Parameters:
RMAX
RMAX is REAL array, dimension (3)
Value of the largest test ratio.
RMAX(1) = largest ratio comparing different calls to CTRSNA
RMAX(2) = largest error in reciprocal condition
numbers taking their conditioning into account
RMAX(3) = largest error in reciprocal condition
numbers not taking their conditioning into
account (may be larger than RMAX(2))
LMAX
LMAX is INTEGER array, dimension (3)
LMAX(i) is example number where largest test ratio
RMAX(i) is achieved. Also:
If CGEHRD returns INFO nonzero on example i, LMAX(1)=i
If CHSEQR returns INFO nonzero on example i, LMAX(2)=i
If CTRSNA returns INFO nonzero on example i, LMAX(3)=i
NINFO
NINFO is INTEGER array, dimension (3)
NINFO(1) = No. of times CGEHRD returned INFO nonzero
NINFO(2) = No. of times CHSEQR returned INFO nonzero
NINFO(3) = No. of times CTRSNA returned INFO nonzero
KNT
KNT is INTEGER
Total number of examples tested.
NIN
NIN is INTEGER
Input logical unit number
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget38 (real, dimension( 3 ) RMAX, integer, dimension( 3 ) LMAX, integer,
dimension( 3 ) NINFO, integer KNT, integer NIN)
CGET38
Purpose:
CGET38 tests CTRSEN, a routine for estimating condition numbers of a
cluster of eigenvalues and/or its associated right invariant subspace
The test matrices are read from a file with logical unit number NIN.
Parameters:
RMAX
RMAX is REAL array, dimension (3)
Values of the largest test ratios.
RMAX(1) = largest residuals from CHST01 or comparing
different calls to CTRSEN
RMAX(2) = largest error in reciprocal condition
numbers taking their conditioning into account
RMAX(3) = largest error in reciprocal condition
numbers not taking their conditioning into
account (may be larger than RMAX(2))
LMAX
LMAX is INTEGER array, dimension (3)
LMAX(i) is example number where largest test ratio
RMAX(i) is achieved. Also:
If CGEHRD returns INFO nonzero on example i, LMAX(1)=i
If CHSEQR returns INFO nonzero on example i, LMAX(2)=i
If CTRSEN returns INFO nonzero on example i, LMAX(3)=i
NINFO
NINFO is INTEGER array, dimension (3)
NINFO(1) = No. of times CGEHRD returned INFO nonzero
NINFO(2) = No. of times CHSEQR returned INFO nonzero
NINFO(3) = No. of times CTRSEN returned INFO nonzero
KNT
KNT is INTEGER
Total number of examples tested.
NIN
NIN is INTEGER
Input logical unit number.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget51 (integer ITYPE, integer N, complex, dimension( lda, * ) A, integer LDA,
complex, dimension( ldb, * ) B, integer LDB, complex, dimension( ldu, * ) U, integer LDU,
complex, dimension( ldv, * ) V, integer LDV, complex, dimension( * ) WORK, real,
dimension( * ) RWORK, real RESULT)
CGET51
Purpose:
CGET51 generally checks a decomposition of the form
A = U B VC>
where * means conjugate transpose and U and V are unitary.
Specifically, if ITYPE=1
RESULT = | A - U B V* | / ( |A| n ulp )
If ITYPE=2, then:
RESULT = | A - B | / ( |A| n ulp )
If ITYPE=3, then:
RESULT = | I - UU* | / ( n ulp )
Parameters:
ITYPE
ITYPE is INTEGER
Specifies the type of tests to be performed.
=1: RESULT = | A - U B V* | / ( |A| n ulp )
=2: RESULT = | A - B | / ( |A| n ulp )
=3: RESULT = | I - UU* | / ( n ulp )
N
N is INTEGER
The size of the matrix. If it is zero, CGET51 does nothing.
It must be at least zero.
A
A is COMPLEX array, dimension (LDA, N)
The original (unfactored) matrix.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 1
and at least N.
B
B is COMPLEX array, dimension (LDB, N)
The factored matrix.
LDB
LDB is INTEGER
The leading dimension of B. It must be at least 1
and at least N.
U
U is COMPLEX array, dimension (LDU, N)
The unitary matrix on the left-hand side in the
decomposition.
Not referenced if ITYPE=2
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N and
at least 1.
V
V is COMPLEX array, dimension (LDV, N)
The unitary matrix on the left-hand side in the
decomposition.
Not referenced if ITYPE=2
LDV
LDV is INTEGER
The leading dimension of V. LDV must be at least N and
at least 1.
WORK
WORK is COMPLEX array, dimension (2*N**2)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL
The values computed by the test specified by ITYPE. The
value is currently limited to 1/ulp, to avoid overflow.
Errors are flagged by RESULT=10/ulp.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget52 (logical LEFT, integer N, complex, dimension( lda, * ) A, integer LDA,
complex, dimension( ldb, * ) B, integer LDB, complex, dimension( lde, * ) E, integer LDE,
complex, dimension( * ) ALPHA, complex, dimension( * ) BETA, complex, dimension( * ) WORK,
real, dimension( * ) RWORK, real, dimension( 2 ) RESULT)
CGET52
Purpose:
CGET52 does an eigenvector check for the generalized eigenvalue
problem.
The basic test for right eigenvectors is:
| b(i) A E(i) - a(i) B E(i) |
RESULT(1) = max -------------------------------
i n ulp max( |b(i) A|, |a(i) B| )
using the 1-norm. Here, a(i)/b(i) = w is the i-th generalized
eigenvalue of A - w B, or, equivalently, b(i)/a(i) = m is the i-th
generalized eigenvalue of m A - B.
H H _ _
For left eigenvectors, A , B , a, and b are used.
CGET52 also tests the normalization of E. Each eigenvector is
supposed to be normalized so that the maximum "absolute value"
of its elements is 1, where in this case, "absolute value"
of a complex value x is |Re(x)| + |Im(x)| ; let us call this
maximum "absolute value" norm of a vector v M(v).
if a(i)=b(i)=0, then the eigenvector is set to be the jth coordinate
vector. The normalization test is:
RESULT(2) = max | M(v(i)) - 1 | / ( n ulp )
eigenvectors v(i)
Parameters:
LEFT
LEFT is LOGICAL
=.TRUE.: The eigenvectors in the columns of E are assumed
to be *left* eigenvectors.
=.FALSE.: The eigenvectors in the columns of E are assumed
to be *right* eigenvectors.
N
N is INTEGER
The size of the matrices. If it is zero, CGET52 does
nothing. It must be at least zero.
A
A is COMPLEX array, dimension (LDA, N)
The matrix A.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 1
and at least N.
B
B is COMPLEX array, dimension (LDB, N)
The matrix B.
LDB
LDB is INTEGER
The leading dimension of B. It must be at least 1
and at least N.
E
E is COMPLEX array, dimension (LDE, N)
The matrix of eigenvectors. It must be O( 1 ).
LDE
LDE is INTEGER
The leading dimension of E. It must be at least 1 and at
least N.
ALPHA
ALPHA is COMPLEX array, dimension (N)
The values a(i) as described above, which, along with b(i),
define the generalized eigenvalues.
BETA
BETA is COMPLEX array, dimension (N)
The values b(i) as described above, which, along with a(i),
define the generalized eigenvalues.
WORK
WORK is COMPLEX array, dimension (N**2)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
The values computed by the test described above. If A E or
B E is likely to overflow, then RESULT(1:2) is set to
10 / ulp.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cget54 (integer N, complex, dimension( lda, * ) A, integer LDA, complex, dimension(
ldb, * ) B, integer LDB, complex, dimension( lds, * ) S, integer LDS, complex, dimension(
ldt, * ) T, integer LDT, complex, dimension( ldu, * ) U, integer LDU, complex, dimension(
ldv, * ) V, integer LDV, complex, dimension( * ) WORK, real RESULT)
CGET54
Purpose:
CGET54 checks a generalized decomposition of the form
A = U*S*V' and B = U*T* V'
where ' means conjugate transpose and U and V are unitary.
Specifically,
RESULT = ||( A - U*S*V', B - U*T*V' )|| / (||( A, B )||*n*ulp )
Parameters:
N
N is INTEGER
The size of the matrix. If it is zero, SGET54 does nothing.
It must be at least zero.
A
A is COMPLEX array, dimension (LDA, N)
The original (unfactored) matrix A.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 1
and at least N.
B
B is COMPLEX array, dimension (LDB, N)
The original (unfactored) matrix B.
LDB
LDB is INTEGER
The leading dimension of B. It must be at least 1
and at least N.
S
S is COMPLEX array, dimension (LDS, N)
The factored matrix S.
LDS
LDS is INTEGER
The leading dimension of S. It must be at least 1
and at least N.
T
T is COMPLEX array, dimension (LDT, N)
The factored matrix T.
LDT
LDT is INTEGER
The leading dimension of T. It must be at least 1
and at least N.
U
U is COMPLEX array, dimension (LDU, N)
The orthogonal matrix on the left-hand side in the
decomposition.
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N and
at least 1.
V
V is COMPLEX array, dimension (LDV, N)
The orthogonal matrix on the left-hand side in the
decomposition.
LDV
LDV is INTEGER
The leading dimension of V. LDV must be at least N and
at least 1.
WORK
WORK is COMPLEX array, dimension (3*N**2)
RESULT
RESULT is REAL
The value RESULT, It is currently limited to 1/ulp, to
avoid overflow. Errors are flagged by RESULT=10/ulp.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cglmts (integer N, integer M, integer P, complex, dimension( lda, * ) A, complex,
dimension( lda, * ) AF, integer LDA, complex, dimension( ldb, * ) B, complex, dimension(
ldb, * ) BF, integer LDB, complex, dimension( * ) D, complex, dimension( * ) DF, complex,
dimension( * ) X, complex, dimension( * ) U, complex, dimension( lwork ) WORK, integer
LWORK, real, dimension( * ) RWORK, real RESULT)
CGLMTS
Purpose:
CGLMTS tests CGGGLM - a subroutine for solving the generalized
linear model problem.
Parameters:
N
N is INTEGER
The number of rows of the matrices A and B. N >= 0.
M
M is INTEGER
The number of columns of the matrix A. M >= 0.
P
P is INTEGER
The number of columns of the matrix B. P >= 0.
A
A is COMPLEX array, dimension (LDA,M)
The N-by-M matrix A.
AF
AF is COMPLEX array, dimension (LDA,M)
LDA
LDA is INTEGER
The leading dimension of the arrays A, AF. LDA >= max(M,N).
B
B is COMPLEX array, dimension (LDB,P)
The N-by-P matrix A.
BF
BF is COMPLEX array, dimension (LDB,P)
LDB
LDB is INTEGER
The leading dimension of the arrays B, BF. LDB >= max(P,N).
D
D is COMPLEX array, dimension( N )
On input, the left hand side of the GLM.
DF
DF is COMPLEX array, dimension( N )
X
X is COMPLEX array, dimension( M )
solution vector X in the GLM problem.
U
U is COMPLEX array, dimension( P )
solution vector U in the GLM problem.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK.
RWORK
RWORK is REAL array, dimension (M)
RESULT
RESULT is REAL
The test ratio:
norm( d - A*x - B*u )
RESULT = -----------------------------------------
(norm(A)+norm(B))*(norm(x)+norm(u))*EPS
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cgqrts (integer N, integer M, integer P, complex, dimension( lda, * ) A, complex,
dimension( lda, * ) AF, complex, dimension( lda, * ) Q, complex, dimension( lda, * ) R,
integer LDA, complex, dimension( * ) TAUA, complex, dimension( ldb, * ) B, complex,
dimension( ldb, * ) BF, complex, dimension( ldb, * ) Z, complex, dimension( ldb, * ) T,
complex, dimension( ldb, * ) BWK, integer LDB, complex, dimension( * ) TAUB, complex,
dimension( lwork ) WORK, integer LWORK, real, dimension( * ) RWORK, real, dimension( 4 )
RESULT)
CGQRTS
Purpose:
CGQRTS tests CGGQRF, which computes the GQR factorization of an
N-by-M matrix A and a N-by-P matrix B: A = Q*R and B = Q*T*Z.
Parameters:
N
N is INTEGER
The number of rows of the matrices A and B. N >= 0.
M
M is INTEGER
The number of columns of the matrix A. M >= 0.
P
P is INTEGER
The number of columns of the matrix B. P >= 0.
A
A is COMPLEX array, dimension (LDA,M)
The N-by-M matrix A.
AF
AF is COMPLEX array, dimension (LDA,N)
Details of the GQR factorization of A and B, as returned
by CGGQRF, see CGGQRF for further details.
Q
Q is COMPLEX array, dimension (LDA,N)
The M-by-M unitary matrix Q.
R
R is COMPLEX array, dimension (LDA,MAX(M,N))
LDA
LDA is INTEGER
The leading dimension of the arrays A, AF, R and Q.
LDA >= max(M,N).
TAUA
TAUA is COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors, as returned
by CGGQRF.
B
B is COMPLEX array, dimension (LDB,P)
On entry, the N-by-P matrix A.
BF
BF is COMPLEX array, dimension (LDB,N)
Details of the GQR factorization of A and B, as returned
by CGGQRF, see CGGQRF for further details.
Z
Z is COMPLEX array, dimension (LDB,P)
The P-by-P unitary matrix Z.
T
T is COMPLEX array, dimension (LDB,max(P,N))
BWK
BWK is COMPLEX array, dimension (LDB,N)
LDB
LDB is INTEGER
The leading dimension of the arrays B, BF, Z and T.
LDB >= max(P,N).
TAUB
TAUB is COMPLEX array, dimension (min(P,N))
The scalar factors of the elementary reflectors, as returned
by SGGRQF.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK, LWORK >= max(N,M,P)**2.
RWORK
RWORK is REAL array, dimension (max(N,M,P))
RESULT
RESULT is REAL array, dimension (4)
The test ratios:
RESULT(1) = norm( R - Q'*A ) / ( MAX(M,N)*norm(A)*ULP)
RESULT(2) = norm( T*Z - Q'*B ) / (MAX(P,N)*norm(B)*ULP)
RESULT(3) = norm( I - Q'*Q ) / ( M*ULP )
RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cgrqts (integer M, integer P, integer N, complex, dimension( lda, * ) A, complex,
dimension( lda, * ) AF, complex, dimension( lda, * ) Q, complex, dimension( lda, * ) R,
integer LDA, complex, dimension( * ) TAUA, complex, dimension( ldb, * ) B, complex,
dimension( ldb, * ) BF, complex, dimension( ldb, * ) Z, complex, dimension( ldb, * ) T,
complex, dimension( ldb, * ) BWK, integer LDB, complex, dimension( * ) TAUB, complex,
dimension( lwork ) WORK, integer LWORK, real, dimension( * ) RWORK, real, dimension( 4 )
RESULT)
CGRQTS
Purpose:
CGRQTS tests CGGRQF, which computes the GRQ factorization of an
M-by-N matrix A and a P-by-N matrix B: A = R*Q and B = Z*T*Q.
Parameters:
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The M-by-N matrix A.
AF
AF is COMPLEX array, dimension (LDA,N)
Details of the GRQ factorization of A and B, as returned
by CGGRQF, see CGGRQF for further details.
Q
Q is COMPLEX array, dimension (LDA,N)
The N-by-N unitary matrix Q.
R
R is COMPLEX array, dimension (LDA,MAX(M,N))
LDA
LDA is INTEGER
The leading dimension of the arrays A, AF, R and Q.
LDA >= max(M,N).
TAUA
TAUA is COMPLEX array, dimension (min(M,N))
The scalar factors of the elementary reflectors, as returned
by SGGQRC.
B
B is COMPLEX array, dimension (LDB,N)
On entry, the P-by-N matrix A.
BF
BF is COMPLEX array, dimension (LDB,N)
Details of the GQR factorization of A and B, as returned
by CGGRQF, see CGGRQF for further details.
Z
Z is REAL array, dimension (LDB,P)
The P-by-P unitary matrix Z.
T
T is COMPLEX array, dimension (LDB,max(P,N))
BWK
BWK is COMPLEX array, dimension (LDB,N)
LDB
LDB is INTEGER
The leading dimension of the arrays B, BF, Z and T.
LDB >= max(P,N).
TAUB
TAUB is COMPLEX array, dimension (min(P,N))
The scalar factors of the elementary reflectors, as returned
by SGGRQF.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK, LWORK >= max(M,P,N)**2.
RWORK
RWORK is REAL array, dimension (M)
RESULT
RESULT is REAL array, dimension (4)
The test ratios:
RESULT(1) = norm( R - A*Q' ) / ( MAX(M,N)*norm(A)*ULP)
RESULT(2) = norm( T*Q - Z'*B ) / (MAX(P,N)*norm(B)*ULP)
RESULT(3) = norm( I - Q'*Q ) / ( N*ULP )
RESULT(4) = norm( I - Z'*Z ) / ( P*ULP )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cgsvts3 (integer M, integer P, integer N, complex, dimension( lda, * ) A, complex,
dimension( lda, * ) AF, integer LDA, complex, dimension( ldb, * ) B, complex, dimension(
ldb, * ) BF, integer LDB, complex, dimension( ldu, * ) U, integer LDU, complex, dimension(
ldv, * ) V, integer LDV, complex, dimension( ldq, * ) Q, integer LDQ, real, dimension( * )
ALPHA, real, dimension( * ) BETA, complex, dimension( ldr, * ) R, integer LDR, integer,
dimension( * ) IWORK, complex, dimension( lwork ) WORK, integer LWORK, real, dimension( *
) RWORK, real, dimension( 6 ) RESULT)
CGSVTS3
Purpose:
CGSVTS3 tests CGGSVD3, which computes the GSVD of an M-by-N matrix A
and a P-by-N matrix B:
U'*A*Q = D1*R and V'*B*Q = D2*R.
Parameters:
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A
A is COMPLEX array, dimension (LDA,M)
The M-by-N matrix A.
AF
AF is COMPLEX array, dimension (LDA,N)
Details of the GSVD of A and B, as returned by CGGSVD3,
see CGGSVD3 for further details.
LDA
LDA is INTEGER
The leading dimension of the arrays A and AF.
LDA >= max( 1,M ).
B
B is COMPLEX array, dimension (LDB,P)
On entry, the P-by-N matrix B.
BF
BF is COMPLEX array, dimension (LDB,N)
Details of the GSVD of A and B, as returned by CGGSVD3,
see CGGSVD3 for further details.
LDB
LDB is INTEGER
The leading dimension of the arrays B and BF.
LDB >= max(1,P).
U
U is COMPLEX array, dimension(LDU,M)
The M by M unitary matrix U.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M).
V
V is COMPLEX array, dimension(LDV,M)
The P by P unitary matrix V.
LDV
LDV is INTEGER
The leading dimension of the array V. LDV >= max(1,P).
Q
Q is COMPLEX array, dimension(LDQ,N)
The N by N unitary matrix Q.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
ALPHA
ALPHA is REAL array, dimension (N)
BETA
BETA is REAL array, dimension (N)
The generalized singular value pairs of A and B, the
``diagonal'' matrices D1 and D2 are constructed from
ALPHA and BETA, see subroutine CGGSVD3 for details.
R
R is COMPLEX array, dimension(LDQ,N)
The upper triangular matrix R.
LDR
LDR is INTEGER
The leading dimension of the array R. LDR >= max(1,N).
IWORK
IWORK is INTEGER array, dimension (N)
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK,
LWORK >= max(M,P,N)*max(M,P,N).
RWORK
RWORK is REAL array, dimension (max(M,P,N))
RESULT
RESULT is REAL array, dimension (6)
The test ratios:
RESULT(1) = norm( U'*A*Q - D1*R ) / ( MAX(M,N)*norm(A)*ULP)
RESULT(2) = norm( V'*B*Q - D2*R ) / ( MAX(P,N)*norm(B)*ULP)
RESULT(3) = norm( I - U'*U ) / ( M*ULP )
RESULT(4) = norm( I - V'*V ) / ( P*ULP )
RESULT(5) = norm( I - Q'*Q ) / ( N*ULP )
RESULT(6) = 0 if ALPHA is in decreasing order;
= ULPINV otherwise.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
August 2015
subroutine chbt21 (character UPLO, integer N, integer KA, integer KS, complex, dimension( lda,
* ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E, complex, dimension(
ldu, * ) U, integer LDU, complex, dimension( * ) WORK, real, dimension( * ) RWORK, real,
dimension( 2 ) RESULT)
CHBT21
Purpose:
CHBT21 generally checks a decomposition of the form
A = U S UC>
where * means conjugate transpose, A is hermitian banded, U is
unitary, and S is diagonal (if KS=0) or symmetric
tridiagonal (if KS=1).
Specifically:
RESULT(1) = | A - U S U* | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU* | / ( n ulp )
Parameters:
UPLO
UPLO is CHARACTER
If UPLO='U', the upper triangle of A and V will be used and
the (strictly) lower triangle will not be referenced.
If UPLO='L', the lower triangle of A and V will be used and
the (strictly) upper triangle will not be referenced.
N
N is INTEGER
The size of the matrix. If it is zero, CHBT21 does nothing.
It must be at least zero.
KA
KA is INTEGER
The bandwidth of the matrix A. It must be at least zero. If
it is larger than N-1, then max( 0, N-1 ) will be used.
KS
KS is INTEGER
The bandwidth of the matrix S. It may only be zero or one.
If zero, then S is diagonal, and E is not referenced. If
one, then S is symmetric tri-diagonal.
A
A is COMPLEX array, dimension (LDA, N)
The original (unfactored) matrix. It is assumed to be
hermitian, and only the upper (UPLO='U') or only the lower
(UPLO='L') will be referenced.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 1
and at least min( KA, N-1 ).
D
D is REAL array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix S.
E
E is REAL array, dimension (N-1)
The off-diagonal of the (symmetric tri-) diagonal matrix S.
E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
(3,2) element, etc.
Not referenced if KS=0.
U
U is COMPLEX array, dimension (LDU, N)
The unitary matrix in the decomposition, expressed as a
dense matrix (i.e., not as a product of Householder
transformations, Givens transformations, etc.)
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N and
at least 1.
WORK
WORK is COMPLEX array, dimension (N**2)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
The values computed by the two tests described above. The
values are currently limited to 1/ulp, to avoid overflow.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine chet21 (integer ITYPE, character UPLO, integer N, integer KBAND, complex,
dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * ) E,
complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer LDV,
complex, dimension( * ) TAU, complex, dimension( * ) WORK, real, dimension( * ) RWORK,
real, dimension( 2 ) RESULT)
CHET21
Purpose:
CHET21 generally checks a decomposition of the form
A = U S UC>
where * means conjugate transpose, A is hermitian, U is unitary, and
S is diagonal (if KBAND=0) or (real) symmetric tridiagonal (if
KBAND=1).
If ITYPE=1, then U is represented as a dense matrix; otherwise U is
expressed as a product of Householder transformations, whose vectors
are stored in the array "V" and whose scaling constants are in "TAU".
We shall use the letter "V" to refer to the product of Householder
transformations (which should be equal to U).
Specifically, if ITYPE=1, then:
RESULT(1) = | A - U S U* | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU* | / ( n ulp )
If ITYPE=2, then:
RESULT(1) = | A - V S V* | / ( |A| n ulp )
If ITYPE=3, then:
RESULT(1) = | I - UV* | / ( n ulp )
For ITYPE > 1, the transformation U is expressed as a product
V = H(1)...H(n-2), where H(j) = I - tau(j) v(j) v(j)C> and each
vector v(j) has its first j elements 0 and the remaining n-j elements
stored in V(j+1:n,j).
Parameters:
ITYPE
ITYPE is INTEGER
Specifies the type of tests to be performed.
1: U expressed as a dense unitary matrix:
RESULT(1) = | A - U S U* | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU* | / ( n ulp )
2: U expressed as a product V of Housholder transformations:
RESULT(1) = | A - V S V* | / ( |A| n ulp )
3: U expressed both as a dense unitary matrix and
as a product of Housholder transformations:
RESULT(1) = | I - UV* | / ( n ulp )
UPLO
UPLO is CHARACTER
If UPLO='U', the upper triangle of A and V will be used and
the (strictly) lower triangle will not be referenced.
If UPLO='L', the lower triangle of A and V will be used and
the (strictly) upper triangle will not be referenced.
N
N is INTEGER
The size of the matrix. If it is zero, CHET21 does nothing.
It must be at least zero.
KBAND
KBAND is INTEGER
The bandwidth of the matrix. It may only be zero or one.
If zero, then S is diagonal, and E is not referenced. If
one, then S is symmetric tri-diagonal.
A
A is COMPLEX array, dimension (LDA, N)
The original (unfactored) matrix. It is assumed to be
hermitian, and only the upper (UPLO='U') or only the lower
(UPLO='L') will be referenced.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least 1
and at least N.
D
D is REAL array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.
E
E is REAL array, dimension (N-1)
The off-diagonal of the (symmetric tri-) diagonal matrix.
E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
(3,2) element, etc.
Not referenced if KBAND=0.
U
U is COMPLEX array, dimension (LDU, N)
If ITYPE=1 or 3, this contains the unitary matrix in
the decomposition, expressed as a dense matrix. If ITYPE=2,
then it is not referenced.
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N and
at least 1.
V
V is COMPLEX array, dimension (LDV, N)
If ITYPE=2 or 3, the columns of this array contain the
Householder vectors used to describe the unitary matrix
in the decomposition. If UPLO='L', then the vectors are in
the lower triangle, if UPLO='U', then in the upper
triangle.
*NOTE* If ITYPE=2 or 3, V is modified and restored. The
subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
is set to one, and later reset to its original value, during
the course of the calculation.
If ITYPE=1, then it is neither referenced nor modified.
LDV
LDV is INTEGER
The leading dimension of V. LDV must be at least N and
at least 1.
TAU
TAU is COMPLEX array, dimension (N)
If ITYPE >= 2, then TAU(j) is the scalar factor of
v(j) v(j)* in the Householder transformation H(j) of
the product U = H(1)...H(n-2)
If ITYPE < 2, then TAU is not referenced.
WORK
WORK is COMPLEX array, dimension (2*N**2)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
The values computed by the two tests described above. The
values are currently limited to 1/ulp, to avoid overflow.
RESULT(1) is always modified. RESULT(2) is modified only
if ITYPE=1.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine chet22 (integer ITYPE, character UPLO, integer N, integer M, integer KBAND,
complex, dimension( lda, * ) A, integer LDA, real, dimension( * ) D, real, dimension( * )
E, complex, dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer
LDV, complex, dimension( * ) TAU, complex, dimension( * ) WORK, real, dimension( * )
RWORK, real, dimension( 2 ) RESULT)
CHET22
Purpose:
CHET22 generally checks a decomposition of the form
A U = U S
where A is complex Hermitian, the columns of U are orthonormal,
and S is diagonal (if KBAND=0) or symmetric tridiagonal (if
KBAND=1). If ITYPE=1, then U is represented as a dense matrix,
otherwise the U is expressed as a product of Householder
transformations, whose vectors are stored in the array "V" and
whose scaling constants are in "TAU"; we shall use the letter
"V" to refer to the product of Householder transformations
(which should be equal to U).
Specifically, if ITYPE=1, then:
RESULT(1) = | U' A U - S | / ( |A| m ulp ) *andC> RESULT(2) = | I - U'U | / ( m ulp )
ITYPE INTEGER
Specifies the type of tests to be performed.
1: U expressed as a dense orthogonal matrix:
RESULT(1) = | A - U S U' | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU' | / ( n ulp )
UPLO CHARACTER
If UPLO='U', the upper triangle of A will be used and the
(strictly) lower triangle will not be referenced. If
UPLO='L', the lower triangle of A will be used and the
(strictly) upper triangle will not be referenced.
Not modified.
N INTEGER
The size of the matrix. If it is zero, CHET22 does nothing.
It must be at least zero.
Not modified.
M INTEGER
The number of columns of U. If it is zero, CHET22 does
nothing. It must be at least zero.
Not modified.
KBAND INTEGER
The bandwidth of the matrix. It may only be zero or one.
If zero, then S is diagonal, and E is not referenced. If
one, then S is symmetric tri-diagonal.
Not modified.
A COMPLEX array, dimension (LDA , N)
The original (unfactored) matrix. It is assumed to be
symmetric, and only the upper (UPLO='U') or only the lower
(UPLO='L') will be referenced.
Not modified.
LDA INTEGER
The leading dimension of A. It must be at least 1
and at least N.
Not modified.
D REAL array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.
Not modified.
E REAL array, dimension (N)
The off-diagonal of the (symmetric tri-) diagonal matrix.
E(1) is ignored, E(2) is the (1,2) and (2,1) element, etc.
Not referenced if KBAND=0.
Not modified.
U COMPLEX array, dimension (LDU, N)
If ITYPE=1, this contains the orthogonal matrix in
the decomposition, expressed as a dense matrix.
Not modified.
LDU INTEGER
The leading dimension of U. LDU must be at least N and
at least 1.
Not modified.
V COMPLEX array, dimension (LDV, N)
If ITYPE=2 or 3, the lower triangle of this array contains
the Householder vectors used to describe the orthogonal
matrix in the decomposition. If ITYPE=1, then it is not
referenced.
Not modified.
LDV INTEGER
The leading dimension of V. LDV must be at least N and
at least 1.
Not modified.
TAU COMPLEX array, dimension (N)
If ITYPE >= 2, then TAU(j) is the scalar factor of
v(j) v(j)' in the Householder transformation H(j) of
the product U = H(1)...H(n-2)
If ITYPE < 2, then TAU is not referenced.
Not modified.
WORK COMPLEX array, dimension (2*N**2)
Workspace.
Modified.
RWORK REAL array, dimension (N)
Workspace.
Modified.
RESULT REAL array, dimension (2)
The values computed by the two tests described above. The
values are currently limited to 1/ulp, to avoid overflow.
RESULT(1) is always modified. RESULT(2) is modified only
if LDU is at least N.
Modified.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine chkxer (character*(*) SRNAMT, integer INFOT, integer NOUT, logical LERR, logical
OK)
CHKXER
Purpose:
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine chpt21 (integer ITYPE, character UPLO, integer N, integer KBAND, complex,
dimension( * ) AP, real, dimension( * ) D, real, dimension( * ) E, complex, dimension(
ldu, * ) U, integer LDU, complex, dimension( * ) VP, complex, dimension( * ) TAU, complex,
dimension( * ) WORK, real, dimension( * ) RWORK, real, dimension( 2 ) RESULT)
CHPT21
Purpose:
CHPT21 generally checks a decomposition of the form
A = U S UC>
where * means conjugate transpose, A is hermitian, U is
unitary, and S is diagonal (if KBAND=0) or (real) symmetric
tridiagonal (if KBAND=1). If ITYPE=1, then U is represented as
a dense matrix, otherwise the U is expressed as a product of
Householder transformations, whose vectors are stored in the
array "V" and whose scaling constants are in "TAU"; we shall
use the letter "V" to refer to the product of Householder
transformations (which should be equal to U).
Specifically, if ITYPE=1, then:
RESULT(1) = | A - U S U* | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU* | / ( n ulp )
If ITYPE=2, then:
RESULT(1) = | A - V S V* | / ( |A| n ulp )
If ITYPE=3, then:
RESULT(1) = | I - UV* | / ( n ulp )
Packed storage means that, for example, if UPLO='U', then the columns
of the upper triangle of A are stored one after another, so that
A(1,j+1) immediately follows A(j,j) in the array AP. Similarly, if
UPLO='L', then the columns of the lower triangle of A are stored one
after another in AP, so that A(j+1,j+1) immediately follows A(n,j)
in the array AP. This means that A(i,j) is stored in:
AP( i + j*(j-1)/2 ) if UPLO='U'
AP( i + (2*n-j)*(j-1)/2 ) if UPLO='L'
The array VP bears the same relation to the matrix V that A does to
AP.
For ITYPE > 1, the transformation U is expressed as a product
of Householder transformations:
If UPLO='U', then V = H(n-1)...H(1), where
H(j) = I - tau(j) v(j) v(j)C>
and the first j-1 elements of v(j) are stored in V(1:j-1,j+1),
(i.e., VP( j*(j+1)/2 + 1 : j*(j+1)/2 + j-1 ) ),
the j-th element is 1, and the last n-j elements are 0.
If UPLO='L', then V = H(1)...H(n-1), where
H(j) = I - tau(j) v(j) v(j)C>
and the first j elements of v(j) are 0, the (j+1)-st is 1, and the
(j+2)-nd through n-th elements are stored in V(j+2:n,j) (i.e.,
in VP( (2*n-j)*(j-1)/2 + j+2 : (2*n-j)*(j-1)/2 + n ) .)
Parameters:
ITYPE
ITYPE is INTEGER
Specifies the type of tests to be performed.
1: U expressed as a dense unitary matrix:
RESULT(1) = | A - U S U* | / ( |A| n ulp ) *andC> RESULT(2) = | I - UU* | / ( n ulp )
2: U expressed as a product V of Housholder transformations:
RESULT(1) = | A - V S V* | / ( |A| n ulp )
3: U expressed both as a dense unitary matrix and
as a product of Housholder transformations:
RESULT(1) = | I - UV* | / ( n ulp )
UPLO
UPLO is CHARACTER
If UPLO='U', the upper triangle of A and V will be used and
the (strictly) lower triangle will not be referenced.
If UPLO='L', the lower triangle of A and V will be used and
the (strictly) upper triangle will not be referenced.
N
N is INTEGER
The size of the matrix. If it is zero, CHPT21 does nothing.
It must be at least zero.
KBAND
KBAND is INTEGER
The bandwidth of the matrix. It may only be zero or one.
If zero, then S is diagonal, and E is not referenced. If
one, then S is symmetric tri-diagonal.
AP
AP is COMPLEX array, dimension (N*(N+1)/2)
The original (unfactored) matrix. It is assumed to be
hermitian, and contains the columns of just the upper
triangle (UPLO='U') or only the lower triangle (UPLO='L'),
packed one after another.
D
D is REAL array, dimension (N)
The diagonal of the (symmetric tri-) diagonal matrix.
E
E is REAL array, dimension (N)
The off-diagonal of the (symmetric tri-) diagonal matrix.
E(1) is the (1,2) and (2,1) element, E(2) is the (2,3) and
(3,2) element, etc.
Not referenced if KBAND=0.
U
U is COMPLEX array, dimension (LDU, N)
If ITYPE=1 or 3, this contains the unitary matrix in
the decomposition, expressed as a dense matrix. If ITYPE=2,
then it is not referenced.
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N and
at least 1.
VP
VP is REAL array, dimension (N*(N+1)/2)
If ITYPE=2 or 3, the columns of this array contain the
Householder vectors used to describe the unitary matrix
in the decomposition, as described in purpose.
*NOTE* If ITYPE=2 or 3, V is modified and restored. The
subdiagonal (if UPLO='L') or the superdiagonal (if UPLO='U')
is set to one, and later reset to its original value, during
the course of the calculation.
If ITYPE=1, then it is neither referenced nor modified.
TAU
TAU is COMPLEX array, dimension (N)
If ITYPE >= 2, then TAU(j) is the scalar factor of
v(j) v(j)* in the Householder transformation H(j) of
the product U = H(1)...H(n-2)
If ITYPE < 2, then TAU is not referenced.
WORK
WORK is COMPLEX array, dimension (N**2)
Workspace.
RWORK
RWORK is REAL array, dimension (N)
Workspace.
RESULT
RESULT is REAL array, dimension (2)
The values computed by the two tests described above. The
values are currently limited to 1/ulp, to avoid overflow.
RESULT(1) is always modified. RESULT(2) is modified only
if ITYPE=1.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine chst01 (integer N, integer ILO, integer IHI, complex, dimension( lda, * ) A,
integer LDA, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( ldq, * ) Q,
integer LDQ, complex, dimension( lwork ) WORK, integer LWORK, real, dimension( * ) RWORK,
real, dimension( 2 ) RESULT)
CHST01
Purpose:
CHST01 tests the reduction of a general matrix A to upper Hessenberg
form: A = Q*H*Q'. Two test ratios are computed;
RESULT(1) = norm( A - Q*H*Q' ) / ( norm(A) * N * EPS )
RESULT(2) = norm( I - Q'*Q ) / ( N * EPS )
The matrix Q is assumed to be given explicitly as it would be
following CGEHRD + CUNGHR.
In this version, ILO and IHI are not used, but they could be used
to save some work if this is desired.
Parameters:
N
N is INTEGER
The order of the matrix A. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
A is assumed to be upper triangular in rows and columns
1:ILO-1 and IHI+1:N, so Q differs from the identity only in
rows and columns ILO+1:IHI.
A
A is COMPLEX array, dimension (LDA,N)
The original n by n matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
H
H is COMPLEX array, dimension (LDH,N)
The upper Hessenberg matrix H from the reduction A = Q*H*Q'
as computed by CGEHRD. H is assumed to be zero below the
first subdiagonal.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
Q
Q is COMPLEX array, dimension (LDQ,N)
The orthogonal matrix Q from the reduction A = Q*H*Q' as
computed by CGEHRD + CUNGHR.
LDQ
LDQ is INTEGER
The leading dimension of the array Q. LDQ >= max(1,N).
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The length of the array WORK. LWORK >= 2*N*N.
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
RESULT(1) = norm( A - Q*H*Q' ) / ( norm(A) * N * EPS )
RESULT(2) = norm( I - Q'*Q ) / ( N * EPS )
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine clarfy (character UPLO, integer N, complex, dimension( * ) V, integer INCV, complex
TAU, complex, dimension( ldc, * ) C, integer LDC, complex, dimension( * ) WORK)
CLARFY
Purpose:
CLARFY applies an elementary reflector, or Householder matrix, H,
to an n x n Hermitian matrix C, from both the left and the right.
H is represented in the form
H = I - tau * v * v'
where tau is a scalar and v is a vector.
If tau is zero, then H is taken to be the unit matrix.
Parameters:
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix C is stored.
= 'U': Upper triangle
= 'L': Lower triangle
N
N is INTEGER
The number of rows and columns of the matrix C. N >= 0.
V
V is COMPLEX array, dimension
(1 + (N-1)*abs(INCV))
The vector v as described above.
INCV
INCV is INTEGER
The increment between successive elements of v. INCV must
not be zero.
TAU
TAU is COMPLEX
The value tau as described above.
C
C is COMPLEX array, dimension (LDC, N)
On entry, the matrix C.
On exit, C is overwritten by H * C * H'.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max( 1, N ).
WORK
WORK is COMPLEX array, dimension (N)
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine clarhs (character*3 PATH, character XTYPE, character UPLO, character TRANS, integer
M, integer N, integer KL, integer KU, integer NRHS, complex, dimension( lda, * ) A,
integer LDA, complex, dimension( ldx, * ) X, integer LDX, complex, dimension( ldb, * ) B,
integer LDB, integer, dimension( 4 ) ISEED, integer INFO)
CLARHS
Purpose:
CLARHS chooses a set of NRHS random solution vectors and sets
up the right hand sides for the linear system
op( A ) * X = B,
where op( A ) may be A, A**T (transpose of A), or A**H (conjugate
transpose of A).
Parameters:
PATH
PATH is CHARACTER*3
The type of the complex matrix A. PATH may be given in any
combination of upper and lower case. Valid paths include
xGE: General m x n matrix
xGB: General banded matrix
xPO: Hermitian positive definite, 2-D storage
xPP: Hermitian positive definite packed
xPB: Hermitian positive definite banded
xHE: Hermitian indefinite, 2-D storage
xHP: Hermitian indefinite packed
xHB: Hermitian indefinite banded
xSY: Symmetric indefinite, 2-D storage
xSP: Symmetric indefinite packed
xSB: Symmetric indefinite banded
xTR: Triangular
xTP: Triangular packed
xTB: Triangular banded
xQR: General m x n matrix
xLQ: General m x n matrix
xQL: General m x n matrix
xRQ: General m x n matrix
where the leading character indicates the precision.
XTYPE
XTYPE is CHARACTER*1
Specifies how the exact solution X will be determined:
= 'N': New solution; generate a random X.
= 'C': Computed; use value of X on entry.
UPLO
UPLO is CHARACTER*1
Used only if A is symmetric or triangular; specifies whether
the upper or lower triangular part of the matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Used only if A is nonsymmetric; specifies the operation
applied to the matrix A.
= 'N': B := A * X
= 'T': B := A**T * X
= 'C': B := A**H * X
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
KL
KL is INTEGER
Used only if A is a band matrix; specifies the number of
subdiagonals of A if A is a general band matrix or if A is
symmetric or triangular and UPLO = 'L'; specifies the number
of superdiagonals of A if A is symmetric or triangular and
UPLO = 'U'. 0 <= KL <= M-1.
KU
KU is INTEGER
Used only if A is a general band matrix or if A is
triangular.
If PATH = xGB, specifies the number of superdiagonals of A,
and 0 <= KU <= N-1.
If PATH = xTR, xTP, or xTB, specifies whether or not the
matrix has unit diagonal:
= 1: matrix has non-unit diagonal (default)
= 2: matrix has unit diagonal
NRHS
NRHS is INTEGER
The number of right hand side vectors in the system A*X = B.
A
A is COMPLEX array, dimension (LDA,N)
The test matrix whose type is given by PATH.
LDA
LDA is INTEGER
The leading dimension of the array A.
If PATH = xGB, LDA >= KL+KU+1.
If PATH = xPB, xSB, xHB, or xTB, LDA >= KL+1.
Otherwise, LDA >= max(1,M).
X
X is or output) COMPLEX array, dimension (LDX,NRHS)
On entry, if XTYPE = 'C' (for 'Computed'), then X contains
the exact solution to the system of linear equations.
On exit, if XTYPE = 'N' (for 'New'), then X is initialized
with random values.
LDX
LDX is INTEGER
The leading dimension of the array X. If TRANS = 'N',
LDX >= max(1,N); if TRANS = 'T', LDX >= max(1,M).
B
B is COMPLEX array, dimension (LDB,NRHS)
The right hand side vector(s) for the system of equations,
computed from B = op(A) * X, where op(A) is determined by
TRANS.
LDB
LDB is INTEGER
The leading dimension of the array B. If TRANS = 'N',
LDB >= max(1,M); if TRANS = 'T', LDB >= max(1,N).
ISEED
ISEED is INTEGER array, dimension (4)
The seed vector for the random number generator (used in
CLATMS). Modified on exit.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine clatm4 (integer ITYPE, integer N, integer NZ1, integer NZ2, logical RSIGN, real
AMAGN, real RCOND, real TRIANG, integer IDIST, integer, dimension( 4 ) ISEED, complex,
dimension( lda, * ) A, integer LDA)
CLATM4
Purpose:
CLATM4 generates basic square matrices, which may later be
multiplied by others in order to produce test matrices. It is
intended mainly to be used to test the generalized eigenvalue
routines.
It first generates the diagonal and (possibly) subdiagonal,
according to the value of ITYPE, NZ1, NZ2, RSIGN, AMAGN, and RCOND.
It then fills in the upper triangle with random numbers, if TRIANG is
non-zero.
Parameters:
ITYPE
ITYPE is INTEGER
The "type" of matrix on the diagonal and sub-diagonal.
If ITYPE < 0, then type abs(ITYPE) is generated and then
swapped end for end (A(I,J) := A'(N-J,N-I).) See also
the description of AMAGN and RSIGN.
Special types:
= 0: the zero matrix.
= 1: the identity.
= 2: a transposed Jordan block.
= 3: If N is odd, then a k+1 x k+1 transposed Jordan block
followed by a k x k identity block, where k=(N-1)/2.
If N is even, then k=(N-2)/2, and a zero diagonal entry
is tacked onto the end.
Diagonal types. The diagonal consists of NZ1 zeros, then
k=N-NZ1-NZ2 nonzeros. The subdiagonal is zero. ITYPE
specifies the nonzero diagonal entries as follows:
= 4: 1, ..., k
= 5: 1, RCOND, ..., RCOND
= 6: 1, ..., 1, RCOND
= 7: 1, a, a^2, ..., a^(k-1)=RCOND
= 8: 1, 1-d, 1-2*d, ..., 1-(k-1)*d=RCOND
= 9: random numbers chosen from (RCOND,1)
= 10: random numbers with distribution IDIST (see CLARND.)
N
N is INTEGER
The order of the matrix.
NZ1
NZ1 is INTEGER
If abs(ITYPE) > 3, then the first NZ1 diagonal entries will
be zero.
NZ2
NZ2 is INTEGER
If abs(ITYPE) > 3, then the last NZ2 diagonal entries will
be zero.
RSIGN
RSIGN is LOGICAL
= .TRUE.: The diagonal and subdiagonal entries will be
multiplied by random numbers of magnitude 1.
= .FALSE.: The diagonal and subdiagonal entries will be
left as they are (usually non-negative real.)
AMAGN
AMAGN is REAL
The diagonal and subdiagonal entries will be multiplied by
AMAGN.
RCOND
RCOND is REAL
If abs(ITYPE) > 4, then the smallest diagonal entry will be
RCOND. RCOND must be between 0 and 1.
TRIANG
TRIANG is REAL
The entries above the diagonal will be random numbers with
magnitude bounded by TRIANG (i.e., random numbers multiplied
by TRIANG.)
IDIST
IDIST is INTEGER
On entry, DIST specifies the type of distribution to be used
to generate a random matrix .
= 1: real and imaginary parts each UNIFORM( 0, 1 )
= 2: real and imaginary parts each UNIFORM( -1, 1 )
= 3: real and imaginary parts each NORMAL( 0, 1 )
= 4: complex number uniform in DISK( 0, 1 )
ISEED
ISEED is INTEGER array, dimension (4)
On entry ISEED specifies the seed of the random number
generator. The values of ISEED are changed on exit, and can
be used in the next call to CLATM4 to continue the same
random number sequence.
Note: ISEED(4) should be odd, for the random number generator
used at present.
A
A is COMPLEX array, dimension (LDA, N)
Array to be computed.
LDA
LDA is INTEGER
Leading dimension of A. Must be at least 1 and at least N.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
logical function clctes (complex Z, complex D)
CLCTES
Purpose:
CLCTES returns .TRUE. if the eigenvalue Z/D is to be selected
(specifically, in this subroutine, if the real part of the
eigenvalue is negative), and otherwise it returns .FALSE..
It is used by the test routine CDRGES to test whether the driver
routine CGGES successfully sorts eigenvalues.
Parameters:
Z
Z is COMPLEX
The numerator part of a complex eigenvalue Z/D.
D
D is COMPLEX
The denominator part of a complex eigenvalue Z/D.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
logical function clctsx (complex ALPHA, complex BETA)
CLCTSX
Purpose:
This function is used to determine what eigenvalues will be
selected. If this is part of the test driver CDRGSX, do not
change the code UNLESS you are testing input examples and not
using the built-in examples.
Parameters:
ALPHA
ALPHA is COMPLEX
BETA
BETA is COMPLEX
parameters to decide whether the pair (ALPHA, BETA) is
selected.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine clsets (integer M, integer P, integer N, complex, dimension( lda, * ) A, complex,
dimension( lda, * ) AF, integer LDA, complex, dimension( ldb, * ) B, complex, dimension(
ldb, * ) BF, integer LDB, complex, dimension( * ) C, complex, dimension( * ) CF, complex,
dimension( * ) D, complex, dimension( * ) DF, complex, dimension( * ) X, complex,
dimension( lwork ) WORK, integer LWORK, real, dimension( * ) RWORK, real, dimension( 2 )
RESULT)
CLSETS
Purpose:
CLSETS tests CGGLSE - a subroutine for solving linear equality
constrained least square problem (LSE).
Parameters:
M
M is INTEGER
The number of rows of the matrix A. M >= 0.
P
P is INTEGER
The number of rows of the matrix B. P >= 0.
N
N is INTEGER
The number of columns of the matrices A and B. N >= 0.
A
A is COMPLEX array, dimension (LDA,N)
The M-by-N matrix A.
AF
AF is COMPLEX array, dimension (LDA,N)
LDA
LDA is INTEGER
The leading dimension of the arrays A, AF, Q and R.
LDA >= max(M,N).
B
B is COMPLEX array, dimension (LDB,N)
The P-by-N matrix A.
BF
BF is COMPLEX array, dimension (LDB,N)
LDB
LDB is INTEGER
The leading dimension of the arrays B, BF, V and S.
LDB >= max(P,N).
C
C is COMPLEX array, dimension( M )
the vector C in the LSE problem.
CF
CF is COMPLEX array, dimension( M )
D
D is COMPLEX array, dimension( P )
the vector D in the LSE problem.
DF
DF is COMPLEX array, dimension( P )
X
X is COMPLEX array, dimension( N )
solution vector X in the LSE problem.
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The dimension of the array WORK.
RWORK
RWORK is REAL array, dimension (M)
RESULT
RESULT is REAL array, dimension (2)
The test ratios:
RESULT(1) = norm( A*x - c )/ norm(A)*norm(X)*EPS
RESULT(2) = norm( B*x - d )/ norm(B)*norm(X)*EPS
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine csbmv (character UPLO, integer N, integer K, complex ALPHA, complex, dimension(
lda, * ) A, integer LDA, complex, dimension( * ) X, integer INCX, complex BETA, complex,
dimension( * ) Y, integer INCY)
CSBMV
Purpose:
CSBMV performs the matrix-vector operation
y := alpha*A*x + beta*y,
where alpha and beta are scalars, x and y are n element vectors and
A is an n by n symmetric band matrix, with k super-diagonals.
UPLO - CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the band matrix A is being supplied as
follows:
UPLO = 'U' or 'u' The upper triangular part of A is
being supplied.
UPLO = 'L' or 'l' The lower triangular part of A is
being supplied.
Unchanged on exit.
N - INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
K - INTEGER
On entry, K specifies the number of super-diagonals of the
matrix A. K must satisfy 0 .le. K.
Unchanged on exit.
ALPHA - COMPLEX
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
A - COMPLEX array, dimension( LDA, N )
Before entry with UPLO = 'U' or 'u', the leading ( k + 1 )
by n part of the array A must contain the upper triangular
band part of the symmetric matrix, supplied column by
column, with the leading diagonal of the matrix in row
( k + 1 ) of the array, the first super-diagonal starting at
position 2 in row k, and so on. The top left k by k triangle
of the array A is not referenced.
The following program segment will transfer the upper
triangular part of a symmetric band matrix from conventional
full matrix storage to band storage:
DO 20, J = 1, N
M = K + 1 - J
DO 10, I = MAX( 1, J - K ), J
A( M + I, J ) = matrix( I, J )
10 CONTINUE
20 CONTINUE
Before entry with UPLO = 'L' or 'l', the leading ( k + 1 )
by n part of the array A must contain the lower triangular
band part of the symmetric matrix, supplied column by
column, with the leading diagonal of the matrix in row 1 of
the array, the first sub-diagonal starting at position 1 in
row 2, and so on. The bottom right k by k triangle of the
array A is not referenced.
The following program segment will transfer the lower
triangular part of a symmetric band matrix from conventional
full matrix storage to band storage:
DO 20, J = 1, N
M = 1 - J
DO 10, I = J, MIN( N, J + K )
A( M + I, J ) = matrix( I, J )
10 CONTINUE
20 CONTINUE
Unchanged on exit.
LDA - INTEGER
On entry, LDA specifies the first dimension of A as declared
in the calling (sub) program. LDA must be at least
( k + 1 ).
Unchanged on exit.
X - COMPLEX array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the
vector x.
Unchanged on exit.
INCX - INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
BETA - COMPLEX
On entry, BETA specifies the scalar beta.
Unchanged on exit.
Y - COMPLEX array, dimension at least
( 1 + ( N - 1 )*abs( INCY ) ).
Before entry, the incremented array Y must contain the
vector y. On exit, Y is overwritten by the updated vector y.
INCY - INTEGER
On entry, INCY specifies the increment for the elements of
Y. INCY must not be zero.
Unchanged on exit.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine csgt01 (integer ITYPE, character UPLO, integer N, integer M, complex, dimension(
lda, * ) A, integer LDA, complex, dimension( ldb, * ) B, integer LDB, complex, dimension(
ldz, * ) Z, integer LDZ, real, dimension( * ) D, complex, dimension( * ) WORK, real,
dimension( * ) RWORK, real, dimension( * ) RESULT)
CSGT01
Purpose:
CSGT01 checks a decomposition of the form
A Z = B Z D or
A B Z = Z D or
B A Z = Z D
where A is a Hermitian matrix, B is Hermitian positive definite,
Z is unitary, and D is diagonal.
One of the following test ratios is computed:
ITYPE = 1: RESULT(1) = | A Z - B Z D | / ( |A| |Z| n ulp )
ITYPE = 2: RESULT(1) = | A B Z - Z D | / ( |A| |Z| n ulp )
ITYPE = 3: RESULT(1) = | B A Z - Z D | / ( |A| |Z| n ulp )
Parameters:
ITYPE
ITYPE is INTEGER
The form of the Hermitian generalized eigenproblem.
= 1: A*z = (lambda)*B*z
= 2: A*B*z = (lambda)*z
= 3: B*A*z = (lambda)*z
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrices A and B is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
M
M is INTEGER
The number of eigenvalues found. M >= 0.
A
A is COMPLEX array, dimension (LDA, N)
The original Hermitian matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B
B is COMPLEX array, dimension (LDB, N)
The original Hermitian positive definite matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Z
Z is COMPLEX array, dimension (LDZ, M)
The computed eigenvectors of the generalized eigenproblem.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
D
D is REAL array, dimension (M)
The computed eigenvalues of the generalized eigenproblem.
WORK
WORK is COMPLEX array, dimension (N*N)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (1)
The test ratio as described above.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
logical function cslect (complex Z)
CSLECT
Purpose:
CSLECT returns .TRUE. if the eigenvalue Z is to be selected,
otherwise it returns .FALSE.
It is used by CCHK41 to test if CGEES successfully sorts eigenvalues,
and by CCHK43 to test if CGEESX successfully sorts eigenvalues.
The common block /SSLCT/ controls how eigenvalues are selected.
If SELOPT = 0, then CSLECT return .TRUE. when real(Z) is less than
zero, and .FALSE. otherwise.
If SELOPT is at least 1, CSLECT returns SELVAL(SELOPT) and adds 1
to SELOPT, cycling back to 1 at SELMAX.
Parameters:
Z
Z is COMPLEX
The eigenvalue Z.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cstt21 (integer N, integer KBAND, real, dimension( * ) AD, real, dimension( * ) AE,
real, dimension( * ) SD, real, dimension( * ) SE, complex, dimension( ldu, * ) U, integer
LDU, complex, dimension( * ) WORK, real, dimension( * ) RWORK, real, dimension( 2 )
RESULT)
CSTT21
Purpose:
CSTT21 checks a decomposition of the form
A = U S UC>
where * means conjugate transpose, A is real symmetric tridiagonal,
U is unitary, and S is real and diagonal (if KBAND=0) or symmetric
tridiagonal (if KBAND=1). Two tests are performed:
RESULT(1) = | A - U S U* | / ( |A| n ulp )
RESULT(2) = | I - UU* | / ( n ulp )
Parameters:
N
N is INTEGER
The size of the matrix. If it is zero, CSTT21 does nothing.
It must be at least zero.
KBAND
KBAND is INTEGER
The bandwidth of the matrix S. It may only be zero or one.
If zero, then S is diagonal, and SE is not referenced. If
one, then S is symmetric tri-diagonal.
AD
AD is REAL array, dimension (N)
The diagonal of the original (unfactored) matrix A. A is
assumed to be real symmetric tridiagonal.
AE
AE is REAL array, dimension (N-1)
The off-diagonal of the original (unfactored) matrix A. A
is assumed to be symmetric tridiagonal. AE(1) is the (1,2)
and (2,1) element, AE(2) is the (2,3) and (3,2) element, etc.
SD
SD is REAL array, dimension (N)
The diagonal of the real (symmetric tri-) diagonal matrix S.
SE
SE is REAL array, dimension (N-1)
The off-diagonal of the (symmetric tri-) diagonal matrix S.
Not referenced if KBSND=0. If KBAND=1, then AE(1) is the
(1,2) and (2,1) element, SE(2) is the (2,3) and (3,2)
element, etc.
U
U is COMPLEX array, dimension (LDU, N)
The unitary matrix in the decomposition.
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N.
WORK
WORK is COMPLEX array, dimension (N**2)
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
The values computed by the two tests described above. The
values are currently limited to 1/ulp, to avoid overflow.
RESULT(1) is always modified.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cstt22 (integer N, integer M, integer KBAND, real, dimension( * ) AD, real,
dimension( * ) AE, real, dimension( * ) SD, real, dimension( * ) SE, complex, dimension(
ldu, * ) U, integer LDU, complex, dimension( ldwork, * ) WORK, integer LDWORK, real,
dimension( * ) RWORK, real, dimension( 2 ) RESULT)
CSTT22
Purpose:
CSTT22 checks a set of M eigenvalues and eigenvectors,
A U = U S
where A is Hermitian tridiagonal, the columns of U are unitary,
and S is diagonal (if KBAND=0) or Hermitian tridiagonal (if KBAND=1).
Two tests are performed:
RESULT(1) = | U* A U - S | / ( |A| m ulp )
RESULT(2) = | I - U*U | / ( m ulp )
Parameters:
N
N is INTEGER
The size of the matrix. If it is zero, CSTT22 does nothing.
It must be at least zero.
M
M is INTEGER
The number of eigenpairs to check. If it is zero, CSTT22
does nothing. It must be at least zero.
KBAND
KBAND is INTEGER
The bandwidth of the matrix S. It may only be zero or one.
If zero, then S is diagonal, and SE is not referenced. If
one, then S is Hermitian tri-diagonal.
AD
AD is REAL array, dimension (N)
The diagonal of the original (unfactored) matrix A. A is
assumed to be Hermitian tridiagonal.
AE
AE is REAL array, dimension (N)
The off-diagonal of the original (unfactored) matrix A. A
is assumed to be Hermitian tridiagonal. AE(1) is ignored,
AE(2) is the (1,2) and (2,1) element, etc.
SD
SD is REAL array, dimension (N)
The diagonal of the (Hermitian tri-) diagonal matrix S.
SE
SE is REAL array, dimension (N)
The off-diagonal of the (Hermitian tri-) diagonal matrix S.
Not referenced if KBSND=0. If KBAND=1, then AE(1) is
ignored, SE(2) is the (1,2) and (2,1) element, etc.
U
U is REAL array, dimension (LDU, N)
The unitary matrix in the decomposition.
LDU
LDU is INTEGER
The leading dimension of U. LDU must be at least N.
WORK
WORK is COMPLEX array, dimension (LDWORK, M+1)
LDWORK
LDWORK is INTEGER
The leading dimension of WORK. LDWORK must be at least
max(1,M).
RWORK
RWORK is REAL array, dimension (N)
RESULT
RESULT is REAL array, dimension (2)
The values computed by the two tests described above. The
values are currently limited to 1/ulp, to avoid overflow.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cunt01 (character ROWCOL, integer M, integer N, complex, dimension( ldu, * ) U,
integer LDU, complex, dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, real
RESID)
CUNT01
Purpose:
CUNT01 checks that the matrix U is unitary by computing the ratio
RESID = norm( I - U*U' ) / ( n * EPS ), if ROWCOL = 'R',
or
RESID = norm( I - U'*U ) / ( m * EPS ), if ROWCOL = 'C'.
Alternatively, if there isn't sufficient workspace to form
I - U*U' or I - U'*U, the ratio is computed as
RESID = abs( I - U*U' ) / ( n * EPS ), if ROWCOL = 'R',
or
RESID = abs( I - U'*U ) / ( m * EPS ), if ROWCOL = 'C'.
where EPS is the machine precision. ROWCOL is used only if m = n;
if m > n, ROWCOL is assumed to be 'C', and if m < n, ROWCOL is
assumed to be 'R'.
Parameters:
ROWCOL
ROWCOL is CHARACTER
Specifies whether the rows or columns of U should be checked
for orthogonality. Used only if M = N.
= 'R': Check for orthogonal rows of U
= 'C': Check for orthogonal columns of U
M
M is INTEGER
The number of rows of the matrix U.
N
N is INTEGER
The number of columns of the matrix U.
U
U is COMPLEX array, dimension (LDU,N)
The unitary matrix U. U is checked for orthogonal columns
if m > n or if m = n and ROWCOL = 'C'. U is checked for
orthogonal rows if m < n or if m = n and ROWCOL = 'R'.
LDU
LDU is INTEGER
The leading dimension of the array U. LDU >= max(1,M).
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The length of the array WORK. For best performance, LWORK
should be at least N*N if ROWCOL = 'C' or M*M if
ROWCOL = 'R', but the test will be done even if LWORK is 0.
RWORK
RWORK is REAL array, dimension (min(M,N))
Used only if LWORK is large enough to use the Level 3 BLAS
code.
RESID
RESID is REAL
RESID = norm( I - U * U' ) / ( n * EPS ), if ROWCOL = 'R', or
RESID = norm( I - U' * U ) / ( m * EPS ), if ROWCOL = 'C'.
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
subroutine cunt03 (character*( * ) RC, integer MU, integer MV, integer N, integer K, complex,
dimension( ldu, * ) U, integer LDU, complex, dimension( ldv, * ) V, integer LDV, complex,
dimension( * ) WORK, integer LWORK, real, dimension( * ) RWORK, real RESULT, integer INFO)
CUNT03
Purpose:
CUNT03 compares two unitary matrices U and V to see if their
corresponding rows or columns span the same spaces. The rows are
checked if RC = 'R', and the columns are checked if RC = 'C'.
RESULT is the maximum of
| V*V' - I | / ( MV ulp ), if RC = 'R', or
| V'*V - I | / ( MV ulp ), if RC = 'C',
and the maximum over rows (or columns) 1 to K of
| U(i) - S*V(i) |/ ( N ulp )
where abs(S) = 1 (chosen to minimize the expression), U(i) is the
i-th row (column) of U, and V(i) is the i-th row (column) of V.
Parameters:
RC
RC is CHARACTER*1
If RC = 'R' the rows of U and V are to be compared.
If RC = 'C' the columns of U and V are to be compared.
MU
MU is INTEGER
The number of rows of U if RC = 'R', and the number of
columns if RC = 'C'. If MU = 0 CUNT03 does nothing.
MU must be at least zero.
MV
MV is INTEGER
The number of rows of V if RC = 'R', and the number of
columns if RC = 'C'. If MV = 0 CUNT03 does nothing.
MV must be at least zero.
N
N is INTEGER
If RC = 'R', the number of columns in the matrices U and V,
and if RC = 'C', the number of rows in U and V. If N = 0
CUNT03 does nothing. N must be at least zero.
K
K is INTEGER
The number of rows or columns of U and V to compare.
0 <= K <= max(MU,MV).
U
U is COMPLEX array, dimension (LDU,N)
The first matrix to compare. If RC = 'R', U is MU by N, and
if RC = 'C', U is N by MU.
LDU
LDU is INTEGER
The leading dimension of U. If RC = 'R', LDU >= max(1,MU),
and if RC = 'C', LDU >= max(1,N).
V
V is COMPLEX array, dimension (LDV,N)
The second matrix to compare. If RC = 'R', V is MV by N, and
if RC = 'C', V is N by MV.
LDV
LDV is INTEGER
The leading dimension of V. If RC = 'R', LDV >= max(1,MV),
and if RC = 'C', LDV >= max(1,N).
WORK
WORK is COMPLEX array, dimension (LWORK)
LWORK
LWORK is INTEGER
The length of the array WORK. For best performance, LWORK
should be at least N*N if RC = 'C' or M*M if RC = 'R', but
the tests will be done even if LWORK is 0.
RWORK
RWORK is REAL array, dimension (max(MV,N))
RESULT
RESULT is REAL
The value computed by the test described above. RESULT is
limited to 1/ulp to avoid overflow.
INFO
INFO is INTEGER
0 indicates a successful exit
-k indicates the k-th parameter had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
November 2011
Author
Generated automatically by Doxygen for LAPACK from the source code.