Provided by: liblapack-doc_3.11.0-2build1_all 
NAME
complex16OTHERauxiliary - complex16
SYNOPSIS
Functions
subroutine clag2z (M, N, SA, LDSA, A, LDA, INFO)
CLAG2Z converts a complex single precision matrix to a complex double precision
matrix.
double precision function dzsum1 (N, CX, INCX)
DZSUM1 forms the 1-norm of the complex vector using the true absolute value.
integer function ilazlc (M, N, A, LDA)
ILAZLC scans a matrix for its last non-zero column.
integer function ilazlr (M, N, A, LDA)
ILAZLR scans a matrix for its last non-zero row.
subroutine zdrscl (N, SA, SX, INCX)
ZDRSCL multiplies a vector by the reciprocal of a real scalar.
subroutine zlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine zlacgv (N, X, INCX)
ZLACGV conjugates a complex vector.
subroutine zlacn2 (N, V, X, EST, KASE, ISAVE)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for
evaluating matrix-vector products.
subroutine zlacon (N, V, X, EST, KASE)
ZLACON estimates the 1-norm of a square matrix, using reverse communication for
evaluating matrix-vector products.
subroutine zlacp2 (UPLO, M, N, A, LDA, B, LDB)
ZLACP2 copies all or part of a real two-dimensional array to a complex array.
subroutine zlacpy (UPLO, M, N, A, LDA, B, LDB)
ZLACPY copies all or part of one two-dimensional array to another.
subroutine zlacrm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
ZLACRM multiplies a complex matrix by a square real matrix.
subroutine zlacrt (N, CX, INCX, CY, INCY, C, S)
ZLACRT performs a linear transformation of a pair of complex vectors.
complex *16 function zladiv (X, Y)
ZLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
subroutine zlaein (RIGHTV, NOINIT, N, H, LDH, W, V, B, LDB, RWORK, EPS3, SMLNUM, INFO)
ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
inverse iteration.
subroutine zlaev2 (A, B, C, RT1, RT2, CS1, SN1)
ZLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian
matrix.
subroutine zlag2c (M, N, A, LDA, SA, LDSA, INFO)
ZLAG2C converts a complex double precision matrix to a complex single precision
matrix.
subroutine zlags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
ZLAGS2
subroutine zlagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
ZLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a
tridiagonal matrix, B and C are rectangular matrices, and α and β are scalars, which
may be 0, 1, or -1.
subroutine zlahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
using the double-shift/single-shift QR algorithm.
subroutine zlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A
so that elements below the specified subdiagonal are zero, and returns auxiliary
matrices which are needed to apply the transformation to the unreduced part of A.
subroutine zlaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
ZLAIC1 applies one step of incremental condition estimation.
double precision function zlangt (NORM, N, DL, D, DU)
ZLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
absolute value of any element of a general tridiagonal matrix.
double precision function zlanhb (NORM, UPLO, N, K, AB, LDAB, WORK)
ZLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a Hermitian band matrix.
double precision function zlanhp (NORM, UPLO, N, AP, WORK)
ZLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a complex Hermitian matrix supplied in
packed form.
double precision function zlanhs (NORM, N, A, LDA, WORK)
ZLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
absolute value of any element of an upper Hessenberg matrix.
double precision function zlanht (NORM, N, D, E)
ZLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a complex Hermitian tridiagonal matrix.
double precision function zlansb (NORM, UPLO, N, K, AB, LDAB, WORK)
ZLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a symmetric band matrix.
double precision function zlansp (NORM, UPLO, N, AP, WORK)
ZLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a symmetric matrix supplied in packed
form.
double precision function zlantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
ZLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a triangular band matrix.
double precision function zlantp (NORM, UPLO, DIAG, N, AP, WORK)
ZLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a triangular matrix supplied in packed
form.
double precision function zlantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
ZLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,
or the element of largest absolute value of a trapezoidal or triangular matrix.
subroutine zlapll (N, X, INCX, Y, INCY, SSMIN)
ZLAPLL measures the linear dependence of two vectors.
subroutine zlapmr (FORWRD, M, N, X, LDX, K)
ZLAPMR rearranges rows of a matrix as specified by a permutation vector.
subroutine zlapmt (FORWRD, M, N, X, LDX, K)
ZLAPMT performs a forward or backward permutation of the columns of a matrix.
subroutine zlaqhb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
ZLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
subroutine zlaqhp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
ZLAQHP scales a Hermitian matrix stored in packed form.
subroutine zlaqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
ZLAQP2 computes a QR factorization with column pivoting of the matrix block.
subroutine zlaqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
ZLAQPS computes a step of QR factorization with column pivoting of a real m-by-n
matrix A by using BLAS level 3.
subroutine zlaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK,
INFO)
ZLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
from the Schur decomposition.
subroutine zlaqr1 (N, H, LDH, S1, S2, V)
ZLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3
matrix H and specified shifts.
subroutine zlaqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect
and deflate fully converged eigenvalues from a trailing principal submatrix
(aggressive early deflation).
subroutine zlaqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
SH, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
ZLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect
and deflate fully converged eigenvalues from a trailing principal submatrix
(aggressive early deflation).
subroutine zlaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK,
INFO)
ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
from the Schur decomposition.
subroutine zlaqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, S, H, LDH, ILOZ, IHIZ, Z,
LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
ZLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine zlaqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
ZLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by
spbequ.
subroutine zlaqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
ZLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
computed by sppequ.
subroutine zlar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT,
ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
ZLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
through bn of the tridiagonal matrix LDLT - λI.
subroutine zlar2v (N, X, Y, Z, INCX, C, S, INCC)
ZLAR2V applies a vector of plane rotations with real cosines and complex sines from
both sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
subroutine zlarcm (M, N, A, LDA, B, LDB, C, LDC, RWORK)
ZLARCM copies all or part of a real two-dimensional array to a complex array.
subroutine zlarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
subroutine zlarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK,
LDWORK)
ZLARFB applies a block reflector or its conjugate-transpose to a general rectangular
matrix.
subroutine zlarfb_gett (IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
ZLARFB_GETT
subroutine zlarfg (N, ALPHA, X, INCX, TAU)
ZLARFG generates an elementary reflector (Householder matrix).
subroutine zlarfgp (N, ALPHA, X, INCX, TAU)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
subroutine zlarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
ZLARFT forms the triangular factor T of a block reflector H = I - vtvH
subroutine zlarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
ZLARFX applies an elementary reflector to a general rectangular matrix, with loop
unrolling when the reflector has order ≤ 10.
subroutine zlarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
ZLARFY
subroutine zlargv (N, X, INCX, Y, INCY, C, INCC)
ZLARGV generates a vector of plane rotations with real cosines and complex sines.
subroutine zlarnv (IDIST, ISEED, N, X)
ZLARNV returns a vector of random numbers from a uniform or normal distribution.
subroutine zlarrv (N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL, DOU, MINRGP, RTOL1, RTOL2, W,
WERR, WGAP, IBLOCK, INDEXW, GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO)
ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and
the eigenvalues of L D LT.
subroutine zlartv (N, X, INCX, Y, INCY, C, S, INCC)
ZLARTV applies a vector of plane rotations with real cosines and complex sines to the
elements of a pair of vectors.
subroutine zlascl (TYPE, KL, KU, CFROM, CTO, M, N, A, LDA, INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
subroutine zlaset (UPLO, M, N, ALPHA, BETA, A, LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to
given values.
subroutine zlasr (SIDE, PIVOT, DIRECT, M, N, C, S, A, LDA)
ZLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine zlaswp (N, A, LDA, K1, K2, IPIV, INCX)
ZLASWP performs a series of row interchanges on a general rectangular matrix.
subroutine zlat2c (UPLO, N, A, LDA, SA, LDSA, INFO)
ZLAT2C converts a double complex triangular matrix to a complex triangular matrix.
subroutine zlatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
ZLATBS solves a triangular banded system of equations.
subroutine zlatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
ZLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes
a contribution to the reciprocal Dif-estimate.
subroutine zlatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
ZLATPS solves a triangular system of equations with the matrix held in packed storage.
subroutine zlatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
tridiagonal form by an unitary similarity transformation.
subroutine zlatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
ZLATRS solves a triangular system of equations with the scale factor set to prevent
overflow.
subroutine zlauu2 (UPLO, N, A, LDA, INFO)
ZLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (unblocked algorithm).
subroutine zlauum (UPLO, N, A, LDA, INFO)
ZLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (blocked algorithm).
subroutine zrot (N, CX, INCX, CY, INCY, C, S)
ZROT applies a plane rotation with real cosine and complex sine to a pair of complex
vectors.
subroutine zspmv (UPLO, N, ALPHA, AP, X, INCX, BETA, Y, INCY)
ZSPMV computes a matrix-vector product for complex vectors using a complex symmetric
packed matrix
subroutine zspr (UPLO, N, ALPHA, X, INCX, AP)
ZSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.
subroutine ztprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B,
LDB, WORK, LDWORK)
ZTPRFB applies a complex 'triangular-pentagonal' block reflector to a complex matrix,
which is composed of two blocks.
Detailed Description
This is the group of complex16 other auxiliary routines
Function Documentation
subroutine clag2z (integer M, integer N, complex, dimension( ldsa, * ) SA, integer LDSA,
complex*16, dimension( lda, * ) A, integer LDA, integer INFO)
CLAG2Z converts a complex single precision matrix to a complex double precision matrix.
Purpose:
CLAG2Z converts a COMPLEX matrix, SA, to a COMPLEX*16 matrix, A.
Note that while it is possible to overflow while converting
from double to single, it is not possible to overflow when
converting from single to double.
This is an auxiliary routine so there is no argument checking.
Parameters
M
M is INTEGER
The number of lines of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
SA
SA is COMPLEX array, dimension (LDSA,N)
On entry, the M-by-N coefficient matrix SA.
LDSA
LDSA is INTEGER
The leading dimension of the array SA. LDSA >= max(1,M).
A
A is COMPLEX*16 array, dimension (LDA,N)
On exit, the M-by-N coefficient matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
INFO
INFO is INTEGER
= 0: successful exit
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function dzsum1 (integer N, complex*16, dimension( * ) CX, integer INCX)
DZSUM1 forms the 1-norm of the complex vector using the true absolute value.
Purpose:
DZSUM1 takes the sum of the absolute values of a complex
vector and returns a double precision result.
Based on DZASUM from the Level 1 BLAS.
The change is to use the 'genuine' absolute value.
Parameters
N
N is INTEGER
The number of elements in the vector CX.
CX
CX is COMPLEX*16 array, dimension (N)
The vector whose elements will be summed.
INCX
INCX is INTEGER
The spacing between successive values of CX. INCX > 0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Nick Higham for use with ZLACON.
integer function ilazlc (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA)
ILAZLC scans a matrix for its last non-zero column.
Purpose:
ILAZLC scans A for its last non-zero column.
Parameters
M
M is INTEGER
The number of rows of the matrix A.
N
N is INTEGER
The number of columns of the matrix A.
A
A is COMPLEX*16 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).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
integer function ilazlr (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA)
ILAZLR scans a matrix for its last non-zero row.
Purpose:
ILAZLR scans A for its last non-zero row.
Parameters
M
M is INTEGER
The number of rows of the matrix A.
N
N is INTEGER
The number of columns of the matrix A.
A
A is COMPLEX*16 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).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zdrscl (integer N, double precision SA, complex*16, dimension( * ) SX, integer
INCX)
ZDRSCL multiplies a vector by the reciprocal of a real scalar.
Purpose:
ZDRSCL multiplies an n-element complex vector x by the real scalar
1/a. This is done without overflow or underflow as long as
the final result x/a does not overflow or underflow.
Parameters
N
N is INTEGER
The number of components of the vector x.
SA
SA is DOUBLE PRECISION
The scalar a which is used to divide each component of x.
SA must be >= 0, or the subroutine will divide by zero.
SX
SX is COMPLEX*16 array, dimension
(1+(N-1)*abs(INCX))
The n-element vector x.
INCX
INCX is INTEGER
The increment between successive values of the vector SX.
> 0: SX(1) = X(1) and SX(1+(i-1)*INCX) = x(i), 1< i<= n
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlabrd (integer M, integer N, integer NB, complex*16, dimension( lda, * ) A,
integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E,
complex*16, dimension( * ) TAUQ, complex*16, dimension( * ) TAUP, complex*16, dimension(
ldx, * ) X, integer LDX, complex*16, dimension( ldy, * ) Y, integer LDY)
ZLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Purpose:
ZLABRD reduces the first NB rows and columns of a complex general
m by n matrix A to upper or lower real bidiagonal form by a unitary
transformation Q**H * A * P, and returns the matrices X and Y which
are needed to apply the transformation to the unreduced part of A.
If m >= n, A is reduced to upper bidiagonal form; if m < n, to lower
bidiagonal form.
This is an auxiliary routine called by ZGEBRD
Parameters
M
M is INTEGER
The number of rows in the matrix A.
N
N is INTEGER
The number of columns in the matrix A.
NB
NB is INTEGER
The number of leading rows and columns of A to be reduced.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n general matrix to be reduced.
On exit, the first NB rows and columns of the matrix are
overwritten; the rest of the array is unchanged.
If m >= n, elements on and below the diagonal in the first NB
columns, with the array TAUQ, represent the unitary
matrix Q as a product of elementary reflectors; and
elements above the diagonal in the first NB rows, with the
array TAUP, represent the unitary matrix P as a product
of elementary reflectors.
If m < n, elements below the diagonal in the first NB
columns, with the array TAUQ, represent the unitary
matrix Q as a product of elementary reflectors, and
elements on and above the diagonal in the first NB rows,
with the array TAUP, represent the unitary matrix P as
a product of elementary reflectors.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
D
D is DOUBLE PRECISION array, dimension (NB)
The diagonal elements of the first NB rows and columns of
the reduced matrix. D(i) = A(i,i).
E
E is DOUBLE PRECISION array, dimension (NB)
The off-diagonal elements of the first NB rows and columns of
the reduced matrix.
TAUQ
TAUQ is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the unitary matrix Q. See Further Details.
TAUP
TAUP is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the unitary matrix P. See Further Details.
X
X is COMPLEX*16 array, dimension (LDX,NB)
The m-by-nb matrix X required to update the unreduced part
of A.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(1,M).
Y
Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y required to update the unreduced part
of A.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= max(1,N).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrices Q and P are represented as products of elementary
reflectors:
Q = H(1) H(2) . . . H(nb) and P = G(1) G(2) . . . G(nb)
Each H(i) and G(i) has the form:
H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
where tauq and taup are complex scalars, and v and u are complex
vectors.
If m >= n, v(1:i-1) = 0, v(i) = 1, and v(i:m) is stored on exit in
A(i:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+1:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
If m < n, v(1:i) = 0, v(i+1) = 1, and v(i+1:m) is stored on exit in
A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i:n) is stored on exit in
A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
The elements of the vectors v and u together form the m-by-nb matrix
V and the nb-by-n matrix U**H which are needed, with X and Y, to apply
the transformation to the unreduced part of the matrix, using a block
update of the form: A := A - V*Y**H - X*U**H.
The contents of A on exit are illustrated by the following examples
with nb = 2:
m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
( 1 1 u1 u1 u1 ) ( 1 u1 u1 u1 u1 u1 )
( v1 1 1 u2 u2 ) ( 1 1 u2 u2 u2 u2 )
( v1 v2 a a a ) ( v1 1 a a a a )
( v1 v2 a a a ) ( v1 v2 a a a a )
( v1 v2 a a a ) ( v1 v2 a a a a )
( v1 v2 a a a )
where a denotes an element of the original matrix which is unchanged,
vi denotes an element of the vector defining H(i), and ui an element
of the vector defining G(i).
subroutine zlacgv (integer N, complex*16, dimension( * ) X, integer INCX)
ZLACGV conjugates a complex vector.
Purpose:
ZLACGV conjugates a complex vector of length N.
Parameters
N
N is INTEGER
The length of the vector X. N >= 0.
X
X is COMPLEX*16 array, dimension
(1+(N-1)*abs(INCX))
On entry, the vector of length N to be conjugated.
On exit, X is overwritten with conjg(X).
INCX
INCX is INTEGER
The spacing between successive elements of X.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlacn2 (integer N, complex*16, dimension( * ) V, complex*16, dimension( * ) X,
double precision EST, integer KASE, integer, dimension( 3 ) ISAVE)
ZLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating
matrix-vector products.
Purpose:
ZLACN2 estimates the 1-norm of a square, complex matrix A.
Reverse communication is used for evaluating matrix-vector products.
Parameters
N
N is INTEGER
The order of the matrix. N >= 1.
V
V is COMPLEX*16 array, dimension (N)
On the final return, V = A*W, where EST = norm(V)/norm(W)
(W is not returned).
X
X is COMPLEX*16 array, dimension (N)
On an intermediate return, X should be overwritten by
A * X, if KASE=1,
A**H * X, if KASE=2,
where A**H is the conjugate transpose of A, and ZLACN2 must be
re-called with all the other parameters unchanged.
EST
EST is DOUBLE PRECISION
On entry with KASE = 1 or 2 and ISAVE(1) = 3, EST should be
unchanged from the previous call to ZLACN2.
On exit, EST is an estimate (a lower bound) for norm(A).
KASE
KASE is INTEGER
On the initial call to ZLACN2, KASE should be 0.
On an intermediate return, KASE will be 1 or 2, indicating
whether X should be overwritten by A * X or A**H * X.
On the final return from ZLACN2, KASE will again be 0.
ISAVE
ISAVE is INTEGER array, dimension (3)
ISAVE is used to save variables between calls to ZLACN2
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Originally named CONEST, dated March 16, 1988.
Last modified: April, 1999
This is a thread safe version of ZLACON, which uses the array ISAVE
in place of a SAVE statement, as follows:
ZLACON ZLACN2
JUMP ISAVE(1)
J ISAVE(2)
ITER ISAVE(3)
Contributors:
Nick Higham, University of Manchester
References:
N.J. Higham, 'FORTRAN codes for estimating the one-norm of
a real or complex matrix, with applications to condition estimation', ACM Trans.
Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
subroutine zlacon (integer N, complex*16, dimension( n ) V, complex*16, dimension( n ) X,
double precision EST, integer KASE)
ZLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating
matrix-vector products.
Purpose:
ZLACON estimates the 1-norm of a square, complex matrix A.
Reverse communication is used for evaluating matrix-vector products.
Parameters
N
N is INTEGER
The order of the matrix. N >= 1.
V
V is COMPLEX*16 array, dimension (N)
On the final return, V = A*W, where EST = norm(V)/norm(W)
(W is not returned).
X
X is COMPLEX*16 array, dimension (N)
On an intermediate return, X should be overwritten by
A * X, if KASE=1,
A**H * X, if KASE=2,
where A**H is the conjugate transpose of A, and ZLACON must be
re-called with all the other parameters unchanged.
EST
EST is DOUBLE PRECISION
On entry with KASE = 1 or 2 and JUMP = 3, EST should be
unchanged from the previous call to ZLACON.
On exit, EST is an estimate (a lower bound) for norm(A).
KASE
KASE is INTEGER
On the initial call to ZLACON, KASE should be 0.
On an intermediate return, KASE will be 1 or 2, indicating
whether X should be overwritten by A * X or A**H * X.
On the final return from ZLACON, KASE will again be 0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Originally named CONEST, dated March 16, 1988.
Last modified: April, 1999
Contributors:
Nick Higham, University of Manchester
References:
N.J. Higham, 'FORTRAN codes for estimating the one-norm of
a real or complex matrix, with applications to condition estimation', ACM Trans.
Math. Soft., vol. 14, no. 4, pp. 381-396, December 1988.
subroutine zlacp2 (character UPLO, integer M, integer N, double precision, dimension( lda, * )
A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB)
ZLACP2 copies all or part of a real two-dimensional array to a complex array.
Purpose:
ZLACP2 copies all or part of a real two-dimensional matrix A to a
complex matrix B.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies the part of the matrix A to be copied to B.
= 'U': Upper triangular part
= 'L': Lower triangular part
Otherwise: All of the matrix 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.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
The m by n matrix A. If UPLO = 'U', only the upper trapezium
is accessed; if UPLO = 'L', only the lower trapezium is
accessed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX*16 array, dimension (LDB,N)
On exit, B = A in the locations specified by UPLO.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlacpy (character UPLO, integer M, integer N, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( ldb, * ) B, integer LDB)
ZLACPY copies all or part of one two-dimensional array to another.
Purpose:
ZLACPY copies all or part of a two-dimensional matrix A to another
matrix B.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies the part of the matrix A to be copied to B.
= 'U': Upper triangular part
= 'L': Lower triangular part
Otherwise: All of the matrix 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.
A
A is COMPLEX*16 array, dimension (LDA,N)
The m by n matrix A. If UPLO = 'U', only the upper trapezium
is accessed; if UPLO = 'L', only the lower trapezium is
accessed.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
B
B is COMPLEX*16 array, dimension (LDB,N)
On exit, B = A in the locations specified by UPLO.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlacrm (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
double precision, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C,
integer LDC, double precision, dimension( * ) RWORK)
ZLACRM multiplies a complex matrix by a square real matrix.
Purpose:
ZLACRM performs a very simple matrix-matrix multiplication:
C := A * B,
where A is M by N and complex; B is N by N and real;
C is M by N and complex.
Parameters
M
M is INTEGER
The number of rows of the matrix A and of the matrix C.
M >= 0.
N
N is INTEGER
The number of columns and rows of the matrix B and
the number of columns of the matrix C.
N >= 0.
A
A is COMPLEX*16 array, dimension (LDA, N)
On entry, A contains the M by N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >=max(1,M).
B
B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, B contains the N by N matrix B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >=max(1,N).
C
C is COMPLEX*16 array, dimension (LDC, N)
On exit, C contains the M by N matrix C.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >=max(1,N).
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*M*N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlacrt (integer N, complex*16, dimension( * ) CX, integer INCX, complex*16,
dimension( * ) CY, integer INCY, complex*16 C, complex*16 S)
ZLACRT performs a linear transformation of a pair of complex vectors.
Purpose:
ZLACRT performs the operation
( c s )( x ) ==> ( x )
( -s c )( y ) ( y )
where c and s are complex and the vectors x and y are complex.
Parameters
N
N is INTEGER
The number of elements in the vectors CX and CY.
CX
CX is COMPLEX*16 array, dimension (N)
On input, the vector x.
On output, CX is overwritten with c*x + s*y.
INCX
INCX is INTEGER
The increment between successive values of CX. INCX <> 0.
CY
CY is COMPLEX*16 array, dimension (N)
On input, the vector y.
On output, CY is overwritten with -s*x + c*y.
INCY
INCY is INTEGER
The increment between successive values of CY. INCY <> 0.
C
C is COMPLEX*16
S
S is COMPLEX*16
C and S define the matrix
[ C S ].
[ -S C ]
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
complex*16 function zladiv (complex*16 X, complex*16 Y)
ZLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
Purpose:
ZLADIV := X / Y, where X and Y are complex. The computation of X / Y
will not overflow on an intermediary step unless the results
overflows.
Parameters
X
X is COMPLEX*16
Y
Y is COMPLEX*16
The complex scalars X and Y.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaein (logical RIGHTV, logical NOINIT, integer N, complex*16, dimension( ldh, * )
H, integer LDH, complex*16 W, complex*16, dimension( * ) V, complex*16, dimension( ldb, *
) B, integer LDB, double precision, dimension( * ) RWORK, double precision EPS3, double
precision SMLNUM, integer INFO)
ZLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
inverse iteration.
Purpose:
ZLAEIN uses inverse iteration to find a right or left eigenvector
corresponding to the eigenvalue W of a complex upper Hessenberg
matrix H.
Parameters
RIGHTV
RIGHTV is LOGICAL
= .TRUE. : compute right eigenvector;
= .FALSE.: compute left eigenvector.
NOINIT
NOINIT is LOGICAL
= .TRUE. : no initial vector supplied in V
= .FALSE.: initial vector supplied in V.
N
N is INTEGER
The order of the matrix H. N >= 0.
H
H is COMPLEX*16 array, dimension (LDH,N)
The upper Hessenberg matrix H.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
W
W is COMPLEX*16
The eigenvalue of H whose corresponding right or left
eigenvector is to be computed.
V
V is COMPLEX*16 array, dimension (N)
On entry, if NOINIT = .FALSE., V must contain a starting
vector for inverse iteration; otherwise V need not be set.
On exit, V contains the computed eigenvector, normalized so
that the component of largest magnitude has magnitude 1; here
the magnitude of a complex number (x,y) is taken to be
|x| + |y|.
B
B is COMPLEX*16 array, dimension (LDB,N)
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,N).
RWORK
RWORK is DOUBLE PRECISION array, dimension (N)
EPS3
EPS3 is DOUBLE PRECISION
A small machine-dependent value which is used to perturb
close eigenvalues, and to replace zero pivots.
SMLNUM
SMLNUM is DOUBLE PRECISION
A machine-dependent value close to the underflow threshold.
INFO
INFO is INTEGER
= 0: successful exit
= 1: inverse iteration did not converge; V is set to the
last iterate.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaev2 (complex*16 A, complex*16 B, complex*16 C, double precision RT1, double
precision RT2, double precision CS1, complex*16 SN1)
ZLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
Purpose:
ZLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
[ A B ]
[ CONJG(B) C ].
On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
eigenvector for RT1, giving the decomposition
[ CS1 CONJG(SN1) ] [ A B ] [ CS1 -CONJG(SN1) ] = [ RT1 0 ]
[-SN1 CS1 ] [ CONJG(B) C ] [ SN1 CS1 ] [ 0 RT2 ].
Parameters
A
A is COMPLEX*16
The (1,1) element of the 2-by-2 matrix.
B
B is COMPLEX*16
The (1,2) element and the conjugate of the (2,1) element of
the 2-by-2 matrix.
C
C is COMPLEX*16
The (2,2) element of the 2-by-2 matrix.
RT1
RT1 is DOUBLE PRECISION
The eigenvalue of larger absolute value.
RT2
RT2 is DOUBLE PRECISION
The eigenvalue of smaller absolute value.
CS1
CS1 is DOUBLE PRECISION
SN1
SN1 is COMPLEX*16
The vector (CS1, SN1) is a unit right eigenvector for RT1.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
RT1 is accurate to a few ulps barring over/underflow.
RT2 may be inaccurate if there is massive cancellation in the
determinant A*C-B*B; higher precision or correctly rounded or
correctly truncated arithmetic would be needed to compute RT2
accurately in all cases.
CS1 and SN1 are accurate to a few ulps barring over/underflow.
Overflow is possible only if RT1 is within a factor of 5 of overflow.
Underflow is harmless if the input data is 0 or exceeds
underflow_threshold / macheps.
subroutine zlag2c (integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex, dimension( ldsa, * ) SA, integer LDSA, integer INFO)
ZLAG2C converts a complex double precision matrix to a complex single precision matrix.
Purpose:
ZLAG2C converts a COMPLEX*16 matrix, SA, to a COMPLEX matrix, A.
RMAX is the overflow for the SINGLE PRECISION arithmetic
ZLAG2C checks that all the entries of A are between -RMAX and
RMAX. If not the conversion is aborted and a flag is raised.
This is an auxiliary routine so there is no argument checking.
Parameters
M
M is INTEGER
The number of lines of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N coefficient matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SA
SA is COMPLEX array, dimension (LDSA,N)
On exit, if INFO=0, the M-by-N coefficient matrix SA; if
INFO>0, the content of SA is unspecified.
LDSA
LDSA is INTEGER
The leading dimension of the array SA. LDSA >= max(1,M).
INFO
INFO is INTEGER
= 0: successful exit.
= 1: an entry of the matrix A is greater than the SINGLE
PRECISION overflow threshold, in this case, the content
of SA in exit is unspecified.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlags2 (logical UPPER, double precision A1, complex*16 A2, double precision A3,
double precision B1, complex*16 B2, double precision B3, double precision CSU, complex*16
SNU, double precision CSV, complex*16 SNV, double precision CSQ, complex*16 SNQ)
ZLAGS2
Purpose:
ZLAGS2 computes 2-by-2 unitary matrices U, V and Q, such
that if ( UPPER ) then
U**H *A*Q = U**H *( A1 A2 )*Q = ( x 0 )
( 0 A3 ) ( x x )
and
V**H*B*Q = V**H *( B1 B2 )*Q = ( x 0 )
( 0 B3 ) ( x x )
or if ( .NOT.UPPER ) then
U**H *A*Q = U**H *( A1 0 )*Q = ( x x )
( A2 A3 ) ( 0 x )
and
V**H *B*Q = V**H *( B1 0 )*Q = ( x x )
( B2 B3 ) ( 0 x )
where
U = ( CSU SNU ), V = ( CSV SNV ),
( -SNU**H CSU ) ( -SNV**H CSV )
Q = ( CSQ SNQ )
( -SNQ**H CSQ )
The rows of the transformed A and B are parallel. Moreover, if the
input 2-by-2 matrix A is not zero, then the transformed (1,1) entry
of A is not zero. If the input matrices A and B are both not zero,
then the transformed (2,2) element of B is not zero, except when the
first rows of input A and B are parallel and the second rows are
zero.
Parameters
UPPER
UPPER is LOGICAL
= .TRUE.: the input matrices A and B are upper triangular.
= .FALSE.: the input matrices A and B are lower triangular.
A1
A1 is DOUBLE PRECISION
A2
A2 is COMPLEX*16
A3
A3 is DOUBLE PRECISION
On entry, A1, A2 and A3 are elements of the input 2-by-2
upper (lower) triangular matrix A.
B1
B1 is DOUBLE PRECISION
B2
B2 is COMPLEX*16
B3
B3 is DOUBLE PRECISION
On entry, B1, B2 and B3 are elements of the input 2-by-2
upper (lower) triangular matrix B.
CSU
CSU is DOUBLE PRECISION
SNU
SNU is COMPLEX*16
The desired unitary matrix U.
CSV
CSV is DOUBLE PRECISION
SNV
SNV is COMPLEX*16
The desired unitary matrix V.
CSQ
CSQ is DOUBLE PRECISION
SNQ
SNQ is COMPLEX*16
The desired unitary matrix Q.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlagtm (character TRANS, integer N, integer NRHS, double precision ALPHA,
complex*16, dimension( * ) DL, complex*16, dimension( * ) D, complex*16, dimension( * )
DU, complex*16, dimension( ldx, * ) X, integer LDX, double precision BETA, complex*16,
dimension( ldb, * ) B, integer LDB)
ZLAGTM performs a matrix-matrix product of the form C = αAB+βC, where A is a tridiagonal
matrix, B and C are rectangular matrices, and α and β are scalars, which may be 0, 1, or
-1.
Purpose:
ZLAGTM performs a matrix-vector product of the form
B := alpha * A * X + beta * B
where A is a tridiagonal matrix of order N, B and X are N by NRHS
matrices, and alpha and beta are real scalars, each of which may be
0., 1., or -1.
Parameters
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': No transpose, B := alpha * A * X + beta * B
= 'T': Transpose, B := alpha * A**T * X + beta * B
= 'C': Conjugate transpose, B := alpha * A**H * X + beta * B
N
N is INTEGER
The order of the matrix A. N >= 0.
NRHS
NRHS is INTEGER
The number of right hand sides, i.e., the number of columns
of the matrices X and B.
ALPHA
ALPHA is DOUBLE PRECISION
The scalar alpha. ALPHA must be 0., 1., or -1.; otherwise,
it is assumed to be 0.
DL
DL is COMPLEX*16 array, dimension (N-1)
The (n-1) sub-diagonal elements of T.
D
D is COMPLEX*16 array, dimension (N)
The diagonal elements of T.
DU
DU is COMPLEX*16 array, dimension (N-1)
The (n-1) super-diagonal elements of T.
X
X is COMPLEX*16 array, dimension (LDX,NRHS)
The N by NRHS matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max(N,1).
BETA
BETA is DOUBLE PRECISION
The scalar beta. BETA must be 0., 1., or -1.; otherwise,
it is assumed to be 1.
B
B is COMPLEX*16 array, dimension (LDB,NRHS)
On entry, the N by NRHS matrix B.
On exit, B is overwritten by the matrix expression
B := alpha * A * X + beta * B.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(N,1).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI,
complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer
ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer INFO)
ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
using the double-shift/single-shift QR algorithm.
Purpose:
ZLAHQR is an auxiliary routine called by CHSEQR to update the
eigenvalues and Schur decomposition already computed by CHSEQR, by
dealing with the Hessenberg submatrix in rows and columns ILO to
IHI.
Parameters
WANTT
WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
WANTZ
WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
N
N is INTEGER
The order of the matrix H. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that H is already upper triangular in rows and
columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
ZLAHQR works primarily with the Hessenberg submatrix in rows
and columns ILO to IHI, but applies transformations to all of
H if WANTT is .TRUE..
1 <= ILO <= max(1,IHI); IHI <= N.
H
H is COMPLEX*16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., then H
is upper triangular in rows and columns ILO:IHI. If INFO
is zero and if WANTT is .FALSE., then the contents of H
are unspecified on exit. The output state of H in case
INF is positive is below under the description of INFO.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
W
W is COMPLEX*16 array, dimension (N)
The computed eigenvalues ILO to IHI are stored in the
corresponding elements of W. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with W(i) = H(i,i).
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by CHSEQR, and on
exit Z has been updated; transformations are applied only to
the submatrix Z(ILOZ:IHIZ,ILO:IHI).
If WANTZ is .FALSE., Z is not referenced.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = i, ZLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of W contain
those eigenvalues which have been successfully
computed.
If INFO > 0 and WANTT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the
eigenvalues of the upper Hessenberg matrix
rows and columns ILO through INFO of the final,
output value of H.
If INFO > 0 and WANTT is .TRUE., then on exit
(*) (initial value of H)*U = U*(final value of H)
where U is an orthogonal matrix. The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.
If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z) = (initial value of Z)*U
where U is the orthogonal matrix in (*)
(regardless of the value of WANTT.)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
02-96 Based on modifications by
David Day, Sandia National Laboratory, USA
12-04 Further modifications by
Ralph Byers, University of Kansas, USA
This is a modified version of ZLAHQR from LAPACK version 3.0.
It is (1) more robust against overflow and underflow and
(2) adopts the more conservative Ahues & Tisseur stopping
criterion (LAWN 122, 1997).
subroutine zlahr2 (integer N, integer K, integer NB, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( nb ) TAU, complex*16, dimension( ldt, nb ) T, integer
LDT, complex*16, dimension( ldy, nb ) Y, integer LDY)
ZLAHR2 reduces the specified number of first columns of a general rectangular matrix A so
that elements below the specified subdiagonal are zero, and returns auxiliary matrices
which are needed to apply the transformation to the unreduced part of A.
Purpose:
ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by an unitary similarity transformation
Q**H * A * Q. The routine returns the matrices V and T which determine
Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
This is an auxiliary routine called by ZGEHRD.
Parameters
N
N is INTEGER
The order of the matrix A.
K
K is INTEGER
The offset for the reduction. Elements below the k-th
subdiagonal in the first NB columns are reduced to zero.
K < N.
NB
NB is INTEGER
The number of columns to be reduced.
A
A is COMPLEX*16 array, dimension (LDA,N-K+1)
On entry, the n-by-(n-k+1) general matrix A.
On exit, the elements on and above the k-th subdiagonal in
the first NB columns are overwritten with the corresponding
elements of the reduced matrix; the elements below the k-th
subdiagonal, with the array TAU, represent the matrix Q as a
product of elementary reflectors. The other columns of A are
unchanged. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
TAU
TAU is COMPLEX*16 array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
T
T is COMPLEX*16 array, dimension (LDT,NB)
The upper triangular matrix T.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= NB.
Y
Y is COMPLEX*16 array, dimension (LDY,NB)
The n-by-nb matrix Y.
LDY
LDY is INTEGER
The leading dimension of the array Y. LDY >= N.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix Q is represented as a product of nb elementary reflectors
Q = H(1) H(2) . . . H(nb).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
A(i+k+1:n,i), and tau in TAU(i).
The elements of the vectors v together form the (n-k+1)-by-nb matrix
V which is needed, with T and Y, to apply the transformation to the
unreduced part of the matrix, using an update of the form:
A := (I - V*T*V**H) * (A - Y*V**H).
The contents of A on exit are illustrated by the following example
with n = 7, k = 3 and nb = 2:
( a a a a a )
( a a a a a )
( a a a a a )
( h h a a a )
( v1 h a a a )
( v1 v2 a a a )
( v1 v2 a a a )
where a denotes an element of the original matrix A, h denotes a
modified element of the upper Hessenberg matrix H, and vi denotes an
element of the vector defining H(i).
This subroutine is a slight modification of LAPACK-3.0's ZLAHRD
incorporating improvements proposed by Quintana-Orti and Van de
Gejin. Note that the entries of A(1:K,2:NB) differ from those
returned by the original LAPACK-3.0's ZLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)
References:
Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical
Software, 32(2):180-194, June 2006.
subroutine zlaic1 (integer JOB, integer J, complex*16, dimension( j ) X, double precision
SEST, complex*16, dimension( j ) W, complex*16 GAMMA, double precision SESTPR, complex*16
S, complex*16 C)
ZLAIC1 applies one step of incremental condition estimation.
Purpose:
ZLAIC1 applies one step of incremental condition estimation in
its simplest version:
Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j
lower triangular matrix L, such that
twonorm(L*x) = sest
Then ZLAIC1 computes sestpr, s, c such that
the vector
[ s*x ]
xhat = [ c ]
is an approximate singular vector of
[ L 0 ]
Lhat = [ w**H gamma ]
in the sense that
twonorm(Lhat*xhat) = sestpr.
Depending on JOB, an estimate for the largest or smallest singular
value is computed.
Note that [s c]**H and sestpr**2 is an eigenpair of the system
diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ]
[ conjg(gamma) ]
where alpha = x**H * w.
Parameters
JOB
JOB is INTEGER
= 1: an estimate for the largest singular value is computed.
= 2: an estimate for the smallest singular value is computed.
J
J is INTEGER
Length of X and W
X
X is COMPLEX*16 array, dimension (J)
The j-vector x.
SEST
SEST is DOUBLE PRECISION
Estimated singular value of j by j matrix L
W
W is COMPLEX*16 array, dimension (J)
The j-vector w.
GAMMA
GAMMA is COMPLEX*16
The diagonal element gamma.
SESTPR
SESTPR is DOUBLE PRECISION
Estimated singular value of (j+1) by (j+1) matrix Lhat.
S
S is COMPLEX*16
Sine needed in forming xhat.
C
C is COMPLEX*16
Cosine needed in forming xhat.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlangt (character NORM, integer N, complex*16, dimension( * ) DL,
complex*16, dimension( * ) D, complex*16, dimension( * ) DU)
ZLANGT returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
absolute value of any element of a general tridiagonal matrix.
Purpose:
ZLANGT returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
complex tridiagonal matrix A.
Returns
ZLANGT
ZLANGT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANGT as described
above.
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANGT is
set to zero.
DL
DL is COMPLEX*16 array, dimension (N-1)
The (n-1) sub-diagonal elements of A.
D
D is COMPLEX*16 array, dimension (N)
The diagonal elements of A.
DU
DU is COMPLEX*16 array, dimension (N-1)
The (n-1) super-diagonal elements of A.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlanhb (character NORM, character UPLO, integer N, integer K,
complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)
ZLANHB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a Hermitian band matrix.
Purpose:
ZLANHB returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of an
n by n hermitian band matrix A, with k super-diagonals.
Returns
ZLANHB
ZLANHB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANHB as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
band matrix A is supplied.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANHB is
set to zero.
K
K is INTEGER
The number of super-diagonals or sub-diagonals of the
band matrix A. K >= 0.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
The upper or lower triangle of the hermitian band matrix A,
stored in the first K+1 rows of AB. The j-th column of A is
stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
Note that the imaginary parts of the diagonal elements need
not be set and are assumed to be zero.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= K+1.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlanhp (character NORM, character UPLO, integer N, complex*16,
dimension( * ) AP, double precision, dimension( * ) WORK)
ZLANHP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a complex Hermitian matrix supplied in packed
form.
Purpose:
ZLANHP returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
complex hermitian matrix A, supplied in packed form.
Returns
ZLANHP
ZLANHP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANHP as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
hermitian matrix A is supplied.
= 'U': Upper triangular part of A is supplied
= 'L': Lower triangular part of A is supplied
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANHP is
set to zero.
AP
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the hermitian matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
Note that the imaginary parts of the diagonal elements need
not be set and are assumed to be zero.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlanhs (character NORM, integer N, complex*16, dimension( lda, * )
A, integer LDA, double precision, dimension( * ) WORK)
ZLANHS returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest
absolute value of any element of an upper Hessenberg matrix.
Purpose:
ZLANHS returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
Hessenberg matrix A.
Returns
ZLANHS
ZLANHS = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANHS as described
above.
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANHS is
set to zero.
A
A is COMPLEX*16 array, dimension (LDA,N)
The n by n upper Hessenberg matrix A; the part of A below the
first sub-diagonal is not referenced.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(N,1).
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I'; otherwise, WORK is not
referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlanht (character NORM, integer N, double precision, dimension( * )
D, complex*16, dimension( * ) E)
ZLANHT returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a complex Hermitian tridiagonal matrix.
Purpose:
ZLANHT returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
complex Hermitian tridiagonal matrix A.
Returns
ZLANHT
ZLANHT = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANHT as described
above.
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANHT is
set to zero.
D
D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of A.
E
E is COMPLEX*16 array, dimension (N-1)
The (n-1) sub-diagonal or super-diagonal elements of A.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlansb (character NORM, character UPLO, integer N, integer K,
complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)
ZLANSB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a symmetric band matrix.
Purpose:
ZLANSB returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of an
n by n symmetric band matrix A, with k super-diagonals.
Returns
ZLANSB
ZLANSB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANSB as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
band matrix A is supplied.
= 'U': Upper triangular part is supplied
= 'L': Lower triangular part is supplied
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANSB is
set to zero.
K
K is INTEGER
The number of super-diagonals or sub-diagonals of the
band matrix A. K >= 0.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
The upper or lower triangle of the symmetric band matrix A,
stored in the first K+1 rows of AB. The j-th column of A is
stored in the j-th column of the array AB as follows:
if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= K+1.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlansp (character NORM, character UPLO, integer N, complex*16,
dimension( * ) AP, double precision, dimension( * ) WORK)
ZLANSP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a symmetric matrix supplied in packed form.
Purpose:
ZLANSP returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
complex symmetric matrix A, supplied in packed form.
Returns
ZLANSP
ZLANSP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANSP as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is supplied.
= 'U': Upper triangular part of A is supplied
= 'L': Lower triangular part of A is supplied
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANSP is
set to zero.
AP
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangle of the symmetric matrix A, packed
columnwise in a linear array. The j-th column of A is stored
in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
WORK is not referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlantb (character NORM, character UPLO, character DIAG, integer N,
integer K, complex*16, dimension( ldab, * ) AB, integer LDAB, double precision, dimension(
* ) WORK)
ZLANTB returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a triangular band matrix.
Purpose:
ZLANTB returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of an
n by n triangular band matrix A, with ( k + 1 ) diagonals.
Returns
ZLANTB
ZLANTB = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANTB as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANTB is
set to zero.
K
K is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals of the matrix A if UPLO = 'L'.
K >= 0.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first k+1 rows of AB. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = 'U', AB(k+1+i-j,j) = A(i,j) for max(1,j-k)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+k).
Note that when DIAG = 'U', the elements of the array AB
corresponding to the diagonal elements of the matrix A are
not referenced, but are assumed to be one.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= K+1.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I'; otherwise, WORK is not
referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlantp (character NORM, character UPLO, character DIAG, integer N,
complex*16, dimension( * ) AP, double precision, dimension( * ) WORK)
ZLANTP returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a triangular matrix supplied in packed form.
Purpose:
ZLANTP returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
triangular matrix A, supplied in packed form.
Returns
ZLANTP
ZLANTP = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANTP as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, ZLANTP is
set to zero.
AP
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array. The j-th column of A is stored in the array
AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
Note that when DIAG = 'U', the elements of the array AP
corresponding to the diagonal elements of the matrix A are
not referenced, but are assumed to be one.
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= N when NORM = 'I'; otherwise, WORK is not
referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function zlantr (character NORM, character UPLO, character DIAG, integer M,
integer N, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( *
) WORK)
ZLANTR returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or
the element of largest absolute value of a trapezoidal or triangular matrix.
Purpose:
ZLANTR returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
trapezoidal or triangular matrix A.
Returns
ZLANTR
ZLANTR = ( max(abs(A(i,j))), NORM = 'M' or 'm'
(
( norm1(A), NORM = '1', 'O' or 'o'
(
( normI(A), NORM = 'I' or 'i'
(
( normF(A), NORM = 'F', 'f', 'E' or 'e'
where norm1 denotes the one norm of a matrix (maximum column sum),
normI denotes the infinity norm of a matrix (maximum row sum) and
normF denotes the Frobenius norm of a matrix (square root of sum of
squares). Note that max(abs(A(i,j))) is not a consistent matrix norm.
Parameters
NORM
NORM is CHARACTER*1
Specifies the value to be returned in ZLANTR as described
above.
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower trapezoidal.
= 'U': Upper trapezoidal
= 'L': Lower trapezoidal
Note that A is triangular instead of trapezoidal if M = N.
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A has unit diagonal.
= 'N': Non-unit diagonal
= 'U': Unit diagonal
M
M is INTEGER
The number of rows of the matrix A. M >= 0, and if
UPLO = 'U', M <= N. When M = 0, ZLANTR is set to zero.
N
N is INTEGER
The number of columns of the matrix A. N >= 0, and if
UPLO = 'L', N <= M. When N = 0, ZLANTR is set to zero.
A
A is COMPLEX*16 array, dimension (LDA,N)
The trapezoidal matrix A (A is triangular if M = N).
If UPLO = 'U', the leading m by n upper trapezoidal part of
the array A contains the upper trapezoidal matrix, and the
strictly lower triangular part of A is not referenced.
If UPLO = 'L', the leading m by n lower trapezoidal part of
the array A contains the lower trapezoidal matrix, and the
strictly upper triangular part of A is not referenced. Note
that when DIAG = 'U', the diagonal elements of A are not
referenced and are assumed to be one.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(M,1).
WORK
WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
where LWORK >= M when NORM = 'I'; otherwise, WORK is not
referenced.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlapll (integer N, complex*16, dimension( * ) X, integer INCX, complex*16,
dimension( * ) Y, integer INCY, double precision SSMIN)
ZLAPLL measures the linear dependence of two vectors.
Purpose:
Given two column vectors X and Y, let
A = ( X Y ).
The subroutine first computes the QR factorization of A = Q*R,
and then computes the SVD of the 2-by-2 upper triangular matrix R.
The smaller singular value of R is returned in SSMIN, which is used
as the measurement of the linear dependency of the vectors X and Y.
Parameters
N
N is INTEGER
The length of the vectors X and Y.
X
X is COMPLEX*16 array, dimension (1+(N-1)*INCX)
On entry, X contains the N-vector X.
On exit, X is overwritten.
INCX
INCX is INTEGER
The increment between successive elements of X. INCX > 0.
Y
Y is COMPLEX*16 array, dimension (1+(N-1)*INCY)
On entry, Y contains the N-vector Y.
On exit, Y is overwritten.
INCY
INCY is INTEGER
The increment between successive elements of Y. INCY > 0.
SSMIN
SSMIN is DOUBLE PRECISION
The smallest singular value of the N-by-2 matrix A = ( X Y ).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlapmr (logical FORWRD, integer M, integer N, complex*16, dimension( ldx, * ) X,
integer LDX, integer, dimension( * ) K)
ZLAPMR rearranges rows of a matrix as specified by a permutation vector.
Purpose:
ZLAPMR rearranges the rows of the M by N matrix X as specified
by the permutation K(1),K(2),...,K(M) of the integers 1,...,M.
If FORWRD = .TRUE., forward permutation:
X(K(I),*) is moved X(I,*) for I = 1,2,...,M.
If FORWRD = .FALSE., backward permutation:
X(I,*) is moved to X(K(I),*) for I = 1,2,...,M.
Parameters
FORWRD
FORWRD is LOGICAL
= .TRUE., forward permutation
= .FALSE., backward permutation
M
M is INTEGER
The number of rows of the matrix X. M >= 0.
N
N is INTEGER
The number of columns of the matrix X. N >= 0.
X
X is COMPLEX*16 array, dimension (LDX,N)
On entry, the M by N matrix X.
On exit, X contains the permuted matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X, LDX >= MAX(1,M).
K
K is INTEGER array, dimension (M)
On entry, K contains the permutation vector. K is used as
internal workspace, but reset to its original value on
output.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlapmt (logical FORWRD, integer M, integer N, complex*16, dimension( ldx, * ) X,
integer LDX, integer, dimension( * ) K)
ZLAPMT performs a forward or backward permutation of the columns of a matrix.
Purpose:
ZLAPMT rearranges the columns of the M by N matrix X as specified
by the permutation K(1),K(2),...,K(N) of the integers 1,...,N.
If FORWRD = .TRUE., forward permutation:
X(*,K(J)) is moved X(*,J) for J = 1,2,...,N.
If FORWRD = .FALSE., backward permutation:
X(*,J) is moved to X(*,K(J)) for J = 1,2,...,N.
Parameters
FORWRD
FORWRD is LOGICAL
= .TRUE., forward permutation
= .FALSE., backward permutation
M
M is INTEGER
The number of rows of the matrix X. M >= 0.
N
N is INTEGER
The number of columns of the matrix X. N >= 0.
X
X is COMPLEX*16 array, dimension (LDX,N)
On entry, the M by N matrix X.
On exit, X contains the permuted matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X, LDX >= MAX(1,M).
K
K is INTEGER array, dimension (N)
On entry, K contains the permutation vector. K is used as
internal workspace, but reset to its original value on
output.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaqhb (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB,
integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision
AMAX, character EQUED)
ZLAQHB scales a Hermitian band matrix, using scaling factors computed by cpbequ.
Purpose:
ZLAQHB equilibrates a Hermitian band matrix A
using the scaling factors in the vector S.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H *U or A = L*L**H of the band
matrix A, in the same storage format as A.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
S
S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.
SCOND
SCOND is DOUBLE PRECISION
Ratio of the smallest S(i) to the largest S(i).
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix entry.
EQUED
EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
Internal Parameters:
THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors. If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaqhp (character UPLO, integer N, complex*16, dimension( * ) AP, double precision,
dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)
ZLAQHP scales a Hermitian matrix stored in packed form.
Purpose:
ZLAQHP equilibrates a Hermitian matrix A using the scaling factors
in the vector S.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Hermitian matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, the equilibrated matrix: diag(S) * A * diag(S), in
the same storage format as A.
S
S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.
SCOND
SCOND is DOUBLE PRECISION
Ratio of the smallest S(i) to the largest S(i).
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix entry.
EQUED
EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
Internal Parameters:
THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors. If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaqp2 (integer M, integer N, integer OFFSET, complex*16, dimension( lda, * ) A,
integer LDA, integer, dimension( * ) JPVT, complex*16, dimension( * ) TAU, double
precision, dimension( * ) VN1, double precision, dimension( * ) VN2, complex*16,
dimension( * ) WORK)
ZLAQP2 computes a QR factorization with column pivoting of the matrix block.
Purpose:
ZLAQP2 computes a QR factorization with column pivoting of
the block A(OFFSET+1:M,1:N).
The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
Parameters
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.
OFFSET
OFFSET is INTEGER
The number of rows of the matrix A that must be pivoted
but no factorized. OFFSET >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the upper triangle of block A(OFFSET+1:M,1:N) is
the triangular factor obtained; the elements in block
A(OFFSET+1:M,1:N) below the diagonal, together with the
array TAU, represent the orthogonal matrix Q as a product of
elementary reflectors. Block A(1:OFFSET,1:N) has been
accordingly pivoted, but no factorized.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT
JPVT is INTEGER array, dimension (N)
On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
to the front of A*P (a leading column); if JPVT(i) = 0,
the i-th column of A is a free column.
On exit, if JPVT(i) = k, then the i-th column of A*P
was the k-th column of A.
TAU
TAU is COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors.
VN1
VN1 is DOUBLE PRECISION array, dimension (N)
The vector with the partial column norms.
VN2
VN2 is DOUBLE PRECISION array, dimension (N)
The vector with the exact column norms.
WORK
WORK is COMPLEX*16 array, dimension (N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer
Science Dept., Duke University, USA
Partial column norm updating strategy modified on April 2011 Z. Drmac and Z.
Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.
References:
LAPACK Working Note 176
subroutine zlaqps (integer M, integer N, integer OFFSET, integer NB, integer KB, complex*16,
dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, complex*16, dimension( *
) TAU, double precision, dimension( * ) VN1, double precision, dimension( * ) VN2,
complex*16, dimension( * ) AUXV, complex*16, dimension( ldf, * ) F, integer LDF)
ZLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A
by using BLAS level 3.
Purpose:
ZLAQPS computes a step of QR factorization with column pivoting
of a complex M-by-N matrix A by using Blas-3. It tries to factorize
NB columns from A starting from the row OFFSET+1, and updates all
of the matrix with Blas-3 xGEMM.
In some cases, due to catastrophic cancellations, it cannot
factorize NB columns. Hence, the actual number of factorized
columns is returned in KB.
Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
Parameters
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
OFFSET
OFFSET is INTEGER
The number of rows of A that have been factorized in
previous steps.
NB
NB is INTEGER
The number of columns to factorize.
KB
KB is INTEGER
The number of columns actually factorized.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, block A(OFFSET+1:M,1:KB) is the triangular
factor obtained and block A(1:OFFSET,1:N) has been
accordingly pivoted, but no factorized.
The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
been updated.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
JPVT
JPVT is INTEGER array, dimension (N)
JPVT(I) = K <==> Column K of the full matrix A has been
permuted into position I in AP.
TAU
TAU is COMPLEX*16 array, dimension (KB)
The scalar factors of the elementary reflectors.
VN1
VN1 is DOUBLE PRECISION array, dimension (N)
The vector with the partial column norms.
VN2
VN2 is DOUBLE PRECISION array, dimension (N)
The vector with the exact column norms.
AUXV
AUXV is COMPLEX*16 array, dimension (NB)
Auxiliary vector.
F
F is COMPLEX*16 array, dimension (LDF,NB)
Matrix F**H = L * Y**H * A.
LDF
LDF is INTEGER
The leading dimension of the array F. LDF >= max(1,N).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer
Science Dept., Duke University, USA
Partial column norm updating strategy modified on April 2011 Z. Drmac and Z.
Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.
References:
LAPACK Working Note 176
subroutine zlaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI,
complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer
ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension(
* ) WORK, integer LWORK, integer INFO)
ZLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
the Schur decomposition.
Purpose:
ZLAQR0 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.
Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
Parameters
WANTT
WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
WANTZ
WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
N
N is INTEGER
The order of the matrix H. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
previous call to ZGEBAL, and then passed to ZGEHRD when the
matrix output by ZGEBAL is reduced to Hessenberg form.
Otherwise, ILO and IHI should be set to 1 and N,
respectively. If N > 0, then 1 <= ILO <= IHI <= N.
If N = 0, then ILO = 1 and IHI = 0.
H
H is COMPLEX*16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and WANTT is .TRUE., then H
contains the upper triangular matrix T from the Schur
decomposition (the Schur form). If INFO = 0 and WANT is
.FALSE., then the contents of H are unspecified on exit.
(The output value of H when INFO > 0 is given under the
description of INFO below.)
This subroutine may explicitly set H(i,j) = 0 for i > j and
j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
W
W is COMPLEX*16 array, dimension (N)
The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
stored in the same order as on the diagonal of the Schur
form returned in H, with W(i) = H(i,i).
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,IHI)
If WANTZ is .FALSE., then Z is not referenced.
If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
(The output value of Z when INFO > 0 is given under
the description of INFO below.)
LDZ
LDZ is INTEGER
The leading dimension of the array Z. if WANTZ is .TRUE.
then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
WORK
WORK is COMPLEX*16 array, dimension LWORK
On exit, if LWORK = -1, WORK(1) returns an estimate of
the optimal value for LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N)
is sufficient, but LWORK typically as large as 6*N may
be required for optimal performance. A workspace query
to determine the optimal workspace size is recommended.
If LWORK = -1, then ZLAQR0 does a workspace query.
In this case, ZLAQR0 checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI. The estimate is returned
in WORK(1). No error message related to LWORK is
issued by XERBLA. Neither H nor Z are accessed.
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = i, ZLAQR0 failed to compute all of
the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed. (Failures are rare.)
If INFO > 0 and WANT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.
If INFO > 0 and WANTT is .TRUE., then on exit
(*) (initial value of H)*U = U*(final value of H)
where U is a unitary matrix. The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.
If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z(ILO:IHI,ILOZ:IHIZ)
= (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
where U is the unitary matrix in (*) (regard-
less of the value of WANTT.)
If INFO > 0 and WANTZ is .FALSE., then Z is not
accessed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive
Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.
subroutine zlaqr1 (integer N, complex*16, dimension( ldh, * ) H, integer LDH, complex*16 S1,
complex*16 S2, complex*16, dimension( * ) V)
ZLAQR1 sets a scalar multiple of the first column of the product of 2-by-2 or 3-by-3
matrix H and specified shifts.
Purpose:
Given a 2-by-2 or 3-by-3 matrix H, ZLAQR1 sets v to a
scalar multiple of the first column of the product
(*) K = (H - s1*I)*(H - s2*I)
scaling to avoid overflows and most underflows.
This is useful for starting double implicit shift bulges
in the QR algorithm.
Parameters
N
N is INTEGER
Order of the matrix H. N must be either 2 or 3.
H
H is COMPLEX*16 array, dimension (LDH,N)
The 2-by-2 or 3-by-3 matrix H in (*).
LDH
LDH is INTEGER
The leading dimension of H as declared in
the calling procedure. LDH >= N
S1
S1 is COMPLEX*16
S2
S2 is COMPLEX*16
S1 and S2 are the shifts defining K in (*) above.
V
V is COMPLEX*16 array, dimension (N)
A scalar multiple of the first column of the
matrix K in (*).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
subroutine zlaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
integer NW, complex*16, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ,
complex*16, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex*16,
dimension( * ) SH, complex*16, dimension( ldv, * ) V, integer LDV, integer NH, complex*16,
dimension( ldt, * ) T, integer LDT, integer NV, complex*16, dimension( ldwv, * ) WV,
integer LDWV, complex*16, dimension( * ) WORK, integer LWORK)
ZLAQR2 performs the unitary similarity transformation of a Hessenberg matrix to detect and
deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early
deflation).
Purpose:
ZLAQR2 is identical to ZLAQR3 except that it avoids
recursion by calling ZLAHQR instead of ZLAQR4.
Aggressive early deflation:
ZLAQR2 accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix. On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
Parameters
WANTT
WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
WANTZ
WANTZ is LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
N
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
KTOP
KTOP is INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
KBOT
KBOT is INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
NW
NW is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
H
H is COMPLEX*16 array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
LDH
LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine. N <= LDH
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine. 1 <= LDZ.
NS
NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND
ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
SH
SH is COMPLEX*16 array, dimension (KBOT)
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND). Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
V
V is COMPLEX*16 array, dimension (LDV,NW)
An NW-by-NW work array.
LDV
LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine. NW <= LDV
NH
NH is INTEGER
The number of columns of T. NH >= NW.
T
T is COMPLEX*16 array, dimension (LDT,NW)
LDT
LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine. NW <= LDT
NV
NV is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.
WV
WV is COMPLEX*16 array, dimension (LDWV,NW)
LDWV
LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine. NW <= LDV
WORK
WORK is COMPLEX*16 array, dimension (LWORK)
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
LWORK
LWORK is INTEGER
The dimension of the work array WORK. LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; ZLAQR2
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT. The estimate is returned
in WORK(1). No error message related to LWORK is issued
by XERBLA. Neither H nor Z are accessed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
subroutine zlaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
integer NW, complex*16, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer IHIZ,
complex*16, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, complex*16,
dimension( * ) SH, complex*16, dimension( ldv, * ) V, integer LDV, integer NH, complex*16,
dimension( ldt, * ) T, integer LDT, integer NV, complex*16, dimension( ldwv, * ) WV,
integer LDWV, complex*16, dimension( * ) WORK, integer LWORK)
ZLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and
deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early
deflation).
Purpose:
Aggressive early deflation:
ZLAQR3 accepts as input an upper Hessenberg matrix
H and performs an unitary similarity transformation
designed to detect and deflate fully converged eigenvalues from
a trailing principal submatrix. On output H has been over-
written by a new Hessenberg matrix that is a perturbation of
an unitary similarity transformation of H. It is to be
hoped that the final version of H has many zero subdiagonal
entries.
Parameters
WANTT
WANTT is LOGICAL
If .TRUE., then the Hessenberg matrix H is fully updated
so that the triangular Schur factor may be
computed (in cooperation with the calling subroutine).
If .FALSE., then only enough of H is updated to preserve
the eigenvalues.
WANTZ
WANTZ is LOGICAL
If .TRUE., then the unitary matrix Z is updated so
so that the unitary Schur factor may be computed
(in cooperation with the calling subroutine).
If .FALSE., then Z is not referenced.
N
N is INTEGER
The order of the matrix H and (if WANTZ is .TRUE.) the
order of the unitary matrix Z.
KTOP
KTOP is INTEGER
It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
KBOT and KTOP together determine an isolated block
along the diagonal of the Hessenberg matrix.
KBOT
KBOT is INTEGER
It is assumed without a check that either
KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together
determine an isolated block along the diagonal of the
Hessenberg matrix.
NW
NW is INTEGER
Deflation window size. 1 <= NW <= (KBOT-KTOP+1).
H
H is COMPLEX*16 array, dimension (LDH,N)
On input the initial N-by-N section of H stores the
Hessenberg matrix undergoing aggressive early deflation.
On output H has been transformed by a unitary
similarity transformation, perturbed, and the returned
to Hessenberg form that (it is to be hoped) has some
zero subdiagonal entries.
LDH
LDH is INTEGER
Leading dimension of H just as declared in the calling
subroutine. N <= LDH
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the unitary
similarity transformation mentioned above has been
accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ is .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
The leading dimension of Z just as declared in the
calling subroutine. 1 <= LDZ.
NS
NS is INTEGER
The number of unconverged (ie approximate) eigenvalues
returned in SR and SI that may be used as shifts by the
calling subroutine.
ND
ND is INTEGER
The number of converged eigenvalues uncovered by this
subroutine.
SH
SH is COMPLEX*16 array, dimension (KBOT)
On output, approximate eigenvalues that may
be used for shifts are stored in SH(KBOT-ND-NS+1)
through SR(KBOT-ND). Converged eigenvalues are
stored in SH(KBOT-ND+1) through SH(KBOT).
V
V is COMPLEX*16 array, dimension (LDV,NW)
An NW-by-NW work array.
LDV
LDV is INTEGER
The leading dimension of V just as declared in the
calling subroutine. NW <= LDV
NH
NH is INTEGER
The number of columns of T. NH >= NW.
T
T is COMPLEX*16 array, dimension (LDT,NW)
LDT
LDT is INTEGER
The leading dimension of T just as declared in the
calling subroutine. NW <= LDT
NV
NV is INTEGER
The number of rows of work array WV available for
workspace. NV >= NW.
WV
WV is COMPLEX*16 array, dimension (LDWV,NW)
LDWV
LDWV is INTEGER
The leading dimension of W just as declared in the
calling subroutine. NW <= LDV
WORK
WORK is COMPLEX*16 array, dimension (LWORK)
On exit, WORK(1) is set to an estimate of the optimal value
of LWORK for the given values of N, NW, KTOP and KBOT.
LWORK
LWORK is INTEGER
The dimension of the work array WORK. LWORK = 2*NW
suffices, but greater efficiency may result from larger
values of LWORK.
If LWORK = -1, then a workspace query is assumed; ZLAQR3
only estimates the optimal workspace size for the given
values of N, NW, KTOP and KBOT. The estimate is returned
in WORK(1). No error message related to LWORK is issued
by XERBLA. Neither H nor Z are accessed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
subroutine zlaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI,
complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer
ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, complex*16, dimension(
* ) WORK, integer LWORK, integer INFO)
ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
the Schur decomposition.
Purpose:
ZLAQR4 implements one level of recursion for ZLAQR0.
It is a complete implementation of the small bulge multi-shift
QR algorithm. It may be called by ZLAQR0 and, for large enough
deflation window size, it may be called by ZLAQR3. This
subroutine is identical to ZLAQR0 except that it calls ZLAQR2
instead of ZLAQR3.
ZLAQR4 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.
Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q: A = Q*H*Q**H = (QZ)*H*(QZ)**H.
Parameters
WANTT
WANTT is LOGICAL
= .TRUE. : the full Schur form T is required;
= .FALSE.: only eigenvalues are required.
WANTZ
WANTZ is LOGICAL
= .TRUE. : the matrix of Schur vectors Z is required;
= .FALSE.: Schur vectors are not required.
N
N is INTEGER
The order of the matrix H. N >= 0.
ILO
ILO is INTEGER
IHI
IHI is INTEGER
It is assumed that H is already upper triangular in rows
and columns 1:ILO-1 and IHI+1:N and, if ILO > 1,
H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
previous call to ZGEBAL, and then passed to ZGEHRD when the
matrix output by ZGEBAL is reduced to Hessenberg form.
Otherwise, ILO and IHI should be set to 1 and N,
respectively. If N > 0, then 1 <= ILO <= IHI <= N.
If N = 0, then ILO = 1 and IHI = 0.
H
H is COMPLEX*16 array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO = 0 and WANTT is .TRUE., then H
contains the upper triangular matrix T from the Schur
decomposition (the Schur form). If INFO = 0 and WANT is
.FALSE., then the contents of H are unspecified on exit.
(The output value of H when INFO > 0 is given under the
description of INFO below.)
This subroutine may explicitly set H(i,j) = 0 for i > j and
j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
W
W is COMPLEX*16 array, dimension (N)
The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are
stored in the same order as on the diagonal of the Schur
form returned in H, with W(i) = H(i,i).
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE..
1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
Z
Z is COMPLEX*16 array, dimension (LDZ,IHI)
If WANTZ is .FALSE., then Z is not referenced.
If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
(The output value of Z when INFO > 0 is given under
the description of INFO below.)
LDZ
LDZ is INTEGER
The leading dimension of the array Z. if WANTZ is .TRUE.
then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1.
WORK
WORK is COMPLEX*16 array, dimension LWORK
On exit, if LWORK = -1, WORK(1) returns an estimate of
the optimal value for LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. LWORK >= max(1,N)
is sufficient, but LWORK typically as large as 6*N may
be required for optimal performance. A workspace query
to determine the optimal workspace size is recommended.
If LWORK = -1, then ZLAQR4 does a workspace query.
In this case, ZLAQR4 checks the input parameters and
estimates the optimal workspace size for the given
values of N, ILO and IHI. The estimate is returned
in WORK(1). No error message related to LWORK is
issued by XERBLA. Neither H nor Z are accessed.
INFO
INFO is INTEGER
= 0: successful exit
> 0: if INFO = i, ZLAQR4 failed to compute all of
the eigenvalues. Elements 1:ilo-1 and i+1:n of WR
and WI contain those eigenvalues which have been
successfully computed. (Failures are rare.)
If INFO > 0 and WANT is .FALSE., then on exit,
the remaining unconverged eigenvalues are the eigen-
values of the upper Hessenberg matrix rows and
columns ILO through INFO of the final, output
value of H.
If INFO > 0 and WANTT is .TRUE., then on exit
(*) (initial value of H)*U = U*(final value of H)
where U is a unitary matrix. The final
value of H is upper Hessenberg and triangular in
rows and columns INFO+1 through IHI.
If INFO > 0 and WANTZ is .TRUE., then on exit
(final value of Z(ILO:IHI,ILOZ:IHIZ)
= (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
where U is the unitary matrix in (*) (regard-
less of the value of WANTT.)
If INFO > 0 and WANTZ is .FALSE., then Z is not
accessed.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive
Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.
subroutine zlaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP,
integer KBOT, integer NSHFTS, complex*16, dimension( * ) S, complex*16, dimension( ldh, *
) H, integer LDH, integer ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer
LDZ, complex*16, dimension( ldv, * ) V, integer LDV, complex*16, dimension( ldu, * ) U,
integer LDU, integer NV, complex*16, dimension( ldwv, * ) WV, integer LDWV, integer NH,
complex*16, dimension( ldwh, * ) WH, integer LDWH)
ZLAQR5 performs a single small-bulge multi-shift QR sweep.
Purpose:
ZLAQR5, called by ZLAQR0, performs a
single small-bulge multi-shift QR sweep.
Parameters
WANTT
WANTT is LOGICAL
WANTT = .true. if the triangular Schur factor
is being computed. WANTT is set to .false. otherwise.
WANTZ
WANTZ is LOGICAL
WANTZ = .true. if the unitary Schur factor is being
computed. WANTZ is set to .false. otherwise.
KACC22
KACC22 is INTEGER with value 0, 1, or 2.
Specifies the computation mode of far-from-diagonal
orthogonal updates.
= 0: ZLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: ZLAQR5 accumulates reflections and uses matrix-matrix
multiply to update the far-from-diagonal matrix entries.
= 2: Same as KACC22 = 1. This option used to enable exploiting
the 2-by-2 structure during matrix multiplications, but
this is no longer supported.
N
N is INTEGER
N is the order of the Hessenberg matrix H upon which this
subroutine operates.
KTOP
KTOP is INTEGER
KBOT
KBOT is INTEGER
These are the first and last rows and columns of an
isolated diagonal block upon which the QR sweep is to be
applied. It is assumed without a check that
either KTOP = 1 or H(KTOP,KTOP-1) = 0
and
either KBOT = N or H(KBOT+1,KBOT) = 0.
NSHFTS
NSHFTS is INTEGER
NSHFTS gives the number of simultaneous shifts. NSHFTS
must be positive and even.
S
S is COMPLEX*16 array, dimension (NSHFTS)
S contains the shifts of origin that define the multi-
shift QR sweep. On output S may be reordered.
H
H is COMPLEX*16 array, dimension (LDH,N)
On input H contains a Hessenberg matrix. On output a
multi-shift QR sweep with shifts SR(J)+i*SI(J) is applied
to the isolated diagonal block in rows and columns KTOP
through KBOT.
LDH
LDH is INTEGER
LDH is the leading dimension of H just as declared in the
calling procedure. LDH >= MAX(1,N).
ILOZ
ILOZ is INTEGER
IHIZ
IHIZ is INTEGER
Specify the rows of Z to which transformations must be
applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N
Z
Z is COMPLEX*16 array, dimension (LDZ,IHIZ)
If WANTZ = .TRUE., then the QR Sweep unitary
similarity transformation is accumulated into
Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
If WANTZ = .FALSE., then Z is unreferenced.
LDZ
LDZ is INTEGER
LDA is the leading dimension of Z just as declared in
the calling procedure. LDZ >= N.
V
V is COMPLEX*16 array, dimension (LDV,NSHFTS/2)
LDV
LDV is INTEGER
LDV is the leading dimension of V as declared in the
calling procedure. LDV >= 3.
U
U is COMPLEX*16 array, dimension (LDU,2*NSHFTS)
LDU
LDU is INTEGER
LDU is the leading dimension of U just as declared in the
in the calling subroutine. LDU >= 2*NSHFTS.
NV
NV is INTEGER
NV is the number of rows in WV agailable for workspace.
NV >= 1.
WV
WV is COMPLEX*16 array, dimension (LDWV,2*NSHFTS)
LDWV
LDWV is INTEGER
LDWV is the leading dimension of WV as declared in the
in the calling subroutine. LDWV >= NV.
NH
NH is INTEGER
NH is the number of columns in array WH available for
workspace. NH >= 1.
WH
WH is COMPLEX*16 array, dimension (LDWH,NH)
LDWH
LDWH is INTEGER
Leading dimension of WH just as declared in the
calling procedure. LDWH >= 2*NSHFTS.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
Lars Karlsson, Daniel Kressner, and Bruno Lang
Thijs Steel, Department of Computer science, KU Leuven, Belgium
References:
K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
23, pages 929--947, 2002.
Lars Karlsson, Daniel Kressner, and Bruno Lang, Optimally packed chains of bulges in
multishift QR algorithms. ACM Trans. Math. Softw. 40, 2, Article 12 (February 2014).
subroutine zlaqsb (character UPLO, integer N, integer KD, complex*16, dimension( ldab, * ) AB,
integer LDAB, double precision, dimension( * ) S, double precision SCOND, double precision
AMAX, character EQUED)
ZLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
Purpose:
ZLAQSB equilibrates a symmetric band matrix A using the scaling
factors in the vector S.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of super-diagonals of the matrix A if UPLO = 'U',
or the number of sub-diagonals if UPLO = 'L'. KD >= 0.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
On entry, the upper or lower triangle of the symmetric band
matrix A, stored in the first KD+1 rows of the array. The
j-th column of A is stored in the j-th column of the array AB
as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
On exit, if INFO = 0, the triangular factor U or L from the
Cholesky factorization A = U**H *U or A = L*L**H of the band
matrix A, in the same storage format as A.
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
S
S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.
SCOND
SCOND is DOUBLE PRECISION
Ratio of the smallest S(i) to the largest S(i).
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix entry.
EQUED
EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
Internal Parameters:
THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors. If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaqsp (character UPLO, integer N, complex*16, dimension( * ) AP, double precision,
dimension( * ) S, double precision SCOND, double precision AMAX, character EQUED)
ZLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
computed by sppequ.
Purpose:
ZLAQSP equilibrates a symmetric matrix A using the scaling factors
in the vector S.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
symmetric matrix A is stored.
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the symmetric matrix
A, packed columnwise in a linear array. The j-th column of A
is stored in the array AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
On exit, the equilibrated matrix: diag(S) * A * diag(S), in
the same storage format as A.
S
S is DOUBLE PRECISION array, dimension (N)
The scale factors for A.
SCOND
SCOND is DOUBLE PRECISION
Ratio of the smallest S(i) to the largest S(i).
AMAX
AMAX is DOUBLE PRECISION
Absolute value of largest matrix entry.
EQUED
EQUED is CHARACTER*1
Specifies whether or not equilibration was done.
= 'N': No equilibration.
= 'Y': Equilibration was done, i.e., A has been replaced by
diag(S) * A * diag(S).
Internal Parameters:
THRESH is a threshold value used to decide if scaling should be done
based on the ratio of the scaling factors. If SCOND < THRESH,
scaling is done.
LARGE and SMALL are threshold values used to decide if scaling should
be done based on the absolute size of the largest matrix element.
If AMAX > LARGE or AMAX < SMALL, scaling is done.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlar1v (integer N, integer B1, integer BN, double precision LAMBDA, double
precision, dimension( * ) D, double precision, dimension( * ) L, double precision,
dimension( * ) LD, double precision, dimension( * ) LLD, double precision PIVMIN, double
precision GAPTOL, complex*16, dimension( * ) Z, logical WANTNC, integer NEGCNT, double
precision ZTZ, double precision MINGMA, integer R, integer, dimension( * ) ISUPPZ, double
precision NRMINV, double precision RESID, double precision RQCORR, double precision,
dimension( * ) WORK)
ZLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
through bn of the tridiagonal matrix LDLT - λI.
Purpose:
ZLAR1V computes the (scaled) r-th column of the inverse of
the sumbmatrix in rows B1 through BN of the tridiagonal matrix
L D L**T - sigma I. When sigma is close to an eigenvalue, the
computed vector is an accurate eigenvector. Usually, r corresponds
to the index where the eigenvector is largest in magnitude.
The following steps accomplish this computation :
(a) Stationary qd transform, L D L**T - sigma I = L(+) D(+) L(+)**T,
(b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
(c) Computation of the diagonal elements of the inverse of
L D L**T - sigma I by combining the above transforms, and choosing
r as the index where the diagonal of the inverse is (one of the)
largest in magnitude.
(d) Computation of the (scaled) r-th column of the inverse using the
twisted factorization obtained by combining the top part of the
the stationary and the bottom part of the progressive transform.
Parameters
N
N is INTEGER
The order of the matrix L D L**T.
B1
B1 is INTEGER
First index of the submatrix of L D L**T.
BN
BN is INTEGER
Last index of the submatrix of L D L**T.
LAMBDA
LAMBDA is DOUBLE PRECISION
The shift. In order to compute an accurate eigenvector,
LAMBDA should be a good approximation to an eigenvalue
of L D L**T.
L
L is DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal matrix
L, in elements 1 to N-1.
D
D is DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D.
LD
LD is DOUBLE PRECISION array, dimension (N-1)
The n-1 elements L(i)*D(i).
LLD
LLD is DOUBLE PRECISION array, dimension (N-1)
The n-1 elements L(i)*L(i)*D(i).
PIVMIN
PIVMIN is DOUBLE PRECISION
The minimum pivot in the Sturm sequence.
GAPTOL
GAPTOL is DOUBLE PRECISION
Tolerance that indicates when eigenvector entries are negligible
w.r.t. their contribution to the residual.
Z
Z is COMPLEX*16 array, dimension (N)
On input, all entries of Z must be set to 0.
On output, Z contains the (scaled) r-th column of the
inverse. The scaling is such that Z(R) equals 1.
WANTNC
WANTNC is LOGICAL
Specifies whether NEGCNT has to be computed.
NEGCNT
NEGCNT is INTEGER
If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
in the matrix factorization L D L**T, and NEGCNT = -1 otherwise.
ZTZ
ZTZ is DOUBLE PRECISION
The square of the 2-norm of Z.
MINGMA
MINGMA is DOUBLE PRECISION
The reciprocal of the largest (in magnitude) diagonal
element of the inverse of L D L**T - sigma I.
R
R is INTEGER
The twist index for the twisted factorization used to
compute Z.
On input, 0 <= R <= N. If R is input as 0, R is set to
the index where (L D L**T - sigma I)^{-1} is largest
in magnitude. If 1 <= R <= N, R is unchanged.
On output, R contains the twist index used to compute Z.
Ideally, R designates the position of the maximum entry in the
eigenvector.
ISUPPZ
ISUPPZ is INTEGER array, dimension (2)
The support of the vector in Z, i.e., the vector Z is
nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
NRMINV
NRMINV is DOUBLE PRECISION
NRMINV = 1/SQRT( ZTZ )
RESID
RESID is DOUBLE PRECISION
The residual of the FP vector.
RESID = ABS( MINGMA )/SQRT( ZTZ )
RQCORR
RQCORR is DOUBLE PRECISION
The Rayleigh Quotient correction to LAMBDA.
RQCORR = MINGMA*TMP
WORK
WORK is DOUBLE PRECISION array, dimension (4*N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
subroutine zlar2v (integer N, complex*16, dimension( * ) X, complex*16, dimension( * ) Y,
complex*16, dimension( * ) Z, integer INCX, double precision, dimension( * ) C,
complex*16, dimension( * ) S, integer INCC)
ZLAR2V applies a vector of plane rotations with real cosines and complex sines from both
sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
Purpose:
ZLAR2V applies a vector of complex plane rotations with real cosines
from both sides to a sequence of 2-by-2 complex Hermitian matrices,
defined by the elements of the vectors x, y and z. For i = 1,2,...,n
( x(i) z(i) ) :=
( conjg(z(i)) y(i) )
( c(i) conjg(s(i)) ) ( x(i) z(i) ) ( c(i) -conjg(s(i)) )
( -s(i) c(i) ) ( conjg(z(i)) y(i) ) ( s(i) c(i) )
Parameters
N
N is INTEGER
The number of plane rotations to be applied.
X
X is COMPLEX*16 array, dimension (1+(N-1)*INCX)
The vector x; the elements of x are assumed to be real.
Y
Y is COMPLEX*16 array, dimension (1+(N-1)*INCX)
The vector y; the elements of y are assumed to be real.
Z
Z is COMPLEX*16 array, dimension (1+(N-1)*INCX)
The vector z.
INCX
INCX is INTEGER
The increment between elements of X, Y and Z. INCX > 0.
C
C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
The cosines of the plane rotations.
S
S is COMPLEX*16 array, dimension (1+(N-1)*INCC)
The sines of the plane rotations.
INCC
INCC is INTEGER
The increment between elements of C and S. INCC > 0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlarcm (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldc, * ) C, integer
LDC, double precision, dimension( * ) RWORK)
ZLARCM copies all or part of a real two-dimensional array to a complex array.
Purpose:
ZLARCM performs a very simple matrix-matrix multiplication:
C := A * B,
where A is M by M and real; B is M by N and complex;
C is M by N and complex.
Parameters
M
M is INTEGER
The number of rows of the matrix A and of the matrix C.
M >= 0.
N
N is INTEGER
The number of columns and rows of the matrix B and
the number of columns of the matrix C.
N >= 0.
A
A is DOUBLE PRECISION array, dimension (LDA, M)
On entry, A contains the M by M matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >=max(1,M).
B
B is COMPLEX*16 array, dimension (LDB, N)
On entry, B contains 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*16 array, dimension (LDC, N)
On exit, C contains the M by N matrix C.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >=max(1,M).
RWORK
RWORK is DOUBLE PRECISION array, dimension (2*M*N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlarf (character SIDE, integer M, integer N, complex*16, dimension( * ) V, integer
INCV, complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16,
dimension( * ) WORK)
ZLARF applies an elementary reflector to a general rectangular matrix.
Purpose:
ZLARF applies a complex elementary reflector H to a complex M-by-N
matrix C, from either the left or the right. H is represented in the
form
H = I - tau * v * v**H
where tau is a complex scalar and v is a complex vector.
If tau = 0, then H is taken to be the unit matrix.
To apply H**H, supply conjg(tau) instead
tau.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': form H * C
= 'R': form C * H
M
M is INTEGER
The number of rows of the matrix C.
N
N is INTEGER
The number of columns of the matrix C.
V
V is COMPLEX*16 array, dimension
(1 + (M-1)*abs(INCV)) if SIDE = 'L'
or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
The vector v in the representation of H. V is not used if
TAU = 0.
INCV
INCV is INTEGER
The increment between elements of v. INCV <> 0.
TAU
TAU is COMPLEX*16
The value tau in the representation of H.
C
C is COMPLEX*16 array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by the matrix H * C if SIDE = 'L',
or C * H if SIDE = 'R'.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is COMPLEX*16 array, dimension
(N) if SIDE = 'L'
or (M) if SIDE = 'R'
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlarfb (character SIDE, character TRANS, character DIRECT, character STOREV,
integer M, integer N, integer K, complex*16, dimension( ldv, * ) V, integer LDV,
complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( ldc, * ) C, integer
LDC, complex*16, dimension( ldwork, * ) WORK, integer LDWORK)
ZLARFB applies a block reflector or its conjugate-transpose to a general rectangular
matrix.
Purpose:
ZLARFB applies a complex block reflector H or its transpose H**H to a
complex M-by-N matrix C, from either the left or the right.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply H or H**H from the Left
= 'R': apply H or H**H from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'C': apply H**H (Conjugate transpose)
DIRECT
DIRECT is CHARACTER*1
Indicates how H is formed from a product of elementary
reflectors
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV
STOREV is CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= 'C': Columnwise
= 'R': Rowwise
M
M is INTEGER
The number of rows of the matrix C.
N
N is INTEGER
The number of columns of the matrix C.
K
K is INTEGER
The order of the matrix T (= the number of elementary
reflectors whose product defines the block reflector).
If SIDE = 'L', M >= K >= 0;
if SIDE = 'R', N >= K >= 0.
V
V is COMPLEX*16 array, dimension
(LDV,K) if STOREV = 'C'
(LDV,M) if STOREV = 'R' and SIDE = 'L'
(LDV,N) if STOREV = 'R' and SIDE = 'R'
See Further Details.
LDV
LDV is INTEGER
The leading dimension of the array V.
If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
if STOREV = 'R', LDV >= K.
T
T is COMPLEX*16 array, dimension (LDT,K)
The triangular K-by-K matrix T in the representation of the
block reflector.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= K.
C
C is COMPLEX*16 array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by H*C or H**H*C or C*H or C*H**H.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is COMPLEX*16 array, dimension (LDWORK,K)
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
If SIDE = 'L', LDWORK >= max(1,N);
if SIDE = 'R', LDWORK >= max(1,M).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored; the corresponding
array elements are modified but restored on exit. The rest of the
array is not used.
DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
( v1 1 ) ( 1 v2 v2 v2 )
( v1 v2 1 ) ( 1 v3 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
V = ( v1 v2 v3 ) V = ( v1 v1 1 )
( v1 v2 v3 ) ( v2 v2 v2 1 )
( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
( 1 v3 )
( 1 )
subroutine zlarfb_gett (character IDENT, integer M, integer N, integer K, complex*16,
dimension( ldt, * ) T, integer LDT, complex*16, dimension( lda, * ) A, integer LDA,
complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension( ldwork, * ) WORK,
integer LDWORK)
ZLARFB_GETT
Purpose:
ZLARFB_GETT applies a complex Householder block reflector H from the
left to a complex (K+M)-by-N 'triangular-pentagonal' matrix
composed of two block matrices: an upper trapezoidal K-by-N matrix A
stored in the array A, and a rectangular M-by-(N-K) matrix B, stored
in the array B. The block reflector H is stored in a compact
WY-representation, where the elementary reflectors are in the
arrays A, B and T. See Further Details section.
Parameters
IDENT
IDENT is CHARACTER*1
If IDENT = not 'I', or not 'i', then V1 is unit
lower-triangular and stored in the left K-by-K block of
the input matrix A,
If IDENT = 'I' or 'i', then V1 is an identity matrix and
not stored.
See Further Details section.
M
M is INTEGER
The number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrices A and B.
N >= 0.
K
K is INTEGER
The number or rows of the matrix A.
K is also order of the matrix T, i.e. the number of
elementary reflectors whose product defines the block
reflector. 0 <= K <= N.
T
T is COMPLEX*16 array, dimension (LDT,K)
The upper-triangular K-by-K matrix T in the representation
of the block reflector.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= K.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry:
a) In the K-by-N upper-trapezoidal part A: input matrix A.
b) In the columns below the diagonal: columns of V1
(ones are not stored on the diagonal).
On exit:
A is overwritten by rectangular K-by-N product H*A.
See Further Details section.
LDA
LDB is INTEGER
The leading dimension of the array A. LDA >= max(1,K).
B
B is COMPLEX*16 array, dimension (LDB,N)
On entry:
a) In the M-by-(N-K) right block: input matrix B.
b) In the M-by-N left block: columns of V2.
On exit:
B is overwritten by rectangular M-by-N product H*B.
See Further Details section.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= max(1,M).
WORK
WORK is COMPLEX*16 array,
dimension (LDWORK,max(K,N-K))
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK. LDWORK>=max(1,K).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
November 2020, Igor Kozachenko,
Computer Science Division,
University of California, Berkeley
Further Details:
(1) Description of the Algebraic Operation.
The matrix A is a K-by-N matrix composed of two column block
matrices, A1, which is K-by-K, and A2, which is K-by-(N-K):
A = ( A1, A2 ).
The matrix B is an M-by-N matrix composed of two column block
matrices, B1, which is M-by-K, and B2, which is M-by-(N-K):
B = ( B1, B2 ).
Perform the operation:
( A_out ) := H * ( A_in ) = ( I - V * T * V**H ) * ( A_in ) =
( B_out ) ( B_in ) ( B_in )
= ( I - ( V1 ) * T * ( V1**H, V2**H ) ) * ( A_in )
( V2 ) ( B_in )
On input:
a) ( A_in ) consists of two block columns:
( B_in )
( A_in ) = (( A1_in ) ( A2_in )) = (( A1_in ) ( A2_in ))
( B_in ) (( B1_in ) ( B2_in )) (( 0 ) ( B2_in )),
where the column blocks are:
( A1_in ) is a K-by-K upper-triangular matrix stored in the
upper triangular part of the array A(1:K,1:K).
( B1_in ) is an M-by-K rectangular ZERO matrix and not stored.
( A2_in ) is a K-by-(N-K) rectangular matrix stored
in the array A(1:K,K+1:N).
( B2_in ) is an M-by-(N-K) rectangular matrix stored
in the array B(1:M,K+1:N).
b) V = ( V1 )
( V2 )
where:
1) if IDENT == 'I',V1 is a K-by-K identity matrix, not stored;
2) if IDENT != 'I',V1 is a K-by-K unit lower-triangular matrix,
stored in the lower-triangular part of the array
A(1:K,1:K) (ones are not stored),
and V2 is an M-by-K rectangular stored the array B(1:M,1:K),
(because on input B1_in is a rectangular zero
matrix that is not stored and the space is
used to store V2).
c) T is a K-by-K upper-triangular matrix stored
in the array T(1:K,1:K).
On output:
a) ( A_out ) consists of two block columns:
( B_out )
( A_out ) = (( A1_out ) ( A2_out ))
( B_out ) (( B1_out ) ( B2_out )),
where the column blocks are:
( A1_out ) is a K-by-K square matrix, or a K-by-K
upper-triangular matrix, if V1 is an
identity matrix. AiOut is stored in
the array A(1:K,1:K).
( B1_out ) is an M-by-K rectangular matrix stored
in the array B(1:M,K:N).
( A2_out ) is a K-by-(N-K) rectangular matrix stored
in the array A(1:K,K+1:N).
( B2_out ) is an M-by-(N-K) rectangular matrix stored
in the array B(1:M,K+1:N).
The operation above can be represented as the same operation
on each block column:
( A1_out ) := H * ( A1_in ) = ( I - V * T * V**H ) * ( A1_in )
( B1_out ) ( 0 ) ( 0 )
( A2_out ) := H * ( A2_in ) = ( I - V * T * V**H ) * ( A2_in )
( B2_out ) ( B2_in ) ( B2_in )
If IDENT != 'I':
The computation for column block 1:
A1_out: = A1_in - V1*T*(V1**H)*A1_in
B1_out: = - V2*T*(V1**H)*A1_in
The computation for column block 2, which exists if N > K:
A2_out: = A2_in - V1*T*( (V1**H)*A2_in + (V2**H)*B2_in )
B2_out: = B2_in - V2*T*( (V1**H)*A2_in + (V2**H)*B2_in )
If IDENT == 'I':
The operation for column block 1:
A1_out: = A1_in - V1*T*A1_in
B1_out: = - V2*T*A1_in
The computation for column block 2, which exists if N > K:
A2_out: = A2_in - T*( A2_in + (V2**H)*B2_in )
B2_out: = B2_in - V2*T*( A2_in + (V2**H)*B2_in )
(2) Description of the Algorithmic Computation.
In the first step, we compute column block 2, i.e. A2 and B2.
Here, we need to use the K-by-(N-K) rectangular workspace
matrix W2 that is of the same size as the matrix A2.
W2 is stored in the array WORK(1:K,1:(N-K)).
In the second step, we compute column block 1, i.e. A1 and B1.
Here, we need to use the K-by-K square workspace matrix W1
that is of the same size as the as the matrix A1.
W1 is stored in the array WORK(1:K,1:K).
NOTE: Hence, in this routine, we need the workspace array WORK
only of size WORK(1:K,1:max(K,N-K)) so it can hold both W2 from
the first step and W1 from the second step.
Case (A), when V1 is unit lower-triangular, i.e. IDENT != 'I',
more computations than in the Case (B).
if( IDENT != 'I' ) then
if ( N > K ) then
(First Step - column block 2)
col2_(1) W2: = A2
col2_(2) W2: = (V1**H) * W2 = (unit_lower_tr_of_(A1)**H) * W2
col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
col2_(4) W2: = T * W2
col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
col2_(7) A2: = A2 - W2
else
(Second Step - column block 1)
col1_(1) W1: = A1
col1_(2) W1: = (V1**H) * W1 = (unit_lower_tr_of_(A1)**H) * W1
col1_(3) W1: = T * W1
col1_(4) B1: = - V2 * W1 = - B1 * W1
col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
col1_(6) square A1: = A1 - W1
end if
end if
Case (B), when V1 is an identity matrix, i.e. IDENT == 'I',
less computations than in the Case (A)
if( IDENT == 'I' ) then
if ( N > K ) then
(First Step - column block 2)
col2_(1) W2: = A2
col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2
col2_(4) W2: = T * W2
col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
col2_(7) A2: = A2 - W2
else
(Second Step - column block 1)
col1_(1) W1: = A1
col1_(3) W1: = T * W1
col1_(4) B1: = - V2 * W1 = - B1 * W1
col1_(6) upper-triangular_of_(A1): = A1 - W1
end if
end if
Combine these cases (A) and (B) together, this is the resulting
algorithm:
if ( N > K ) then
(First Step - column block 2)
col2_(1) W2: = A2
if( IDENT != 'I' ) then
col2_(2) W2: = (V1**H) * W2
= (unit_lower_tr_of_(A1)**H) * W2
end if
col2_(3) W2: = W2 + (V2**H) * B2 = W2 + (B1**H) * B2]
col2_(4) W2: = T * W2
col2_(5) B2: = B2 - V2 * W2 = B2 - B1 * W2
if( IDENT != 'I' ) then
col2_(6) W2: = V1 * W2 = unit_lower_tr_of_(A1) * W2
end if
col2_(7) A2: = A2 - W2
else
(Second Step - column block 1)
col1_(1) W1: = A1
if( IDENT != 'I' ) then
col1_(2) W1: = (V1**H) * W1
= (unit_lower_tr_of_(A1)**H) * W1
end if
col1_(3) W1: = T * W1
col1_(4) B1: = - V2 * W1 = - B1 * W1
if( IDENT != 'I' ) then
col1_(5) square W1: = V1 * W1 = unit_lower_tr_of_(A1) * W1
col1_(6_a) below_diag_of_(A1): = - below_diag_of_(W1)
end if
col1_(6_b) up_tr_of_(A1): = up_tr_of_(A1) - up_tr_of_(W1)
end if
subroutine zlarfg (integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX,
complex*16 TAU)
ZLARFG generates an elementary reflector (Householder matrix).
Purpose:
ZLARFG generates a complex elementary reflector H of order n, such
that
H**H * ( alpha ) = ( beta ), H**H * H = I.
( x ) ( 0 )
where alpha and beta are scalars, with beta real, and x is an
(n-1)-element complex vector. H is represented in the form
H = I - tau * ( 1 ) * ( 1 v**H ) ,
( v )
where tau is a complex scalar and v is a complex (n-1)-element
vector. Note that H is not hermitian.
If the elements of x are all zero and alpha is real, then tau = 0
and H is taken to be the unit matrix.
Otherwise 1 <= real(tau) <= 2 and abs(tau-1) <= 1 .
Parameters
N
N is INTEGER
The order of the elementary reflector.
ALPHA
ALPHA is COMPLEX*16
On entry, the value alpha.
On exit, it is overwritten with the value beta.
X
X is COMPLEX*16 array, dimension
(1+(N-2)*abs(INCX))
On entry, the vector x.
On exit, it is overwritten with the vector v.
INCX
INCX is INTEGER
The increment between elements of X. INCX > 0.
TAU
TAU is COMPLEX*16
The value tau.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlarfgp (integer N, complex*16 ALPHA, complex*16, dimension( * ) X, integer INCX,
complex*16 TAU)
ZLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Purpose:
ZLARFGP generates a complex elementary reflector H of order n, such
that
H**H * ( alpha ) = ( beta ), H**H * H = I.
( x ) ( 0 )
where alpha and beta are scalars, beta is real and non-negative, and
x is an (n-1)-element complex vector. H is represented in the form
H = I - tau * ( 1 ) * ( 1 v**H ) ,
( v )
where tau is a complex scalar and v is a complex (n-1)-element
vector. Note that H is not hermitian.
If the elements of x are all zero and alpha is real, then tau = 0
and H is taken to be the unit matrix.
Parameters
N
N is INTEGER
The order of the elementary reflector.
ALPHA
ALPHA is COMPLEX*16
On entry, the value alpha.
On exit, it is overwritten with the value beta.
X
X is COMPLEX*16 array, dimension
(1+(N-2)*abs(INCX))
On entry, the vector x.
On exit, it is overwritten with the vector v.
INCX
INCX is INTEGER
The increment between elements of X. INCX > 0.
TAU
TAU is COMPLEX*16
The value tau.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlarft (character DIRECT, character STOREV, integer N, integer K, complex*16,
dimension( ldv, * ) V, integer LDV, complex*16, dimension( * ) TAU, complex*16, dimension(
ldt, * ) T, integer LDT)
ZLARFT forms the triangular factor T of a block reflector H = I - vtvH
Purpose:
ZLARFT forms the triangular factor T of a complex block reflector H
of order n, which is defined as a product of k elementary reflectors.
If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
If STOREV = 'C', the vector which defines the elementary reflector
H(i) is stored in the i-th column of the array V, and
H = I - V * T * V**H
If STOREV = 'R', the vector which defines the elementary reflector
H(i) is stored in the i-th row of the array V, and
H = I - V**H * T * V
Parameters
DIRECT
DIRECT is CHARACTER*1
Specifies the order in which the elementary reflectors are
multiplied to form the block reflector:
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV
STOREV is CHARACTER*1
Specifies how the vectors which define the elementary
reflectors are stored (see also Further Details):
= 'C': columnwise
= 'R': rowwise
N
N is INTEGER
The order of the block reflector H. N >= 0.
K
K is INTEGER
The order of the triangular factor T (= the number of
elementary reflectors). K >= 1.
V
V is COMPLEX*16 array, dimension
(LDV,K) if STOREV = 'C'
(LDV,N) if STOREV = 'R'
The matrix V. See further details.
LDV
LDV is INTEGER
The leading dimension of the array V.
If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
TAU
TAU is COMPLEX*16 array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).
T
T is COMPLEX*16 array, dimension (LDT,K)
The k by k triangular factor T of the block reflector.
If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
lower triangular. The rest of the array is not used.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= K.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The shape of the matrix V and the storage of the vectors which define
the H(i) is best illustrated by the following example with n = 5 and
k = 3. The elements equal to 1 are not stored.
DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
( v1 1 ) ( 1 v2 v2 v2 )
( v1 v2 1 ) ( 1 v3 v3 )
( v1 v2 v3 )
( v1 v2 v3 )
DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
V = ( v1 v2 v3 ) V = ( v1 v1 1 )
( v1 v2 v3 ) ( v2 v2 v2 1 )
( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
( 1 v3 )
( 1 )
subroutine zlarfx (character SIDE, integer M, integer N, complex*16, dimension( * ) V,
complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * )
WORK)
ZLARFX applies an elementary reflector to a general rectangular matrix, with loop
unrolling when the reflector has order ≤ 10.
Purpose:
ZLARFX applies a complex elementary reflector H to a complex m by n
matrix C, from either the left or the right. H is represented in the
form
H = I - tau * v * v**H
where tau is a complex scalar and v is a complex vector.
If tau = 0, then H is taken to be the unit matrix
This version uses inline code if H has order < 11.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': form H * C
= 'R': form C * H
M
M is INTEGER
The number of rows of the matrix C.
N
N is INTEGER
The number of columns of the matrix C.
V
V is COMPLEX*16 array, dimension (M) if SIDE = 'L'
or (N) if SIDE = 'R'
The vector v in the representation of H.
TAU
TAU is COMPLEX*16
The value tau in the representation of H.
C
C is COMPLEX*16 array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by the matrix H * C if SIDE = 'L',
or C * H if SIDE = 'R'.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is COMPLEX*16 array, dimension (N) if SIDE = 'L'
or (M) if SIDE = 'R'
WORK is not referenced if H has order < 11.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlarfy (character UPLO, integer N, complex*16, dimension( * ) V, integer INCV,
complex*16 TAU, complex*16, dimension( ldc, * ) C, integer LDC, complex*16, dimension( * )
WORK)
ZLARFY
Purpose:
ZLARFY 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*16 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*16
The value tau as described above.
C
C is COMPLEX*16 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*16 array, dimension (N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlargv (integer N, complex*16, dimension( * ) X, integer INCX, complex*16,
dimension( * ) Y, integer INCY, double precision, dimension( * ) C, integer INCC)
ZLARGV generates a vector of plane rotations with real cosines and complex sines.
Purpose:
ZLARGV generates a vector of complex plane rotations with real
cosines, determined by elements of the complex vectors x and y.
For i = 1,2,...,n
( c(i) s(i) ) ( x(i) ) = ( r(i) )
( -conjg(s(i)) c(i) ) ( y(i) ) = ( 0 )
where c(i)**2 + ABS(s(i))**2 = 1
The following conventions are used (these are the same as in ZLARTG,
but differ from the BLAS1 routine ZROTG):
If y(i)=0, then c(i)=1 and s(i)=0.
If x(i)=0, then c(i)=0 and s(i) is chosen so that r(i) is real.
Parameters
N
N is INTEGER
The number of plane rotations to be generated.
X
X is COMPLEX*16 array, dimension (1+(N-1)*INCX)
On entry, the vector x.
On exit, x(i) is overwritten by r(i), for i = 1,...,n.
INCX
INCX is INTEGER
The increment between elements of X. INCX > 0.
Y
Y is COMPLEX*16 array, dimension (1+(N-1)*INCY)
On entry, the vector y.
On exit, the sines of the plane rotations.
INCY
INCY is INTEGER
The increment between elements of Y. INCY > 0.
C
C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
The cosines of the plane rotations.
INCC
INCC is INTEGER
The increment between elements of C. INCC > 0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
6-6-96 - Modified with a new algorithm by W. Kahan and J. Demmel
This version has a few statements commented out for thread safety
(machine parameters are computed on each entry). 10 feb 03, SJH.
subroutine zlarnv (integer IDIST, integer, dimension( 4 ) ISEED, integer N, complex*16,
dimension( * ) X)
ZLARNV returns a vector of random numbers from a uniform or normal distribution.
Purpose:
ZLARNV returns a vector of n random complex numbers from a uniform or
normal distribution.
Parameters
IDIST
IDIST is INTEGER
Specifies the distribution of the random numbers:
= 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: uniformly distributed on the disc abs(z) < 1
= 5: uniformly distributed on the circle abs(z) = 1
ISEED
ISEED is INTEGER array, dimension (4)
On entry, the seed of the random number generator; the array
elements must be between 0 and 4095, and ISEED(4) must be
odd.
On exit, the seed is updated.
N
N is INTEGER
The number of random numbers to be generated.
X
X is COMPLEX*16 array, dimension (N)
The generated random numbers.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
This routine calls the auxiliary routine DLARUV to generate random
real numbers from a uniform (0,1) distribution, in batches of up to
128 using vectorisable code. The Box-Muller method is used to
transform numbers from a uniform to a normal distribution.
subroutine zlarrv (integer N, double precision VL, double precision VU, double precision,
dimension( * ) D, double precision, dimension( * ) L, double precision PIVMIN, integer,
dimension( * ) ISPLIT, integer M, integer DOL, integer DOU, double precision MINRGP,
double precision RTOL1, double precision RTOL2, double precision, dimension( * ) W, double
precision, dimension( * ) WERR, double precision, dimension( * ) WGAP, integer, dimension(
* ) IBLOCK, integer, dimension( * ) INDEXW, double precision, dimension( * ) GERS,
complex*16, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double
precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
ZLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the
eigenvalues of L D LT.
Purpose:
ZLARRV computes the eigenvectors of the tridiagonal matrix
T = L D L**T given L, D and APPROXIMATIONS to the eigenvalues of L D L**T.
The input eigenvalues should have been computed by DLARRE.
Parameters
N
N is INTEGER
The order of the matrix. N >= 0.
VL
VL is DOUBLE PRECISION
Lower bound of the interval that contains the desired
eigenvalues. VL < VU. Needed to compute gaps on the left or right
end of the extremal eigenvalues in the desired RANGE.
VU
VU is DOUBLE PRECISION
Upper bound of the interval that contains the desired
eigenvalues. VL < VU. Needed to compute gaps on the left or right
end of the extremal eigenvalues in the desired RANGE.
D
D is DOUBLE PRECISION array, dimension (N)
On entry, the N diagonal elements of the diagonal matrix D.
On exit, D may be overwritten.
L
L is DOUBLE PRECISION array, dimension (N)
On entry, the (N-1) subdiagonal elements of the unit
bidiagonal matrix L are in elements 1 to N-1 of L
(if the matrix is not split.) At the end of each block
is stored the corresponding shift as given by DLARRE.
On exit, L is overwritten.
PIVMIN
PIVMIN is DOUBLE PRECISION
The minimum pivot allowed in the Sturm sequence.
ISPLIT
ISPLIT is INTEGER array, dimension (N)
The splitting points, at which T breaks up into blocks.
The first block consists of rows/columns 1 to
ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
through ISPLIT( 2 ), etc.
M
M is INTEGER
The total number of input eigenvalues. 0 <= M <= N.
DOL
DOL is INTEGER
DOU
DOU is INTEGER
If the user wants to compute only selected eigenvectors from all
the eigenvalues supplied, he can specify an index range DOL:DOU.
Or else the setting DOL=1, DOU=M should be applied.
Note that DOL and DOU refer to the order in which the eigenvalues
are stored in W.
If the user wants to compute only selected eigenpairs, then
the columns DOL-1 to DOU+1 of the eigenvector space Z contain the
computed eigenvectors. All other columns of Z are set to zero.
MINRGP
MINRGP is DOUBLE PRECISION
RTOL1
RTOL1 is DOUBLE PRECISION
RTOL2
RTOL2 is DOUBLE PRECISION
Parameters for bisection.
An interval [LEFT,RIGHT] has converged if
RIGHT-LEFT < MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )
W
W is DOUBLE PRECISION array, dimension (N)
The first M elements of W contain the APPROXIMATE eigenvalues for
which eigenvectors are to be computed. The eigenvalues
should be grouped by split-off block and ordered from
smallest to largest within the block ( The output array
W from DLARRE is expected here ). Furthermore, they are with
respect to the shift of the corresponding root representation
for their block. On exit, W holds the eigenvalues of the
UNshifted matrix.
WERR
WERR is DOUBLE PRECISION array, dimension (N)
The first M elements contain the semiwidth of the uncertainty
interval of the corresponding eigenvalue in W
WGAP
WGAP is DOUBLE PRECISION array, dimension (N)
The separation from the right neighbor eigenvalue in W.
IBLOCK
IBLOCK is INTEGER array, dimension (N)
The indices of the blocks (submatrices) associated with the
corresponding eigenvalues in W; IBLOCK(i)=1 if eigenvalue
W(i) belongs to the first block from the top, =2 if W(i)
belongs to the second block, etc.
INDEXW
INDEXW is INTEGER array, dimension (N)
The indices of the eigenvalues within each block (submatrix);
for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the
i-th eigenvalue W(i) is the 10-th eigenvalue in the second block.
GERS
GERS is DOUBLE PRECISION array, dimension (2*N)
The N Gerschgorin intervals (the i-th Gerschgorin interval
is (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals should
be computed from the original UNshifted matrix.
Z
Z is COMPLEX*16 array, dimension (LDZ, max(1,M) )
If INFO = 0, the first M columns of Z contain the
orthonormal eigenvectors of the matrix T
corresponding to the input eigenvalues, with the i-th
column of Z holding the eigenvector associated with W(i).
Note: the user must ensure that at least max(1,M) columns are
supplied in the array Z.
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDZ >= 1, and if
JOBZ = 'V', LDZ >= max(1,N).
ISUPPZ
ISUPPZ is INTEGER array, dimension ( 2*max(1,M) )
The support of the eigenvectors in Z, i.e., the indices
indicating the nonzero elements in Z. The I-th eigenvector
is nonzero only in elements ISUPPZ( 2*I-1 ) through
ISUPPZ( 2*I ).
WORK
WORK is DOUBLE PRECISION array, dimension (12*N)
IWORK
IWORK is INTEGER array, dimension (7*N)
INFO
INFO is INTEGER
= 0: successful exit
> 0: A problem occurred in ZLARRV.
< 0: One of the called subroutines signaled an internal problem.
Needs inspection of the corresponding parameter IINFO
for further information.
=-1: Problem in DLARRB when refining a child's eigenvalues.
=-2: Problem in DLARRF when computing the RRR of a child.
When a child is inside a tight cluster, it can be difficult
to find an RRR. A partial remedy from the user's point of
view is to make the parameter MINRGP smaller and recompile.
However, as the orthogonality of the computed vectors is
proportional to 1/MINRGP, the user should be aware that
he might be trading in precision when he decreases MINRGP.
=-3: Problem in DLARRB when refining a single eigenvalue
after the Rayleigh correction was rejected.
= 5: The Rayleigh Quotient Iteration failed to converge to
full accuracy in MAXITR steps.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Beresford Parlett, University of California, Berkeley, USA
Jim Demmel, University of California, Berkeley, USA
Inderjit Dhillon, University of Texas, Austin, USA
Osni Marques, LBNL/NERSC, USA
Christof Voemel, University of California, Berkeley, USA
subroutine zlartv (integer N, complex*16, dimension( * ) X, integer INCX, complex*16,
dimension( * ) Y, integer INCY, double precision, dimension( * ) C, complex*16, dimension(
* ) S, integer INCC)
ZLARTV applies a vector of plane rotations with real cosines and complex sines to the
elements of a pair of vectors.
Purpose:
ZLARTV applies a vector of complex plane rotations with real cosines
to elements of the complex vectors x and y. For i = 1,2,...,n
( x(i) ) := ( c(i) s(i) ) ( x(i) )
( y(i) ) ( -conjg(s(i)) c(i) ) ( y(i) )
Parameters
N
N is INTEGER
The number of plane rotations to be applied.
X
X is COMPLEX*16 array, dimension (1+(N-1)*INCX)
The vector x.
INCX
INCX is INTEGER
The increment between elements of X. INCX > 0.
Y
Y is COMPLEX*16 array, dimension (1+(N-1)*INCY)
The vector y.
INCY
INCY is INTEGER
The increment between elements of Y. INCY > 0.
C
C is DOUBLE PRECISION array, dimension (1+(N-1)*INCC)
The cosines of the plane rotations.
S
S is COMPLEX*16 array, dimension (1+(N-1)*INCC)
The sines of the plane rotations.
INCC
INCC is INTEGER
The increment between elements of C and S. INCC > 0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlascl (character TYPE, integer KL, integer KU, double precision CFROM, double
precision CTO, integer M, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer INFO)
ZLASCL multiplies a general rectangular matrix by a real scalar defined as cto/cfrom.
Purpose:
ZLASCL multiplies the M by N complex matrix A by the real scalar
CTO/CFROM. This is done without over/underflow as long as the final
result CTO*A(I,J)/CFROM does not over/underflow. TYPE specifies that
A may be full, upper triangular, lower triangular, upper Hessenberg,
or banded.
Parameters
TYPE
TYPE is CHARACTER*1
TYPE indices the storage type of the input matrix.
= 'G': A is a full matrix.
= 'L': A is a lower triangular matrix.
= 'U': A is an upper triangular matrix.
= 'H': A is an upper Hessenberg matrix.
= 'B': A is a symmetric band matrix with lower bandwidth KL
and upper bandwidth KU and with the only the lower
half stored.
= 'Q': A is a symmetric band matrix with lower bandwidth KL
and upper bandwidth KU and with the only the upper
half stored.
= 'Z': A is a band matrix with lower bandwidth KL and upper
bandwidth KU. See ZGBTRF for storage details.
KL
KL is INTEGER
The lower bandwidth of A. Referenced only if TYPE = 'B',
'Q' or 'Z'.
KU
KU is INTEGER
The upper bandwidth of A. Referenced only if TYPE = 'B',
'Q' or 'Z'.
CFROM
CFROM is DOUBLE PRECISION
CTO
CTO is DOUBLE PRECISION
The matrix A is multiplied by CTO/CFROM. A(I,J) is computed
without over/underflow if the final result CTO*A(I,J)/CFROM
can be represented without over/underflow. CFROM must be
nonzero.
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.
A
A is COMPLEX*16 array, dimension (LDA,N)
The matrix to be multiplied by CTO/CFROM. See TYPE for the
storage type.
LDA
LDA is INTEGER
The leading dimension of the array A.
If TYPE = 'G', 'L', 'U', 'H', LDA >= max(1,M);
TYPE = 'B', LDA >= KL+1;
TYPE = 'Q', LDA >= KU+1;
TYPE = 'Z', LDA >= 2*KL+KU+1.
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.
subroutine zlaset (character UPLO, integer M, integer N, complex*16 ALPHA, complex*16 BETA,
complex*16, dimension( lda, * ) A, integer LDA)
ZLASET initializes the off-diagonal elements and the diagonal elements of a matrix to
given values.
Purpose:
ZLASET initializes a 2-D array A to BETA on the diagonal and
ALPHA on the offdiagonals.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies the part of the matrix A to be set.
= 'U': Upper triangular part is set. The lower triangle
is unchanged.
= 'L': Lower triangular part is set. The upper triangle
is unchanged.
Otherwise: All of the matrix A is set.
M
M is INTEGER
On entry, M specifies the number of rows of A.
N
N is INTEGER
On entry, N specifies the number of columns of A.
ALPHA
ALPHA is COMPLEX*16
All the offdiagonal array elements are set to ALPHA.
BETA
BETA is COMPLEX*16
All the diagonal array elements are set to BETA.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the m by n matrix A.
On exit, A(i,j) = ALPHA, 1 <= i <= m, 1 <= j <= n, i.ne.j;
A(i,i) = BETA , 1 <= i <= min(m,n)
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlasr (character SIDE, character PIVOT, character DIRECT, integer M, integer N,
double precision, dimension( * ) C, double precision, dimension( * ) S, complex*16,
dimension( lda, * ) A, integer LDA)
ZLASR applies a sequence of plane rotations to a general rectangular matrix.
Purpose:
ZLASR applies a sequence of real plane rotations to a complex matrix
A, from either the left or the right.
When SIDE = 'L', the transformation takes the form
A := P*A
and when SIDE = 'R', the transformation takes the form
A := A*P**T
where P is an orthogonal matrix consisting of a sequence of z plane
rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
and P**T is the transpose of P.
When DIRECT = 'F' (Forward sequence), then
P = P(z-1) * ... * P(2) * P(1)
and when DIRECT = 'B' (Backward sequence), then
P = P(1) * P(2) * ... * P(z-1)
where P(k) is a plane rotation matrix defined by the 2-by-2 rotation
R(k) = ( c(k) s(k) )
= ( -s(k) c(k) ).
When PIVOT = 'V' (Variable pivot), the rotation is performed
for the plane (k,k+1), i.e., P(k) has the form
P(k) = ( 1 )
( ... )
( 1 )
( c(k) s(k) )
( -s(k) c(k) )
( 1 )
( ... )
( 1 )
where R(k) appears as a rank-2 modification to the identity matrix in
rows and columns k and k+1.
When PIVOT = 'T' (Top pivot), the rotation is performed for the
plane (1,k+1), so P(k) has the form
P(k) = ( c(k) s(k) )
( 1 )
( ... )
( 1 )
( -s(k) c(k) )
( 1 )
( ... )
( 1 )
where R(k) appears in rows and columns 1 and k+1.
Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
performed for the plane (k,z), giving P(k) the form
P(k) = ( 1 )
( ... )
( 1 )
( c(k) s(k) )
( 1 )
( ... )
( 1 )
( -s(k) c(k) )
where R(k) appears in rows and columns k and z. The rotations are
performed without ever forming P(k) explicitly.
Parameters
SIDE
SIDE is CHARACTER*1
Specifies whether the plane rotation matrix P is applied to
A on the left or the right.
= 'L': Left, compute A := P*A
= 'R': Right, compute A:= A*P**T
PIVOT
PIVOT is CHARACTER*1
Specifies the plane for which P(k) is a plane rotation
matrix.
= 'V': Variable pivot, the plane (k,k+1)
= 'T': Top pivot, the plane (1,k+1)
= 'B': Bottom pivot, the plane (k,z)
DIRECT
DIRECT is CHARACTER*1
Specifies whether P is a forward or backward sequence of
plane rotations.
= 'F': Forward, P = P(z-1)*...*P(2)*P(1)
= 'B': Backward, P = P(1)*P(2)*...*P(z-1)
M
M is INTEGER
The number of rows of the matrix A. If m <= 1, an immediate
return is effected.
N
N is INTEGER
The number of columns of the matrix A. If n <= 1, an
immediate return is effected.
C
C is DOUBLE PRECISION array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The cosines c(k) of the plane rotations.
S
S is DOUBLE PRECISION array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The sines s(k) of the plane rotations. The 2-by-2 plane
rotation part of the matrix P(k), R(k), has the form
R(k) = ( c(k) s(k) )
( -s(k) c(k) ).
A
A is COMPLEX*16 array, dimension (LDA,N)
The M-by-N matrix A. On exit, A is overwritten by P*A if
SIDE = 'R' or by A*P**T if SIDE = 'L'.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlaswp (integer N, complex*16, dimension( lda, * ) A, integer LDA, integer K1,
integer K2, integer, dimension( * ) IPIV, integer INCX)
ZLASWP performs a series of row interchanges on a general rectangular matrix.
Purpose:
ZLASWP performs a series of row interchanges on the matrix A.
One row interchange is initiated for each of rows K1 through K2 of A.
Parameters
N
N is INTEGER
The number of columns of the matrix A.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the matrix of column dimension N to which the row
interchanges will be applied.
On exit, the permuted matrix.
LDA
LDA is INTEGER
The leading dimension of the array A.
K1
K1 is INTEGER
The first element of IPIV for which a row interchange will
be done.
K2
K2 is INTEGER
(K2-K1+1) is the number of elements of IPIV for which a row
interchange will be done.
IPIV
IPIV is INTEGER array, dimension (K1+(K2-K1)*abs(INCX))
The vector of pivot indices. Only the elements in positions
K1 through K1+(K2-K1)*abs(INCX) of IPIV are accessed.
IPIV(K1+(K-K1)*abs(INCX)) = L implies rows K and L are to be
interchanged.
INCX
INCX is INTEGER
The increment between successive values of IPIV. If INCX
is negative, the pivots are applied in reverse order.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Modified by
R. C. Whaley, Computer Science Dept., Univ. of Tenn., Knoxville, USA
subroutine zlat2c (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA,
complex, dimension( ldsa, * ) SA, integer LDSA, integer INFO)
ZLAT2C converts a double complex triangular matrix to a complex triangular matrix.
Purpose:
ZLAT2C converts a COMPLEX*16 triangular matrix, SA, to a COMPLEX
triangular matrix, A.
RMAX is the overflow for the SINGLE PRECISION arithmetic
ZLAT2C checks that all the entries of A are between -RMAX and
RMAX. If not the conversion is aborted and a flag is raised.
This is an auxiliary routine so there is no argument checking.
Parameters
UPLO
UPLO is CHARACTER*1
= 'U': A is upper triangular;
= 'L': A is lower triangular.
N
N is INTEGER
The number of rows and columns of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N triangular coefficient matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
SA
SA is COMPLEX array, dimension (LDSA,N)
Only the UPLO part of SA is referenced. On exit, if INFO=0,
the N-by-N coefficient matrix SA; if INFO>0, the content of
the UPLO part of SA is unspecified.
LDSA
LDSA is INTEGER
The leading dimension of the array SA. LDSA >= max(1,M).
INFO
INFO is INTEGER
= 0: successful exit.
= 1: an entry of the matrix A is greater than the SINGLE
PRECISION overflow threshold, in this case, the content
of the UPLO part of SA in exit is unspecified.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, integer KD, complex*16, dimension( ldab, * ) AB, integer LDAB, complex*16, dimension( *
) X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)
ZLATBS solves a triangular banded system of equations.
Purpose:
ZLATBS solves one of the triangular systems
A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
with scaling to prevent overflow, where A is an upper or lower
triangular band matrix. Here A**T denotes the transpose of A, x and b
are n-element vectors, and s is a scaling factor, usually less than
or equal to 1, chosen so that the components of x will be less than
the overflow threshold. If the unscaled problem will not cause
overflow, the Level 2 BLAS routine ZTBSV is called. If the matrix A
is singular (A(j,j) = 0 for some j), then s is set to 0 and a
non-trivial solution to A*x = 0 is returned.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T * x = s*b (Transpose)
= 'C': Solve A**H * x = s*b (Conjugate transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of subdiagonals or superdiagonals in the
triangular matrix A. KD >= 0.
AB
AB is COMPLEX*16 array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first KD+1 rows of the array. The j-th column of A is stored
in the j-th column of the array AB as follows:
if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
X
X is COMPLEX*16 array, dimension (N)
On entry, the right hand side b of the triangular system.
On exit, X is overwritten by the solution vector x.
SCALE
SCALE is DOUBLE PRECISION
The scaling factor s for the triangular system
A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
If SCALE = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
CNORM
CNORM is DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
A rough bound on x is computed; if that is less than overflow, ZTBSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b. The basic algorithm
if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTBSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the
columnwise method can be performed without fear of overflow. If
the computed bound is greater than a large constant, x is scaled to
prevent overflow, but if the bound overflows, x is set to 0, x(j) to
1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b or
A**H *x = b. The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
subroutine zlatdf (integer IJOB, integer N, complex*16, dimension( ldz, * ) Z, integer LDZ,
complex*16, dimension( * ) RHS, double precision RDSUM, double precision RDSCAL, integer,
dimension( * ) IPIV, integer, dimension( * ) JPIV)
ZLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a
contribution to the reciprocal Dif-estimate.
Purpose:
ZLATDF computes the contribution to the reciprocal Dif-estimate
by solving for x in Z * x = b, where b is chosen such that the norm
of x is as large as possible. It is assumed that LU decomposition
of Z has been computed by ZGETC2. On entry RHS = f holds the
contribution from earlier solved sub-systems, and on return RHS = x.
The factorization of Z returned by ZGETC2 has the form
Z = P * L * U * Q, where P and Q are permutation matrices. L is lower
triangular with unit diagonal elements and U is upper triangular.
Parameters
IJOB
IJOB is INTEGER
IJOB = 2: First compute an approximative null-vector e
of Z using ZGECON, e is normalized and solve for
Zx = +-e - f with the sign giving the greater value of
2-norm(x). About 5 times as expensive as Default.
IJOB .ne. 2: Local look ahead strategy where
all entries of the r.h.s. b is chosen as either +1 or
-1. Default.
N
N is INTEGER
The number of columns of the matrix Z.
Z
Z is COMPLEX*16 array, dimension (LDZ, N)
On entry, the LU part of the factorization of the n-by-n
matrix Z computed by ZGETC2: Z = P * L * U * Q
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDA >= max(1, N).
RHS
RHS is COMPLEX*16 array, dimension (N).
On entry, RHS contains contributions from other subsystems.
On exit, RHS contains the solution of the subsystem with
entries according to the value of IJOB (see above).
RDSUM
RDSUM is DOUBLE PRECISION
On entry, the sum of squares of computed contributions to
the Dif-estimate under computation by ZTGSYL, where the
scaling factor RDSCAL (see below) has been factored out.
On exit, the corresponding sum of squares updated with the
contributions from the current sub-system.
If TRANS = 'T' RDSUM is not touched.
NOTE: RDSUM only makes sense when ZTGSY2 is called by CTGSYL.
RDSCAL
RDSCAL is DOUBLE PRECISION
On entry, scaling factor used to prevent overflow in RDSUM.
On exit, RDSCAL is updated w.r.t. the current contributions
in RDSUM.
If TRANS = 'T', RDSCAL is not touched.
NOTE: RDSCAL only makes sense when ZTGSY2 is called by
ZTGSYL.
IPIV
IPIV is INTEGER array, dimension (N).
The pivot indices; for 1 <= i <= N, row i of the
matrix has been interchanged with row IPIV(i).
JPIV
JPIV is INTEGER array, dimension (N).
The pivot indices; for 1 <= j <= N, column j of the
matrix has been interchanged with column JPIV(j).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
This routine is a further developed implementation of algorithm BSOLVE in [1] using
complete pivoting in the LU factorization.
Contributors:
Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901
87 Umea, Sweden.
References:
[1] Bo Kagstrom and Lars Westin, Generalized Schur Methods with Condition Estimators
for Solving the Generalized Sylvester Equation, IEEE Transactions on Automatic
Control, Vol. 34, No. 7, July 1989, pp 745-751.
[2] Peter Poromaa, On Efficient and Robust Estimators for the Separation between two
Regular Matrix Pairs with Applications in Condition Estimation. Report UMINF-95.05,
Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
subroutine zlatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, complex*16, dimension( * ) AP, complex*16, dimension( * ) X, double precision SCALE,
double precision, dimension( * ) CNORM, integer INFO)
ZLATPS solves a triangular system of equations with the matrix held in packed storage.
Purpose:
ZLATPS solves one of the triangular systems
A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
with scaling to prevent overflow, where A is an upper or lower
triangular matrix stored in packed form. Here A**T denotes the
transpose of A, A**H denotes the conjugate transpose of A, x and b
are n-element vectors, and s is a scaling factor, usually less than
or equal to 1, chosen so that the components of x will be less than
the overflow threshold. If the unscaled problem will not cause
overflow, the Level 2 BLAS routine ZTPSV is called. If the matrix A
is singular (A(j,j) = 0 for some j), then s is set to 0 and a
non-trivial solution to A*x = 0 is returned.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T * x = s*b (Transpose)
= 'C': Solve A**H * x = s*b (Conjugate transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
AP
AP is COMPLEX*16 array, dimension (N*(N+1)/2)
The upper or lower triangular matrix A, packed columnwise in
a linear array. The j-th column of A is stored in the array
AP as follows:
if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
X
X is COMPLEX*16 array, dimension (N)
On entry, the right hand side b of the triangular system.
On exit, X is overwritten by the solution vector x.
SCALE
SCALE is DOUBLE PRECISION
The scaling factor s for the triangular system
A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
If SCALE = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
CNORM
CNORM is DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
A rough bound on x is computed; if that is less than overflow, ZTPSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b. The basic algorithm
if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTPSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the
columnwise method can be performed without fear of overflow. If
the computed bound is greater than a large constant, x is scaled to
prevent overflow, but if the bound overflows, x is set to 0, x(j) to
1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b or
A**H *x = b. The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call ZTPSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
subroutine zlatrd (character UPLO, integer N, integer NB, complex*16, dimension( lda, * ) A,
integer LDA, double precision, dimension( * ) E, complex*16, dimension( * ) TAU,
complex*16, dimension( ldw, * ) W, integer LDW)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
tridiagonal form by an unitary similarity transformation.
Purpose:
ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
Hermitian tridiagonal form by a unitary similarity
transformation Q**H * A * Q, and returns the matrices V and W which are
needed to apply the transformation to the unreduced part of A.
If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by ZHETRD.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the upper or lower triangular part of the
Hermitian matrix A is stored:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the matrix A.
NB
NB is INTEGER
The number of rows and columns to be reduced.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the Hermitian matrix A. If UPLO = 'U', the leading
n-by-n upper triangular part of A contains the upper
triangular part of the matrix A, and the strictly lower
triangular part of A is not referenced. If UPLO = 'L', the
leading n-by-n lower triangular part of A contains the lower
triangular part of the matrix A, and the strictly upper
triangular part of A is not referenced.
On exit:
if UPLO = 'U', the last NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements above the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors;
if UPLO = 'L', the first NB columns have been reduced to
tridiagonal form, with the diagonal elements overwriting
the diagonal elements of A; the elements below the diagonal
with the array TAU, represent the unitary matrix Q as a
product of elementary reflectors.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
E
E is DOUBLE PRECISION array, dimension (N-1)
If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
elements of the last NB columns of the reduced matrix;
if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
the first NB columns of the reduced matrix.
TAU
TAU is COMPLEX*16 array, dimension (N-1)
The scalar factors of the elementary reflectors, stored in
TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
See Further Details.
W
W is COMPLEX*16 array, dimension (LDW,NB)
The n-by-nb matrix W required to update the unreduced part
of A.
LDW
LDW is INTEGER
The leading dimension of the array W. LDW >= max(1,N).
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
If UPLO = 'U', the matrix Q is represented as a product of elementary
reflectors
Q = H(n) H(n-1) . . . H(n-nb+1).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
and tau in TAU(i-1).
If UPLO = 'L', the matrix Q is represented as a product of elementary
reflectors
Q = H(1) H(2) . . . H(nb).
Each H(i) has the form
H(i) = I - tau * v * v**H
where tau is a complex scalar, and v is a complex vector with
v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
and tau in TAU(i).
The elements of the vectors v together form the n-by-nb matrix V
which is needed, with W, to apply the transformation to the unreduced
part of the matrix, using a Hermitian rank-2k update of the form:
A := A - V*W**H - W*V**H.
The contents of A on exit are illustrated by the following examples
with n = 5 and nb = 2:
if UPLO = 'U': if UPLO = 'L':
( a a a v4 v5 ) ( d )
( a a v4 v5 ) ( 1 d )
( a 1 v5 ) ( v1 1 a )
( d 1 ) ( v1 v2 a a )
( d ) ( v1 v2 a a a )
where d denotes a diagonal element of the reduced matrix, a denotes
an element of the original matrix that is unchanged, and vi denotes
an element of the vector defining H(i).
subroutine zlatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) X, double
precision SCALE, double precision, dimension( * ) CNORM, integer INFO)
ZLATRS solves a triangular system of equations with the scale factor set to prevent
overflow.
Purpose:
ZLATRS solves one of the triangular systems
A * x = s*b, A**T * x = s*b, or A**H * x = s*b,
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A, A**H denotes the
conjugate transpose of A, x and b are n-element vectors, and s is a
scaling factor, usually less than or equal to 1, chosen so that the
components of x will be less than the overflow threshold. If the
unscaled problem will not cause overflow, the Level 2 BLAS routine
ZTRSV is called. If the matrix A is singular (A(j,j) = 0 for some j),
then s is set to 0 and a non-trivial solution to A*x = 0 is returned.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is CHARACTER*1
Specifies the operation applied to A.
= 'N': Solve A * x = s*b (No transpose)
= 'T': Solve A**T * x = s*b (Transpose)
= 'C': Solve A**H * x = s*b (Conjugate transpose)
DIAG
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Non-unit triangular
= 'U': Unit triangular
NORMIN
NORMIN is CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y': CNORM contains the column norms on entry
= 'N': CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
N
N is INTEGER
The order of the matrix A. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
The triangular matrix A. If UPLO = 'U', the leading n by n
upper triangular part of the array A contains the upper
triangular matrix, and the strictly lower triangular part of
A is not referenced. If UPLO = 'L', the leading n by n lower
triangular part of the array A contains the lower triangular
matrix, and the strictly upper triangular part of A is not
referenced. If DIAG = 'U', the diagonal elements of A are
also not referenced and are assumed to be 1.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max (1,N).
X
X is COMPLEX*16 array, dimension (N)
On entry, the right hand side b of the triangular system.
On exit, X is overwritten by the solution vector x.
SCALE
SCALE is DOUBLE PRECISION
The scaling factor s for the triangular system
A * x = s*b, A**T * x = s*b, or A**H * x = s*b.
If SCALE = 0, the matrix A is singular or badly scaled, and
the vector x is an exact or approximate solution to A*x = 0.
CNORM
CNORM is DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j)
contains the norm of the off-diagonal part of the j-th column
of A. If TRANS = 'N', CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
A rough bound on x is computed; if that is less than overflow, ZTRSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
A columnwise scheme is used for solving A*x = b. The basic algorithm
if A is lower triangular is
x[1:n] := b[1:n]
for j = 1, ..., n
x(j) := x(j) / A(j,j)
x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
end
Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
G(j) = bound on x[j+1:n]
Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
Then for iteration j+1 we have
M(j+1) <= G(j) / | A(j+1,j+1) |
G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal. Hence
G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
1<=i<=j
and
|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
Since |x(j)| <= M(j), we use the Level 2 BLAS routine ZTRSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
max(underflow, 1/overflow).
The bound on x(j) is also used to determine when a step in the
columnwise method can be performed without fear of overflow. If
the computed bound is greater than a large constant, x is scaled to
prevent overflow, but if the bound overflows, x is set to 0, x(j) to
1, and scale to 0, and a non-trivial solution to A*x = 0 is found.
Similarly, a row-wise scheme is used to solve A**T *x = b or
A**H *x = b. The basic algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we
add the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.
Then the bound on x(j) is
M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
subroutine zlauu2 (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer INFO)
ZLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (unblocked algorithm).
Purpose:
ZLAUU2 computes the product U * U**H or L**H * L, where the triangular
factor U or L is stored in the upper or lower triangular part of
the array A.
If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
overwriting the factor U in A.
If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
overwriting the factor L in A.
This is the unblocked form of the algorithm, calling Level 2 BLAS.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the triangular factor stored in the array A
is upper or lower triangular:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the triangular factor U or L. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the triangular factor U or L.
On exit, if UPLO = 'U', the upper triangle of A is
overwritten with the upper triangle of the product U * U**H;
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L**H * L.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zlauum (character UPLO, integer N, complex*16, dimension( lda, * ) A, integer LDA,
integer INFO)
ZLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (blocked algorithm).
Purpose:
ZLAUUM computes the product U * U**H or L**H * L, where the triangular
factor U or L is stored in the upper or lower triangular part of
the array A.
If UPLO = 'U' or 'u' then the upper triangle of the result is stored,
overwriting the factor U in A.
If UPLO = 'L' or 'l' then the lower triangle of the result is stored,
overwriting the factor L in A.
This is the blocked form of the algorithm, calling Level 3 BLAS.
Parameters
UPLO
UPLO is CHARACTER*1
Specifies whether the triangular factor stored in the array A
is upper or lower triangular:
= 'U': Upper triangular
= 'L': Lower triangular
N
N is INTEGER
The order of the triangular factor U or L. N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the triangular factor U or L.
On exit, if UPLO = 'U', the upper triangle of A is
overwritten with the upper triangle of the product U * U**H;
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L**H * L.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,N).
INFO
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -k, the k-th argument had an illegal value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zrot (integer N, complex*16, dimension( * ) CX, integer INCX, complex*16,
dimension( * ) CY, integer INCY, double precision C, complex*16 S)
ZROT applies a plane rotation with real cosine and complex sine to a pair of complex
vectors.
Purpose:
ZROT applies a plane rotation, where the cos (C) is real and the
sin (S) is complex, and the vectors CX and CY are complex.
Parameters
N
N is INTEGER
The number of elements in the vectors CX and CY.
CX
CX is COMPLEX*16 array, dimension (N)
On input, the vector X.
On output, CX is overwritten with C*X + S*Y.
INCX
INCX is INTEGER
The increment between successive values of CX. INCX <> 0.
CY
CY is COMPLEX*16 array, dimension (N)
On input, the vector Y.
On output, CY is overwritten with -CONJG(S)*X + C*Y.
INCY
INCY is INTEGER
The increment between successive values of CY. INCX <> 0.
C
C is DOUBLE PRECISION
S
S is COMPLEX*16
C and S define a rotation
[ C S ]
[ -conjg(S) C ]
where C*C + S*CONJG(S) = 1.0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zspmv (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( * ) AP,
complex*16, dimension( * ) X, integer INCX, complex*16 BETA, complex*16, dimension( * ) Y,
integer INCY)
ZSPMV computes a matrix-vector product for complex vectors using a complex symmetric
packed matrix
Purpose:
ZSPMV 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 matrix, supplied in packed form.
Parameters
UPLO
UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the matrix A is supplied in the packed
array AP as follows:
UPLO = 'U' or 'u' The upper triangular part of A is
supplied in AP.
UPLO = 'L' or 'l' The lower triangular part of A is
supplied in AP.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
AP
AP is COMPLEX*16 array, dimension at least
( ( N*( N + 1 ) )/2 ).
Before entry, with UPLO = 'U' or 'u', the array AP must
contain the upper triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
and a( 2, 2 ) respectively, and so on.
Before entry, with UPLO = 'L' or 'l', the array AP must
contain the lower triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
and a( 3, 1 ) respectively, and so on.
Unchanged on exit.
X
X is COMPLEX*16 array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the N-
element vector x.
Unchanged on exit.
INCX
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
BETA
BETA is COMPLEX*16
On entry, BETA specifies the scalar beta. When BETA is
supplied as zero then Y need not be set on input.
Unchanged on exit.
Y
Y is COMPLEX*16 array, dimension at least
( 1 + ( N - 1 )*abs( INCY ) ).
Before entry, the incremented array Y must contain the n
element vector y. On exit, Y is overwritten by the updated
vector y.
INCY
INCY is 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.
subroutine zspr (character UPLO, integer N, complex*16 ALPHA, complex*16, dimension( * ) X,
integer INCX, complex*16, dimension( * ) AP)
ZSPR performs the symmetrical rank-1 update of a complex symmetric packed matrix.
Purpose:
ZSPR performs the symmetric rank 1 operation
A := alpha*x*x**H + A,
where alpha is a complex scalar, x is an n element vector and A is an
n by n symmetric matrix, supplied in packed form.
Parameters
UPLO
UPLO is CHARACTER*1
On entry, UPLO specifies whether the upper or lower
triangular part of the matrix A is supplied in the packed
array AP as follows:
UPLO = 'U' or 'u' The upper triangular part of A is
supplied in AP.
UPLO = 'L' or 'l' The lower triangular part of A is
supplied in AP.
Unchanged on exit.
N
N is INTEGER
On entry, N specifies the order of the matrix A.
N must be at least zero.
Unchanged on exit.
ALPHA
ALPHA is COMPLEX*16
On entry, ALPHA specifies the scalar alpha.
Unchanged on exit.
X
X is COMPLEX*16 array, dimension at least
( 1 + ( N - 1 )*abs( INCX ) ).
Before entry, the incremented array X must contain the N-
element vector x.
Unchanged on exit.
INCX
INCX is INTEGER
On entry, INCX specifies the increment for the elements of
X. INCX must not be zero.
Unchanged on exit.
AP
AP is COMPLEX*16 array, dimension at least
( ( N*( N + 1 ) )/2 ).
Before entry, with UPLO = 'U' or 'u', the array AP must
contain the upper triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 1, 2 )
and a( 2, 2 ) respectively, and so on. On exit, the array
AP is overwritten by the upper triangular part of the
updated matrix.
Before entry, with UPLO = 'L' or 'l', the array AP must
contain the lower triangular part of the symmetric matrix
packed sequentially, column by column, so that AP( 1 )
contains a( 1, 1 ), AP( 2 ) and AP( 3 ) contain a( 2, 1 )
and a( 3, 1 ) respectively, and so on. On exit, the array
AP is overwritten by the lower triangular part of the
updated matrix.
Note that the imaginary parts of the diagonal elements need
not be set, they are assumed to be zero, and on exit they
are set to zero.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine ztprfb (character SIDE, character TRANS, character DIRECT, character STOREV,
integer M, integer N, integer K, integer L, complex*16, dimension( ldv, * ) V, integer
LDV, complex*16, dimension( ldt, * ) T, integer LDT, complex*16, dimension( lda, * ) A,
integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, dimension(
ldwork, * ) WORK, integer LDWORK)
ZTPRFB applies a complex 'triangular-pentagonal' block reflector to a complex matrix,
which is composed of two blocks.
Purpose:
ZTPRFB applies a complex 'triangular-pentagonal' block reflector H or its
conjugate transpose H**H to a complex matrix C, which is composed of two
blocks A and B, either from the left or right.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply H or H**H from the Left
= 'R': apply H or H**H from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'C': apply H**H (Conjugate transpose)
DIRECT
DIRECT is CHARACTER*1
Indicates how H is formed from a product of elementary
reflectors
= 'F': H = H(1) H(2) . . . H(k) (Forward)
= 'B': H = H(k) . . . H(2) H(1) (Backward)
STOREV
STOREV is CHARACTER*1
Indicates how the vectors which define the elementary
reflectors are stored:
= 'C': Columns
= 'R': Rows
M
M is INTEGER
The number of rows of the matrix B.
M >= 0.
N
N is INTEGER
The number of columns of the matrix B.
N >= 0.
K
K is INTEGER
The order of the matrix T, i.e. the number of elementary
reflectors whose product defines the block reflector.
K >= 0.
L
L is INTEGER
The order of the trapezoidal part of V.
K >= L >= 0. See Further Details.
V
V is COMPLEX*16 array, dimension
(LDV,K) if STOREV = 'C'
(LDV,M) if STOREV = 'R' and SIDE = 'L'
(LDV,N) if STOREV = 'R' and SIDE = 'R'
The pentagonal matrix V, which contains the elementary reflectors
H(1), H(2), ..., H(K). See Further Details.
LDV
LDV is INTEGER
The leading dimension of the array V.
If STOREV = 'C' and SIDE = 'L', LDV >= max(1,M);
if STOREV = 'C' and SIDE = 'R', LDV >= max(1,N);
if STOREV = 'R', LDV >= K.
T
T is COMPLEX*16 array, dimension (LDT,K)
The triangular K-by-K matrix T in the representation of the
block reflector.
LDT
LDT is INTEGER
The leading dimension of the array T.
LDT >= K.
A
A is COMPLEX*16 array, dimension
(LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R'
On entry, the K-by-N or M-by-K matrix A.
On exit, A is overwritten by the corresponding block of
H*C or H**H*C or C*H or C*H**H. See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDA >= max(1,K);
If SIDE = 'R', LDA >= max(1,M).
B
B is COMPLEX*16 array, dimension (LDB,N)
On entry, the M-by-N matrix B.
On exit, B is overwritten by the corresponding block of
H*C or H**H*C or C*H or C*H**H. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).
WORK
WORK is COMPLEX*16 array, dimension
(LDWORK,N) if SIDE = 'L',
(LDWORK,K) if SIDE = 'R'.
LDWORK
LDWORK is INTEGER
The leading dimension of the array WORK.
If SIDE = 'L', LDWORK >= K;
if SIDE = 'R', LDWORK >= M.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
The matrix C is a composite matrix formed from blocks A and B.
The block B is of size M-by-N; if SIDE = 'R', A is of size M-by-K,
and if SIDE = 'L', A is of size K-by-N.
If SIDE = 'R' and DIRECT = 'F', C = [A B].
If SIDE = 'L' and DIRECT = 'F', C = [A]
[B].
If SIDE = 'R' and DIRECT = 'B', C = [B A].
If SIDE = 'L' and DIRECT = 'B', C = [B]
[A].
The pentagonal matrix V is composed of a rectangular block V1 and a
trapezoidal block V2. The size of the trapezoidal block is determined by
the parameter L, where 0<=L<=K. If L=K, the V2 block of V is triangular;
if L=0, there is no trapezoidal block, thus V = V1 is rectangular.
If DIRECT = 'F' and STOREV = 'C': V = [V1]
[V2]
- V2 is upper trapezoidal (first L rows of K-by-K upper triangular)
If DIRECT = 'F' and STOREV = 'R': V = [V1 V2]
- V2 is lower trapezoidal (first L columns of K-by-K lower triangular)
If DIRECT = 'B' and STOREV = 'C': V = [V2]
[V1]
- V2 is lower trapezoidal (last L rows of K-by-K lower triangular)
If DIRECT = 'B' and STOREV = 'R': V = [V2 V1]
- V2 is upper trapezoidal (last L columns of K-by-K upper triangular)
If STOREV = 'C' and SIDE = 'L', V is M-by-K with V2 L-by-K.
If STOREV = 'C' and SIDE = 'R', V is N-by-K with V2 L-by-K.
If STOREV = 'R' and SIDE = 'L', V is K-by-M with V2 K-by-L.
If STOREV = 'R' and SIDE = 'R', V is K-by-N with V2 K-by-L.
Author
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