Provided by: liblapack-doc_3.11.0-2build1_all 
NAME
doubleOTHERauxiliary - double
SYNOPSIS
Functions
subroutine clatrs3 (UPLO, TRANS, DIAG, NORMIN, N, NRHS, A, LDA, X, LDX, SCALE, CNORM,
WORK, LWORK, INFO)
CLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
subroutine dlabrd (M, N, NB, A, LDA, D, E, TAUQ, TAUP, X, LDX, Y, LDY)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
subroutine dlacn2 (N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for
evaluating matrix-vector products.
subroutine dlacon (N, V, X, ISGN, EST, KASE)
DLACON estimates the 1-norm of a square matrix, using reverse communication for
evaluating matrix-vector products.
subroutine dladiv (A, B, C, D, P, Q)
DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
subroutine dladiv1 (A, B, C, D, P, Q)
double precision function dladiv2 (A, B, C, D, R, T)
subroutine dlaein (RIGHTV, NOINIT, N, H, LDH, WR, WI, VR, VI, B, LDB, WORK, EPS3, SMLNUM,
BIGNUM, INFO)
DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
inverse iteration.
subroutine dlaexc (WANTQ, N, T, LDT, Q, LDQ, J1, N1, N2, WORK, INFO)
DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur
canonical form, by an orthogonal similarity transformation.
subroutine dlag2 (A, LDA, B, LDB, SAFMIN, SCALE1, SCALE2, WR1, WR2, WI)
DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with
scaling as necessary to avoid over-/underflow.
subroutine dlag2s (M, N, A, LDA, SA, LDSA, INFO)
DLAG2S converts a double precision matrix to a single precision matrix.
subroutine dlags2 (UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU, CSV, SNV, CSQ, SNQ)
DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A
and B such that the rows of the transformed A and B are parallel.
subroutine dlagtm (TRANS, N, NRHS, ALPHA, DL, D, DU, X, LDX, BETA, B, LDB)
DLAGTM 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 dlagv2 (A, LDA, B, LDB, ALPHAR, ALPHAI, BETA, CSL, SNL, CSR, SNR)
DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil
(A,B) where B is upper triangular.
subroutine dlahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)
DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
using the double-shift/single-shift QR algorithm.
subroutine dlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
DLAHR2 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 dlaic1 (JOB, J, X, SEST, W, GAMMA, SESTPR, S, C)
DLAIC1 applies one step of incremental condition estimation.
subroutine dlaln2 (LTRANS, NA, NW, SMIN, CA, A, LDA, D1, D2, B, LDB, WR, WI, X, LDX,
SCALE, XNORM, INFO)
DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
double precision function dlangt (NORM, N, DL, D, DU)
DLANGT 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 dlanhs (NORM, N, A, LDA, WORK)
DLANHS 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 dlansb (NORM, UPLO, N, K, AB, LDAB, WORK)
DLANSB 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 dlansp (NORM, UPLO, N, AP, WORK)
DLANSP 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 dlantb (NORM, UPLO, DIAG, N, K, AB, LDAB, WORK)
DLANTB 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 dlantp (NORM, UPLO, DIAG, N, AP, WORK)
DLANTP 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 dlantr (NORM, UPLO, DIAG, M, N, A, LDA, WORK)
DLANTR 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 dlanv2 (A, B, C, D, RT1R, RT1I, RT2R, RT2I, CS, SN)
DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in
standard form.
subroutine dlapll (N, X, INCX, Y, INCY, SSMIN)
DLAPLL measures the linear dependence of two vectors.
subroutine dlapmr (FORWRD, M, N, X, LDX, K)
DLAPMR rearranges rows of a matrix as specified by a permutation vector.
subroutine dlapmt (FORWRD, M, N, X, LDX, K)
DLAPMT performs a forward or backward permutation of the columns of a matrix.
subroutine dlaqp2 (M, N, OFFSET, A, LDA, JPVT, TAU, VN1, VN2, WORK)
DLAQP2 computes a QR factorization with column pivoting of the matrix block.
subroutine dlaqps (M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1, VN2, AUXV, F, LDF)
DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n
matrix A by using BLAS level 3.
subroutine dlaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
LWORK, INFO)
DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
from the Schur decomposition.
subroutine dlaqr1 (N, H, LDH, SR1, SI1, SR2, SI2, V)
DLAQR1 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 dlaqr2 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to
detect and deflate fully converged eigenvalues from a trailing principal submatrix
(aggressive early deflation).
subroutine dlaqr3 (WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND,
SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)
DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to
detect and deflate fully converged eigenvalues from a trailing principal submatrix
(aggressive early deflation).
subroutine dlaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
LWORK, INFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
from the Schur decomposition.
subroutine dlaqr5 (WANTT, WANTZ, KACC22, N, KTOP, KBOT, NSHFTS, SR, SI, H, LDH, ILOZ,
IHIZ, Z, LDZ, V, LDV, U, LDU, NV, WV, LDWV, NH, WH, LDWH)
DLAQR5 performs a single small-bulge multi-shift QR sweep.
subroutine dlaqsb (UPLO, N, KD, AB, LDAB, S, SCOND, AMAX, EQUED)
DLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by
spbequ.
subroutine dlaqsp (UPLO, N, AP, S, SCOND, AMAX, EQUED)
DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
computed by sppequ.
subroutine dlaqtr (LTRAN, LREAL, N, T, LDT, B, W, SCALE, X, WORK, INFO)
DLAQTR solves a real quasi-triangular system of equations, or a complex quasi-
triangular system of special form, in real arithmetic.
subroutine dlar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT,
ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
through bn of the tridiagonal matrix LDLT - λI.
subroutine dlar2v (N, X, Y, Z, INCX, C, S, INCC)
DLAR2V applies a vector of plane rotations with real cosines and real sines from both
sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
subroutine dlarf (SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
subroutine dlarfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK,
LDWORK)
DLARFB applies a block reflector or its transpose to a general rectangular matrix.
subroutine dlarfb_gett (IDENT, M, N, K, T, LDT, A, LDA, B, LDB, WORK, LDWORK)
DLARFB_GETT
subroutine dlarfg (N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
subroutine dlarfgp (N, ALPHA, X, INCX, TAU)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
subroutine dlarft (DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
DLARFT forms the triangular factor T of a block reflector H = I - vtvH
subroutine dlarfx (SIDE, M, N, V, TAU, C, LDC, WORK)
DLARFX applies an elementary reflector to a general rectangular matrix, with loop
unrolling when the reflector has order ≤ 10.
subroutine dlarfy (UPLO, N, V, INCV, TAU, C, LDC, WORK)
DLARFY
subroutine dlargv (N, X, INCX, Y, INCY, C, INCC)
DLARGV generates a vector of plane rotations with real cosines and real sines.
subroutine dlarrv (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)
DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and
the eigenvalues of L D LT.
subroutine dlartv (N, X, INCX, Y, INCY, C, S, INCC)
DLARTV applies a vector of plane rotations with real cosines and real sines to the
elements of a pair of vectors.
subroutine dlaswp (N, A, LDA, K1, K2, IPIV, INCX)
DLASWP performs a series of row interchanges on a general rectangular matrix.
subroutine dlat2s (UPLO, N, A, LDA, SA, LDSA, INFO)
DLAT2S converts a double-precision triangular matrix to a single-precision triangular
matrix.
subroutine dlatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
DLATBS solves a triangular banded system of equations.
subroutine dlatdf (IJOB, N, Z, LDZ, RHS, RDSUM, RDSCAL, IPIV, JPIV)
DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes
a contribution to the reciprocal Dif-estimate.
subroutine dlatps (UPLO, TRANS, DIAG, NORMIN, N, AP, X, SCALE, CNORM, INFO)
DLATPS solves a triangular system of equations with the matrix held in packed storage.
subroutine dlatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
tridiagonal form by an orthogonal similarity transformation.
subroutine dlatrs (UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM, INFO)
DLATRS solves a triangular system of equations with the scale factor set to prevent
overflow.
subroutine dlatrs3 (UPLO, TRANS, DIAG, NORMIN, N, NRHS, A, LDA, X, LDX, SCALE, CNORM,
WORK, LWORK, INFO)
DLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
subroutine dlauu2 (UPLO, N, A, LDA, INFO)
DLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (unblocked algorithm).
subroutine dlauum (UPLO, N, A, LDA, INFO)
DLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (blocked algorithm).
subroutine drscl (N, SA, SX, INCX)
DRSCL multiplies a vector by the reciprocal of a real scalar.
subroutine dtprfb (SIDE, TRANS, DIRECT, STOREV, M, N, K, L, V, LDV, T, LDT, A, LDA, B,
LDB, WORK, LDWORK)
DTPRFB applies a real 'triangular-pentagonal' block reflector to a real matrix, which
is composed of two blocks.
subroutine slatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
tridiagonal form by an orthogonal similarity transformation.
subroutine slatrs3 (UPLO, TRANS, DIAG, NORMIN, N, NRHS, A, LDA, X, LDX, SCALE, CNORM,
WORK, LWORK, INFO)
SLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
subroutine zlatrs3 (UPLO, TRANS, DIAG, NORMIN, N, NRHS, A, LDA, X, LDX, SCALE, CNORM,
WORK, LWORK, INFO)
ZLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
Detailed Description
This is the group of double other auxiliary routines
Function Documentation
subroutine clatrs3 (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, integer NRHS, complex, dimension( lda, * ) A, integer LDA, complex, dimension( ldx, * )
X, integer LDX, real, dimension( * ) SCALE, real, dimension( * ) CNORM, real, dimension( *
) WORK, integer LWORK, integer INFO)
CLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
Purpose:
CLATRS3 solves one of the triangular systems
A * X = B * diag(scale), A**T * X = B * diag(scale), or
A**H * X = B * diag(scale)
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-by-nrhs matrices and scale
is an nrhs-element vector of scaling factors. A scaling factor scale(j)
is usually less than or equal to 1, chosen such that X(:,j) is less
than the overflow threshold. If the matrix A is singular (A(j,j) = 0
for some j), then a non-trivial solution to A*X = 0 is returned. If
the system is so badly scaled that the solution cannot be represented
as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
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**T* 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.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is COMPLEX 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 array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is REAL array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
CNORM
CNORM is REAL 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.
WORK
WORK is REAL array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N
+ NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal
dimensions of the WORK array, returns this value as the first entry of the WORK array, and
no error message related to LWORK is issued by XERBLA.
Parameters
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:
subroutine dlabrd (integer M, integer N, integer NB, double precision, dimension( lda, * ) A,
integer LDA, double precision, dimension( * ) D, double precision, dimension( * ) E,
double precision, dimension( * ) TAUQ, double precision, dimension( * ) TAUP, double
precision, dimension( ldx, * ) X, integer LDX, double precision, dimension( ldy, * ) Y,
integer LDY)
DLABRD reduces the first nb rows and columns of a general matrix to a bidiagonal form.
Purpose:
DLABRD reduces the first NB rows and columns of a real general
m by n matrix A to upper or lower bidiagonal form by an orthogonal
transformation Q**T * 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 DGEBRD
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 DOUBLE PRECISION 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 orthogonal
matrix Q as a product of elementary reflectors; and
elements above the diagonal in the first NB rows, with the
array TAUP, represent the orthogonal 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 orthogonal
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 orthogonal 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 DOUBLE PRECISION array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the orthogonal matrix Q. See Further Details.
TAUP
TAUP is DOUBLE PRECISION array, dimension (NB)
The scalar factors of the elementary reflectors which
represent the orthogonal matrix P. See Further Details.
X
X is DOUBLE PRECISION 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 DOUBLE PRECISION 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**T and G(i) = I - taup * u * u**T
where tauq and taup are real scalars, and v and u are real 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**T 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**T - X*U**T.
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 dlacn2 (integer N, double precision, dimension( * ) V, double precision, dimension(
* ) X, integer, dimension( * ) ISGN, double precision EST, integer KASE, integer,
dimension( 3 ) ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating
matrix-vector products.
Purpose:
DLACN2 estimates the 1-norm of a square, real 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 DOUBLE PRECISION array, dimension (N)
On the final return, V = A*W, where EST = norm(V)/norm(W)
(W is not returned).
X
X is DOUBLE PRECISION array, dimension (N)
On an intermediate return, X should be overwritten by
A * X, if KASE=1,
A**T * X, if KASE=2,
and DLACN2 must be re-called with all the other parameters
unchanged.
ISGN
ISGN is INTEGER array, dimension (N)
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 DLACN2.
On exit, EST is an estimate (a lower bound) for norm(A).
KASE
KASE is INTEGER
On the initial call to DLACN2, 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**T * X.
On the final return from DLACN2, KASE will again be 0.
ISAVE
ISAVE is INTEGER array, dimension (3)
ISAVE is used to save variables between calls to DLACN2
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Originally named SONEST, dated March 16, 1988.
This is a thread safe version of DLACON, which uses the array ISAVE
in place of a SAVE statement, as follows:
DLACON DLACN2
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 dlacon (integer N, double precision, dimension( * ) V, double precision, dimension(
* ) X, integer, dimension( * ) ISGN, double precision EST, integer KASE)
DLACON estimates the 1-norm of a square matrix, using reverse communication for evaluating
matrix-vector products.
Purpose:
DLACON estimates the 1-norm of a square, real 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 DOUBLE PRECISION array, dimension (N)
On the final return, V = A*W, where EST = norm(V)/norm(W)
(W is not returned).
X
X is DOUBLE PRECISION array, dimension (N)
On an intermediate return, X should be overwritten by
A * X, if KASE=1,
A**T * X, if KASE=2,
and DLACON must be re-called with all the other parameters
unchanged.
ISGN
ISGN is INTEGER array, dimension (N)
EST
EST is DOUBLE PRECISION
On entry with KASE = 1 or 2 and JUMP = 3, EST should be
unchanged from the previous call to DLACON.
On exit, EST is an estimate (a lower bound) for norm(A).
KASE
KASE is INTEGER
On the initial call to DLACON, 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**T * X.
On the final return from DLACON, KASE will again be 0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Nick Higham, University of Manchester.
Originally named SONEST, dated March 16, 1988.
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 dladiv (double precision A, double precision B, double precision C, double
precision D, double precision P, double precision Q)
DLADIV performs complex division in real arithmetic, avoiding unnecessary overflow.
Purpose:
DLADIV performs complex division in real arithmetic
a + i*b
p + i*q = ---------
c + i*d
The algorithm is due to Michael Baudin and Robert L. Smith
and can be found in the paper
'A Robust Complex Division in Scilab'
Parameters
A
A is DOUBLE PRECISION
B
B is DOUBLE PRECISION
C
C is DOUBLE PRECISION
D
D is DOUBLE PRECISION
The scalars a, b, c, and d in the above expression.
P
P is DOUBLE PRECISION
Q
Q is DOUBLE PRECISION
The scalars p and q in the above expression.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlaein (logical RIGHTV, logical NOINIT, integer N, double precision, dimension(
ldh, * ) H, integer LDH, double precision WR, double precision WI, double precision,
dimension( * ) VR, double precision, dimension( * ) VI, double precision, dimension( ldb,
* ) B, integer LDB, double precision, dimension( * ) WORK, double precision EPS3, double
precision SMLNUM, double precision BIGNUM, integer INFO)
DLAEIN computes a specified right or left eigenvector of an upper Hessenberg matrix by
inverse iteration.
Purpose:
DLAEIN uses inverse iteration to find a right or left eigenvector
corresponding to the eigenvalue (WR,WI) of a real 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 (VR,VI).
= .FALSE.: initial vector supplied in (VR,VI).
N
N is INTEGER
The order of the matrix H. N >= 0.
H
H is DOUBLE PRECISION array, dimension (LDH,N)
The upper Hessenberg matrix H.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
WR
WR is DOUBLE PRECISION
WI
WI is DOUBLE PRECISION
The real and imaginary parts of the eigenvalue of H whose
corresponding right or left eigenvector is to be computed.
VR
VR is DOUBLE PRECISION array, dimension (N)
VI
VI is DOUBLE PRECISION array, dimension (N)
On entry, if NOINIT = .FALSE. and WI = 0.0, VR must contain
a real starting vector for inverse iteration using the real
eigenvalue WR; if NOINIT = .FALSE. and WI.ne.0.0, VR and VI
must contain the real and imaginary parts of a complex
starting vector for inverse iteration using the complex
eigenvalue (WR,WI); otherwise VR and VI need not be set.
On exit, if WI = 0.0 (real eigenvalue), VR contains the
computed real eigenvector; if WI.ne.0.0 (complex eigenvalue),
VR and VI contain the real and imaginary parts of the
computed complex eigenvector. The eigenvector is 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|.
VI is not referenced if WI = 0.0.
B
B is DOUBLE PRECISION array, dimension (LDB,N)
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= N+1.
WORK
WORK 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.
BIGNUM
BIGNUM is DOUBLE PRECISION
A machine-dependent value close to the overflow threshold.
INFO
INFO is INTEGER
= 0: successful exit
= 1: inverse iteration did not converge; VR is set to the
last iterate, and so is VI if WI.ne.0.0.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlaexc (logical WANTQ, integer N, double precision, dimension( ldt, * ) T, integer
LDT, double precision, dimension( ldq, * ) Q, integer LDQ, integer J1, integer N1, integer
N2, double precision, dimension( * ) WORK, integer INFO)
DLAEXC swaps adjacent diagonal blocks of a real upper quasi-triangular matrix in Schur
canonical form, by an orthogonal similarity transformation.
Purpose:
DLAEXC swaps adjacent diagonal blocks T11 and T22 of order 1 or 2 in
an upper quasi-triangular matrix T by an orthogonal similarity
transformation.
T must be in Schur canonical form, that is, block upper triangular
with 1-by-1 and 2-by-2 diagonal blocks; each 2-by-2 diagonal block
has its diagonal elements equal and its off-diagonal elements of
opposite sign.
Parameters
WANTQ
WANTQ is LOGICAL
= .TRUE. : accumulate the transformation in the matrix Q;
= .FALSE.: do not accumulate the transformation.
N
N is INTEGER
The order of the matrix T. N >= 0.
T
T is DOUBLE PRECISION array, dimension (LDT,N)
On entry, the upper quasi-triangular matrix T, in Schur
canonical form.
On exit, the updated matrix T, again in Schur canonical form.
LDT
LDT is INTEGER
The leading dimension of the array T. LDT >= max(1,N).
Q
Q is DOUBLE PRECISION array, dimension (LDQ,N)
On entry, if WANTQ is .TRUE., the orthogonal matrix Q.
On exit, if WANTQ is .TRUE., the updated matrix Q.
If WANTQ is .FALSE., Q is not referenced.
LDQ
LDQ is INTEGER
The leading dimension of the array Q.
LDQ >= 1; and if WANTQ is .TRUE., LDQ >= N.
J1
J1 is INTEGER
The index of the first row of the first block T11.
N1
N1 is INTEGER
The order of the first block T11. N1 = 0, 1 or 2.
N2
N2 is INTEGER
The order of the second block T22. N2 = 0, 1 or 2.
WORK
WORK is DOUBLE PRECISION array, dimension (N)
INFO
INFO is INTEGER
= 0: successful exit
= 1: the transformed matrix T would be too far from Schur
form; the blocks are not swapped and T and Q are
unchanged.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlag2 (double precision, dimension( lda, * ) A, integer LDA, double precision,
dimension( ldb, * ) B, integer LDB, double precision SAFMIN, double precision SCALE1,
double precision SCALE2, double precision WR1, double precision WR2, double precision WI)
DLAG2 computes the eigenvalues of a 2-by-2 generalized eigenvalue problem, with scaling as
necessary to avoid over-/underflow.
Purpose:
DLAG2 computes the eigenvalues of a 2 x 2 generalized eigenvalue
problem A - w B, with scaling as necessary to avoid over-/underflow.
The scaling factor 's' results in a modified eigenvalue equation
s A - w B
where s is a non-negative scaling factor chosen so that w, w B,
and s A do not overflow and, if possible, do not underflow, either.
Parameters
A
A is DOUBLE PRECISION array, dimension (LDA, 2)
On entry, the 2 x 2 matrix A. It is assumed that its 1-norm
is less than 1/SAFMIN. Entries less than
sqrt(SAFMIN)*norm(A) are subject to being treated as zero.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= 2.
B
B is DOUBLE PRECISION array, dimension (LDB, 2)
On entry, the 2 x 2 upper triangular matrix B. It is
assumed that the one-norm of B is less than 1/SAFMIN. The
diagonals should be at least sqrt(SAFMIN) times the largest
element of B (in absolute value); if a diagonal is smaller
than that, then +/- sqrt(SAFMIN) will be used instead of
that diagonal.
LDB
LDB is INTEGER
The leading dimension of the array B. LDB >= 2.
SAFMIN
SAFMIN is DOUBLE PRECISION
The smallest positive number s.t. 1/SAFMIN does not
overflow. (This should always be DLAMCH('S') -- it is an
argument in order to avoid having to call DLAMCH frequently.)
SCALE1
SCALE1 is DOUBLE PRECISION
A scaling factor used to avoid over-/underflow in the
eigenvalue equation which defines the first eigenvalue. If
the eigenvalues are complex, then the eigenvalues are
( WR1 +/- WI i ) / SCALE1 (which may lie outside the
exponent range of the machine), SCALE1=SCALE2, and SCALE1
will always be positive. If the eigenvalues are real, then
the first (real) eigenvalue is WR1 / SCALE1 , but this may
overflow or underflow, and in fact, SCALE1 may be zero or
less than the underflow threshold if the exact eigenvalue
is sufficiently large.
SCALE2
SCALE2 is DOUBLE PRECISION
A scaling factor used to avoid over-/underflow in the
eigenvalue equation which defines the second eigenvalue. If
the eigenvalues are complex, then SCALE2=SCALE1. If the
eigenvalues are real, then the second (real) eigenvalue is
WR2 / SCALE2 , but this may overflow or underflow, and in
fact, SCALE2 may be zero or less than the underflow
threshold if the exact eigenvalue is sufficiently large.
WR1
WR1 is DOUBLE PRECISION
If the eigenvalue is real, then WR1 is SCALE1 times the
eigenvalue closest to the (2,2) element of A B**(-1). If the
eigenvalue is complex, then WR1=WR2 is SCALE1 times the real
part of the eigenvalues.
WR2
WR2 is DOUBLE PRECISION
If the eigenvalue is real, then WR2 is SCALE2 times the
other eigenvalue. If the eigenvalue is complex, then
WR1=WR2 is SCALE1 times the real part of the eigenvalues.
WI
WI is DOUBLE PRECISION
If the eigenvalue is real, then WI is zero. If the
eigenvalue is complex, then WI is SCALE1 times the imaginary
part of the eigenvalues. WI will always be non-negative.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlag2s (integer M, integer N, double precision, dimension( lda, * ) A, integer LDA,
real, dimension( ldsa, * ) SA, integer LDSA, integer INFO)
DLAG2S converts a double precision matrix to a single precision matrix.
Purpose:
DLAG2S converts a DOUBLE PRECISION matrix, A, to a SINGLE
PRECISION matrix, SA.
RMAX is the overflow for the SINGLE PRECISION arithmetic
DLAG2S 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 DOUBLE PRECISION 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 REAL 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 dlags2 (logical UPPER, double precision A1, double precision A2, double precision
A3, double precision B1, double precision B2, double precision B3, double precision CSU,
double precision SNU, double precision CSV, double precision SNV, double precision CSQ,
double precision SNQ)
DLAGS2 computes 2-by-2 orthogonal matrices U, V, and Q, and applies them to matrices A and
B such that the rows of the transformed A and B are parallel.
Purpose:
DLAGS2 computes 2-by-2 orthogonal matrices U, V and Q, such
that if ( UPPER ) then
U**T *A*Q = U**T *( A1 A2 )*Q = ( x 0 )
( 0 A3 ) ( x x )
and
V**T*B*Q = V**T *( B1 B2 )*Q = ( x 0 )
( 0 B3 ) ( x x )
or if ( .NOT.UPPER ) then
U**T *A*Q = U**T *( A1 0 )*Q = ( x x )
( A2 A3 ) ( 0 x )
and
V**T*B*Q = V**T*( B1 0 )*Q = ( x x )
( B2 B3 ) ( 0 x )
The rows of the transformed A and B are parallel, where
U = ( CSU SNU ), V = ( CSV SNV ), Q = ( CSQ SNQ )
( -SNU CSU ) ( -SNV CSV ) ( -SNQ CSQ )
Z**T denotes the transpose of Z.
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 DOUBLE PRECISION
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 DOUBLE PRECISION
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 DOUBLE PRECISION
The desired orthogonal matrix U.
CSV
CSV is DOUBLE PRECISION
SNV
SNV is DOUBLE PRECISION
The desired orthogonal matrix V.
CSQ
CSQ is DOUBLE PRECISION
SNQ
SNQ is DOUBLE PRECISION
The desired orthogonal matrix Q.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlagtm (character TRANS, integer N, integer NRHS, double precision ALPHA, double
precision, dimension( * ) DL, double precision, dimension( * ) D, double precision,
dimension( * ) DU, double precision, dimension( ldx, * ) X, integer LDX, double precision
BETA, double precision, dimension( ldb, * ) B, integer LDB)
DLAGTM 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:
DLAGTM 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'* X + beta * B
= 'C': Conjugate transpose = Transpose
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 DOUBLE PRECISION array, dimension (N-1)
The (n-1) sub-diagonal elements of T.
D
D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of T.
DU
DU is DOUBLE PRECISION array, dimension (N-1)
The (n-1) super-diagonal elements of T.
X
X is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlagv2 (double precision, dimension( lda, * ) A, integer LDA, double precision,
dimension( ldb, * ) B, integer LDB, double precision, dimension( 2 ) ALPHAR, double
precision, dimension( 2 ) ALPHAI, double precision, dimension( 2 ) BETA, double precision
CSL, double precision SNL, double precision CSR, double precision SNR)
DLAGV2 computes the Generalized Schur factorization of a real 2-by-2 matrix pencil (A,B)
where B is upper triangular.
Purpose:
DLAGV2 computes the Generalized Schur factorization of a real 2-by-2
matrix pencil (A,B) where B is upper triangular. This routine
computes orthogonal (rotation) matrices given by CSL, SNL and CSR,
SNR such that
1) if the pencil (A,B) has two real eigenvalues (include 0/0 or 1/0
types), then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ 0 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 b12 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
[ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ],
2) if the pencil (A,B) has a pair of complex conjugate eigenvalues,
then
[ a11 a12 ] := [ CSL SNL ] [ a11 a12 ] [ CSR -SNR ]
[ a21 a22 ] [ -SNL CSL ] [ a21 a22 ] [ SNR CSR ]
[ b11 0 ] := [ CSL SNL ] [ b11 b12 ] [ CSR -SNR ]
[ 0 b22 ] [ -SNL CSL ] [ 0 b22 ] [ SNR CSR ]
where b11 >= b22 > 0.
Parameters
A
A is DOUBLE PRECISION array, dimension (LDA, 2)
On entry, the 2 x 2 matrix A.
On exit, A is overwritten by the ``A-part'' of the
generalized Schur form.
LDA
LDA is INTEGER
THe leading dimension of the array A. LDA >= 2.
B
B is DOUBLE PRECISION array, dimension (LDB, 2)
On entry, the upper triangular 2 x 2 matrix B.
On exit, B is overwritten by the ``B-part'' of the
generalized Schur form.
LDB
LDB is INTEGER
THe leading dimension of the array B. LDB >= 2.
ALPHAR
ALPHAR is DOUBLE PRECISION array, dimension (2)
ALPHAI
ALPHAI is DOUBLE PRECISION array, dimension (2)
BETA
BETA is DOUBLE PRECISION array, dimension (2)
(ALPHAR(k)+i*ALPHAI(k))/BETA(k) are the eigenvalues of the
pencil (A,B), k=1,2, i = sqrt(-1). Note that BETA(k) may
be zero.
CSL
CSL is DOUBLE PRECISION
The cosine of the left rotation matrix.
SNL
SNL is DOUBLE PRECISION
The sine of the left rotation matrix.
CSR
CSR is DOUBLE PRECISION
The cosine of the right rotation matrix.
SNR
SNR is DOUBLE PRECISION
The sine of the right rotation matrix.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
subroutine dlahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double
precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double
precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension(
ldz, * ) Z, integer LDZ, integer INFO)
DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
using the double-shift/single-shift QR algorithm.
Purpose:
DLAHQR is an auxiliary routine called by DHSEQR to update the
eigenvalues and Schur decomposition already computed by DHSEQR, 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 quasi-triangular in
rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
ILO = 1). DLAHQR 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 DOUBLE PRECISION array, dimension (LDH,N)
On entry, the upper Hessenberg matrix H.
On exit, if INFO is zero and if WANTT is .TRUE., H is upper
quasi-triangular in rows and columns ILO:IHI, with any
2-by-2 diagonal blocks in standard form. If INFO is zero
and WANTT is .FALSE., the contents of H are unspecified on
exit. The output state of H if INFO is nonzero is given
below under the description of INFO.
LDH
LDH is INTEGER
The leading dimension of the array H. LDH >= max(1,N).
WR
WR is DOUBLE PRECISION array, dimension (N)
WI
WI is DOUBLE PRECISION array, dimension (N)
The real and imaginary parts, respectively, of the computed
eigenvalues ILO to IHI are stored in the corresponding
elements of WR and WI. If two eigenvalues are computed as a
complex conjugate pair, they are stored in consecutive
elements of WR and WI, say the i-th and (i+1)th, with
WI(i) > 0 and WI(i+1) < 0. If WANTT is .TRUE., the
eigenvalues are stored in the same order as on the diagonal
of the Schur form returned in H, with WR(i) = H(i,i), and, if
H(i:i+1,i:i+1) is a 2-by-2 diagonal block,
WI(i) = sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(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 DOUBLE PRECISION array, dimension (LDZ,N)
If WANTZ is .TRUE., on entry Z must contain the current
matrix Z of transformations accumulated by DHSEQR, 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, DLAHQR failed to compute all the
eigenvalues ILO to IHI in a total of 30 iterations
per eigenvalue; elements i+1:ihi of WR and WI
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.
Further Details:
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 DLAHQR 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 dlahr2 (integer N, integer K, integer NB, double precision, dimension( lda, * ) A,
integer LDA, double precision, dimension( nb ) TAU, double precision, dimension( ldt, nb )
T, integer LDT, double precision, dimension( ldy, nb ) Y, integer LDY)
DLAHR2 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:
DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)
matrix A so that elements below the k-th subdiagonal are zero. The
reduction is performed by an orthogonal similarity transformation
Q**T * A * Q. The routine returns the matrices V and T which determine
Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.
This is an auxiliary routine called by DGEHRD.
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
T
T is DOUBLE PRECISION 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 DOUBLE PRECISION 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**T
where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DLAHRD
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 DLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
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 dlaic1 (integer JOB, integer J, double precision, dimension( j ) X, double
precision SEST, double precision, dimension( j ) W, double precision GAMMA, double
precision SESTPR, double precision S, double precision C)
DLAIC1 applies one step of incremental condition estimation.
Purpose:
DLAIC1 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 DLAIC1 computes sestpr, s, c such that
the vector
[ s*x ]
xhat = [ c ]
is an approximate singular vector of
[ L 0 ]
Lhat = [ w**T 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]**T and sestpr**2 is an eigenpair of the system
diag(sest*sest, 0) + [alpha gamma] * [ alpha ]
[ gamma ]
where alpha = x**T*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 DOUBLE PRECISION array, dimension (J)
The j-vector x.
SEST
SEST is DOUBLE PRECISION
Estimated singular value of j by j matrix L
W
W is DOUBLE PRECISION array, dimension (J)
The j-vector w.
GAMMA
GAMMA is DOUBLE PRECISION
The diagonal element gamma.
SESTPR
SESTPR is DOUBLE PRECISION
Estimated singular value of (j+1) by (j+1) matrix Lhat.
S
S is DOUBLE PRECISION
Sine needed in forming xhat.
C
C is DOUBLE PRECISION
Cosine needed in forming xhat.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlaln2 (logical LTRANS, integer NA, integer NW, double precision SMIN, double
precision CA, double precision, dimension( lda, * ) A, integer LDA, double precision D1,
double precision D2, double precision, dimension( ldb, * ) B, integer LDB, double
precision WR, double precision WI, double precision, dimension( ldx, * ) X, integer LDX,
double precision SCALE, double precision XNORM, integer INFO)
DLALN2 solves a 1-by-1 or 2-by-2 linear system of equations of the specified form.
Purpose:
DLALN2 solves a system of the form (ca A - w D ) X = s B
or (ca A**T - w D) X = s B with possible scaling ('s') and
perturbation of A. (A**T means A-transpose.)
A is an NA x NA real matrix, ca is a real scalar, D is an NA x NA
real diagonal matrix, w is a real or complex value, and X and B are
NA x 1 matrices -- real if w is real, complex if w is complex. NA
may be 1 or 2.
If w is complex, X and B are represented as NA x 2 matrices,
the first column of each being the real part and the second
being the imaginary part.
's' is a scaling factor (<= 1), computed by DLALN2, which is
so chosen that X can be computed without overflow. X is further
scaled if necessary to assure that norm(ca A - w D)*norm(X) is less
than overflow.
If both singular values of (ca A - w D) are less than SMIN,
SMIN*identity will be used instead of (ca A - w D). If only one
singular value is less than SMIN, one element of (ca A - w D) will be
perturbed enough to make the smallest singular value roughly SMIN.
If both singular values are at least SMIN, (ca A - w D) will not be
perturbed. In any case, the perturbation will be at most some small
multiple of max( SMIN, ulp*norm(ca A - w D) ). The singular values
are computed by infinity-norm approximations, and thus will only be
correct to a factor of 2 or so.
Note: all input quantities are assumed to be smaller than overflow
by a reasonable factor. (See BIGNUM.)
Parameters
LTRANS
LTRANS is LOGICAL
=.TRUE.: A-transpose will be used.
=.FALSE.: A will be used (not transposed.)
NA
NA is INTEGER
The size of the matrix A. It may (only) be 1 or 2.
NW
NW is INTEGER
1 if 'w' is real, 2 if 'w' is complex. It may only be 1
or 2.
SMIN
SMIN is DOUBLE PRECISION
The desired lower bound on the singular values of A. This
should be a safe distance away from underflow or overflow,
say, between (underflow/machine precision) and (machine
precision * overflow ). (See BIGNUM and ULP.)
CA
CA is DOUBLE PRECISION
The coefficient c, which A is multiplied by.
A
A is DOUBLE PRECISION array, dimension (LDA,NA)
The NA x NA matrix A.
LDA
LDA is INTEGER
The leading dimension of A. It must be at least NA.
D1
D1 is DOUBLE PRECISION
The 1,1 element in the diagonal matrix D.
D2
D2 is DOUBLE PRECISION
The 2,2 element in the diagonal matrix D. Not used if NA=1.
B
B is DOUBLE PRECISION array, dimension (LDB,NW)
The NA x NW matrix B (right-hand side). If NW=2 ('w' is
complex), column 1 contains the real part of B and column 2
contains the imaginary part.
LDB
LDB is INTEGER
The leading dimension of B. It must be at least NA.
WR
WR is DOUBLE PRECISION
The real part of the scalar 'w'.
WI
WI is DOUBLE PRECISION
The imaginary part of the scalar 'w'. Not used if NW=1.
X
X is DOUBLE PRECISION array, dimension (LDX,NW)
The NA x NW matrix X (unknowns), as computed by DLALN2.
If NW=2 ('w' is complex), on exit, column 1 will contain
the real part of X and column 2 will contain the imaginary
part.
LDX
LDX is INTEGER
The leading dimension of X. It must be at least NA.
SCALE
SCALE is DOUBLE PRECISION
The scale factor that B must be multiplied by to insure
that overflow does not occur when computing X. Thus,
(ca A - w D) X will be SCALE*B, not B (ignoring
perturbations of A.) It will be at most 1.
XNORM
XNORM is DOUBLE PRECISION
The infinity-norm of X, when X is regarded as an NA x NW
real matrix.
INFO
INFO is INTEGER
An error flag. It will be set to zero if no error occurs,
a negative number if an argument is in error, or a positive
number if ca A - w D had to be perturbed.
The possible values are:
= 0: No error occurred, and (ca A - w D) did not have to be
perturbed.
= 1: (ca A - w D) had to be perturbed to make its smallest
(or only) singular value greater than SMIN.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
double precision function dlangt (character NORM, integer N, double precision, dimension( * )
DL, double precision, dimension( * ) D, double precision, dimension( * ) DU)
DLANGT 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:
DLANGT returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
real tridiagonal matrix A.
Returns
DLANGT
DLANGT = ( 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 DLANGT as described
above.
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, DLANGT is
set to zero.
DL
DL is DOUBLE PRECISION array, dimension (N-1)
The (n-1) sub-diagonal elements of A.
D
D is DOUBLE PRECISION array, dimension (N)
The diagonal elements of A.
DU
DU is DOUBLE PRECISION 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 dlanhs (character NORM, integer N, double precision, dimension( lda,
* ) A, integer LDA, double precision, dimension( * ) WORK)
DLANHS 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:
DLANHS 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
DLANHS
DLANHS = ( 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 DLANHS as described
above.
N
N is INTEGER
The order of the matrix A. N >= 0. When N = 0, DLANHS is
set to zero.
A
A is DOUBLE PRECISION 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 dlansb (character NORM, character UPLO, integer N, integer K, double
precision, dimension( ldab, * ) AB, integer LDAB, double precision, dimension( * ) WORK)
DLANSB 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:
DLANSB 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
DLANSB
DLANSB = ( 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 DLANSB 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, DLANSB 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 DOUBLE PRECISION 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 dlansp (character NORM, character UPLO, integer N, double precision,
dimension( * ) AP, double precision, dimension( * ) WORK)
DLANSP 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:
DLANSP returns the value of the one norm, or the Frobenius norm, or
the infinity norm, or the element of largest absolute value of a
real symmetric matrix A, supplied in packed form.
Returns
DLANSP
DLANSP = ( 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 DLANSP 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, DLANSP is
set to zero.
AP
AP is DOUBLE PRECISION 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 dlantb (character NORM, character UPLO, character DIAG, integer N,
integer K, double precision, dimension( ldab, * ) AB, integer LDAB, double precision,
dimension( * ) WORK)
DLANTB 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:
DLANTB 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
DLANTB
DLANTB = ( 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 DLANTB 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, DLANTB 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 DOUBLE PRECISION 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 dlantp (character NORM, character UPLO, character DIAG, integer N,
double precision, dimension( * ) AP, double precision, dimension( * ) WORK)
DLANTP 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:
DLANTP 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
DLANTP
DLANTP = ( 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 DLANTP 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, DLANTP is
set to zero.
AP
AP is DOUBLE PRECISION 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 dlantr (character NORM, character UPLO, character DIAG, integer M,
integer N, double precision, dimension( lda, * ) A, integer LDA, double precision,
dimension( * ) WORK)
DLANTR 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:
DLANTR 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
DLANTR
DLANTR = ( 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 DLANTR 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, DLANTR 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, DLANTR is set to zero.
A
A is DOUBLE PRECISION 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 dlanv2 (double precision A, double precision B, double precision C, double
precision D, double precision RT1R, double precision RT1I, double precision RT2R, double
precision RT2I, double precision CS, double precision SN)
DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric matrix in standard
form.
Purpose:
DLANV2 computes the Schur factorization of a real 2-by-2 nonsymmetric
matrix in standard form:
[ A B ] = [ CS -SN ] [ AA BB ] [ CS SN ]
[ C D ] [ SN CS ] [ CC DD ] [-SN CS ]
where either
1) CC = 0 so that AA and DD are real eigenvalues of the matrix, or
2) AA = DD and BB*CC < 0, so that AA + or - sqrt(BB*CC) are complex
conjugate eigenvalues.
Parameters
A
A is DOUBLE PRECISION
B
B is DOUBLE PRECISION
C
C is DOUBLE PRECISION
D
D is DOUBLE PRECISION
On entry, the elements of the input matrix.
On exit, they are overwritten by the elements of the
standardised Schur form.
RT1R
RT1R is DOUBLE PRECISION
RT1I
RT1I is DOUBLE PRECISION
RT2R
RT2R is DOUBLE PRECISION
RT2I
RT2I is DOUBLE PRECISION
The real and imaginary parts of the eigenvalues. If the
eigenvalues are a complex conjugate pair, RT1I > 0.
CS
CS is DOUBLE PRECISION
SN
SN is DOUBLE PRECISION
Parameters of the rotation matrix.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
Modified by V. Sima, Research Institute for Informatics, Bucharest,
Romania, to reduce the risk of cancellation errors,
when computing real eigenvalues, and to ensure, if possible, that
abs(RT1R) >= abs(RT2R).
subroutine dlapll (integer N, double precision, dimension( * ) X, integer INCX, double
precision, dimension( * ) Y, integer INCY, double precision SSMIN)
DLAPLL 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlapmr (logical FORWRD, integer M, integer N, double precision, dimension( ldx, * )
X, integer LDX, integer, dimension( * ) K)
DLAPMR rearranges rows of a matrix as specified by a permutation vector.
Purpose:
DLAPMR 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 DOUBLE PRECISION 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 dlapmt (logical FORWRD, integer M, integer N, double precision, dimension( ldx, * )
X, integer LDX, integer, dimension( * ) K)
DLAPMT performs a forward or backward permutation of the columns of a matrix.
Purpose:
DLAPMT 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 DOUBLE PRECISION 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 dlaqp2 (integer M, integer N, integer OFFSET, double precision, dimension( lda, * )
A, integer LDA, integer, dimension( * ) JPVT, double precision, dimension( * ) TAU, double
precision, dimension( * ) VN1, double precision, dimension( * ) VN2, double precision,
dimension( * ) WORK)
DLAQP2 computes a QR factorization with column pivoting of the matrix block.
Purpose:
DLAQP2 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlaqps (integer M, integer N, integer OFFSET, integer NB, integer KB, double
precision, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, double
precision, dimension( * ) TAU, double precision, dimension( * ) VN1, double precision,
dimension( * ) VN2, double precision, dimension( * ) AUXV, double precision, dimension(
ldf, * ) F, integer LDF)
DLAQPS computes a step of QR factorization with column pivoting of a real m-by-n matrix A
by using BLAS level 3.
Purpose:
DLAQPS computes a step of QR factorization with column pivoting
of a real 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)
Auxiliary vector.
F
F is DOUBLE PRECISION array, dimension (LDF,NB)
Matrix F**T = L*Y**T*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 dlaqr0 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double
precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double
precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension(
ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer
INFO)
DLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
the Schur decomposition.
Purpose:
DLAQR0 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.
Optionally Z may be postmultiplied into an input orthogonal
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 orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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 DGEBAL, and then passed to DGEHRD when the
matrix output by DGEBAL 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 DOUBLE PRECISION 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 quasi-triangular matrix T from the Schur
decomposition (the Schur form); 2-by-2 diagonal blocks
(corresponding to complex conjugate pairs of eigenvalues)
are returned in standard form, with H(i,i) = H(i+1,i+1)
and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT 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).
WR
WR is DOUBLE PRECISION array, dimension (IHI)
WI
WI is DOUBLE PRECISION array, dimension (IHI)
The real and imaginary parts, respectively, of the computed
eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
and WI(ILO:IHI). If two eigenvalues are computed as a
complex conjugate pair, they are stored in consecutive
elements of WR and WI, say the i-th and (i+1)th, with
WI(i) > 0 and WI(i+1) < 0. 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
WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
WI(i+1) = -WI(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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAQR0 does a workspace query.
In this case, DLAQR0 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, DLAQR0 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 an orthogonal matrix. The final
value of H is upper Hessenberg and quasi-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 orthogonal matrix in (*) (regard-
less of the value of WANTT.)
If INFO > 0 and WANTZ is .FALSE., then Z is not
accessed.
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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlaqr1 (integer N, double precision, dimension( ldh, * ) H, integer LDH, double
precision SR1, double precision SI1, double precision SR2, double precision SI2, double
precision, dimension( * ) V)
DLAQR1 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, DLAQR1 sets v to a
scalar multiple of the first column of the product
(*) K = (H - (sr1 + i*si1)*I)*(H - (sr2 + i*si2)*I)
scaling to avoid overflows and most underflows. It
is assumed that either
1) sr1 = sr2 and si1 = -si2
or
2) si1 = si2 = 0.
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 DOUBLE PRECISION 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
SR1
SR1 is DOUBLE PRECISION
SI1
SI1 is DOUBLE PRECISION
SR2
SR2 is DOUBLE PRECISION
SI2
SI2 is DOUBLE PRECISION
The shifts in (*).
V
V is DOUBLE PRECISION 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 dlaqr2 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
integer NW, double precision, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer
IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, double
precision, dimension( * ) SR, double precision, dimension( * ) SI, double precision,
dimension( ldv, * ) V, integer LDV, integer NH, double precision, dimension( ldt, * ) T,
integer LDT, integer NV, double precision, dimension( ldwv, * ) WV, integer LDWV, double
precision, dimension( * ) WORK, integer LWORK)
DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect
and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive
early deflation).
Purpose:
DLAQR2 is identical to DLAQR3 except that it avoids
recursion by calling DLAHQR instead of DLAQR4.
Aggressive early deflation:
This subroutine accepts as input an upper Hessenberg matrix
H and performs an orthogonal 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 orthogonal 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 quasi-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 orthogonal matrix Z is updated so
so that the orthogonal 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 orthogonal 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 DOUBLE PRECISION 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 an orthogonal
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 DOUBLE PRECISION array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the orthogonal
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.
SR
SR is DOUBLE PRECISION array, dimension (KBOT)
SI
SI is DOUBLE PRECISION array, dimension (KBOT)
On output, the real and imaginary parts of approximate
eigenvalues that may be used for shifts are stored in
SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
The real and imaginary parts of converged eigenvalues
are stored in SR(KBOT-ND+1) through SR(KBOT) and
SI(KBOT-ND+1) through SI(KBOT), respectively.
V
V is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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; DLAQR2
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 dlaqr3 (logical WANTT, logical WANTZ, integer N, integer KTOP, integer KBOT,
integer NW, double precision, dimension( ldh, * ) H, integer LDH, integer ILOZ, integer
IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, integer NS, integer ND, double
precision, dimension( * ) SR, double precision, dimension( * ) SI, double precision,
dimension( ldv, * ) V, integer LDV, integer NH, double precision, dimension( ldt, * ) T,
integer LDT, integer NV, double precision, dimension( ldwv, * ) WV, integer LDWV, double
precision, dimension( * ) WORK, integer LWORK)
DLAQR3 performs the orthogonal 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:
DLAQR3 accepts as input an upper Hessenberg matrix
H and performs an orthogonal 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 orthogonal 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 quasi-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 orthogonal matrix Z is updated so
so that the orthogonal 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 orthogonal 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 DOUBLE PRECISION 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 an orthogonal
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 DOUBLE PRECISION array, dimension (LDZ,N)
IF WANTZ is .TRUE., then on output, the orthogonal
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.
SR
SR is DOUBLE PRECISION array, dimension (KBOT)
SI
SI is DOUBLE PRECISION array, dimension (KBOT)
On output, the real and imaginary parts of approximate
eigenvalues that may be used for shifts are stored in
SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
The real and imaginary parts of converged eigenvalues
are stored in SR(KBOT-ND+1) through SR(KBOT) and
SI(KBOT-ND+1) through SI(KBOT), respectively.
V
V is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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; DLAQR3
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 dlaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double
precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double
precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension(
ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer
INFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
the Schur decomposition.
Purpose:
DLAQR4 implements one level of recursion for DLAQR0.
It is a complete implementation of the small bulge multi-shift
QR algorithm. It may be called by DLAQR0 and, for large enough
deflation window size, it may be called by DLAQR3. This
subroutine is identical to DLAQR0 except that it calls DLAQR2
instead of DLAQR3.
DLAQR4 computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**T, where T is an upper quasi-triangular matrix (the
Schur form), and Z is the orthogonal matrix of Schur vectors.
Optionally Z may be postmultiplied into an input orthogonal
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 orthogonal matrix Q: A = Q*H*Q**T = (QZ)*T*(QZ)**T.
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 DGEBAL, and then passed to DGEHRD when the
matrix output by DGEBAL 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 DOUBLE PRECISION 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 quasi-triangular matrix T from the Schur
decomposition (the Schur form); 2-by-2 diagonal blocks
(corresponding to complex conjugate pairs of eigenvalues)
are returned in standard form, with H(i,i) = H(i+1,i+1)
and H(i+1,i)*H(i,i+1) < 0. If INFO = 0 and WANTT 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).
WR
WR is DOUBLE PRECISION array, dimension (IHI)
WI
WI is DOUBLE PRECISION array, dimension (IHI)
The real and imaginary parts, respectively, of the computed
eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
and WI(ILO:IHI). If two eigenvalues are computed as a
complex conjugate pair, they are stored in consecutive
elements of WR and WI, say the i-th and (i+1)th, with
WI(i) > 0 and WI(i+1) < 0. 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
WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
WI(i+1) = -WI(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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DLAQR4 does a workspace query.
In this case, DLAQR4 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, DLAQR4 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 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(ILO:IHI,ILOZ:IHIZ)
= (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U
where U is the orthogonal 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 dlaqr5 (logical WANTT, logical WANTZ, integer KACC22, integer N, integer KTOP,
integer KBOT, integer NSHFTS, double precision, dimension( * ) SR, double precision,
dimension( * ) SI, double precision, dimension( ldh, * ) H, integer LDH, integer ILOZ,
integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, double precision,
dimension( ldv, * ) V, integer LDV, double precision, dimension( ldu, * ) U, integer LDU,
integer NV, double precision, dimension( ldwv, * ) WV, integer LDWV, integer NH, double
precision, dimension( ldwh, * ) WH, integer LDWH)
DLAQR5 performs a single small-bulge multi-shift QR sweep.
Purpose:
DLAQR5, called by DLAQR0, performs a
single small-bulge multi-shift QR sweep.
Parameters
WANTT
WANTT is LOGICAL
WANTT = .true. if the quasi-triangular Schur factor
is being computed. WANTT is set to .false. otherwise.
WANTZ
WANTZ is LOGICAL
WANTZ = .true. if the orthogonal 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: DLAQR5 does not accumulate reflections and does not
use matrix-matrix multiply to update far-from-diagonal
matrix entries.
= 1: DLAQR5 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.
SR
SR is DOUBLE PRECISION array, dimension (NSHFTS)
SI
SI is DOUBLE PRECISION array, dimension (NSHFTS)
SR contains the real parts and SI contains the imaginary
parts of the NSHFTS shifts of origin that define the
multi-shift QR sweep. On output SR and SI may be
reordered.
H
H is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDZ,IHIZ)
If WANTZ = .TRUE., then the QR Sweep orthogonal
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlaqsb (character UPLO, integer N, integer KD, double precision, dimension( ldab, *
) AB, integer LDAB, double precision, dimension( * ) S, double precision SCOND, double
precision AMAX, character EQUED)
DLAQSB scales a symmetric/Hermitian band matrix, using scaling factors computed by spbequ.
Purpose:
DLAQSB 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 DOUBLE PRECISION 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**T*U or A = L*L**T 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 dlaqsp (character UPLO, integer N, double precision, dimension( * ) AP, double
precision, dimension( * ) S, double precision SCOND, double precision AMAX, character
EQUED)
DLAQSP scales a symmetric/Hermitian matrix in packed storage, using scaling factors
computed by sppequ.
Purpose:
DLAQSP 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 DOUBLE PRECISION 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 dlaqtr (logical LTRAN, logical LREAL, integer N, double precision, dimension( ldt,
* ) T, integer LDT, double precision, dimension( * ) B, double precision W, double
precision SCALE, double precision, dimension( * ) X, double precision, dimension( * )
WORK, integer INFO)
DLAQTR solves a real quasi-triangular system of equations, or a complex quasi-triangular
system of special form, in real arithmetic.
Purpose:
DLAQTR solves the real quasi-triangular system
op(T)*p = scale*c, if LREAL = .TRUE.
or the complex quasi-triangular systems
op(T + iB)*(p+iq) = scale*(c+id), if LREAL = .FALSE.
in real arithmetic, where T is upper quasi-triangular.
If LREAL = .FALSE., then the first diagonal block of T must be
1 by 1, B is the specially structured matrix
B = [ b(1) b(2) ... b(n) ]
[ w ]
[ w ]
[ . ]
[ w ]
op(A) = A or A**T, A**T denotes the transpose of
matrix A.
On input, X = [ c ]. On output, X = [ p ].
[ d ] [ q ]
This subroutine is designed for the condition number estimation
in routine DTRSNA.
Parameters
LTRAN
LTRAN is LOGICAL
On entry, LTRAN specifies the option of conjugate transpose:
= .FALSE., op(T+i*B) = T+i*B,
= .TRUE., op(T+i*B) = (T+i*B)**T.
LREAL
LREAL is LOGICAL
On entry, LREAL specifies the input matrix structure:
= .FALSE., the input is complex
= .TRUE., the input is real
N
N is INTEGER
On entry, N specifies the order of T+i*B. N >= 0.
T
T is DOUBLE PRECISION array, dimension (LDT,N)
On entry, T contains a matrix in Schur canonical form.
If LREAL = .FALSE., then the first diagonal block of T mu
be 1 by 1.
LDT
LDT is INTEGER
The leading dimension of the matrix T. LDT >= max(1,N).
B
B is DOUBLE PRECISION array, dimension (N)
On entry, B contains the elements to form the matrix
B as described above.
If LREAL = .TRUE., B is not referenced.
W
W is DOUBLE PRECISION
On entry, W is the diagonal element of the matrix B.
If LREAL = .TRUE., W is not referenced.
SCALE
SCALE is DOUBLE PRECISION
On exit, SCALE is the scale factor.
X
X is DOUBLE PRECISION array, dimension (2*N)
On entry, X contains the right hand side of the system.
On exit, X is overwritten by the solution.
WORK
WORK is DOUBLE PRECISION array, dimension (N)
INFO
INFO is INTEGER
On exit, INFO is set to
0: successful exit.
1: the some diagonal 1 by 1 block has been perturbed by
a small number SMIN to keep nonsingularity.
2: the some diagonal 2 by 2 block has been perturbed by
a small number in DLALN2 to keep nonsingularity.
NOTE: In the interests of speed, this routine does not
check the inputs for errors.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlar1v (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, double precision, 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)
DLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1
through bn of the tridiagonal matrix LDLT - λI.
Purpose:
DLAR1V 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 DOUBLE PRECISION 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 dlar2v (integer N, double precision, dimension( * ) X, double precision, dimension(
* ) Y, double precision, dimension( * ) Z, integer INCX, double precision, dimension( * )
C, double precision, dimension( * ) S, integer INCC)
DLAR2V applies a vector of plane rotations with real cosines and real sines from both
sides to a sequence of 2-by-2 symmetric/Hermitian matrices.
Purpose:
DLAR2V applies a vector of real plane rotations from both sides to
a sequence of 2-by-2 real symmetric matrices, defined by the elements
of the vectors x, y and z. For i = 1,2,...,n
( x(i) z(i) ) := ( c(i) s(i) ) ( x(i) z(i) ) ( c(i) -s(i) )
( z(i) y(i) ) ( -s(i) c(i) ) ( z(i) y(i) ) ( s(i) c(i) )
Parameters
N
N is INTEGER
The number of plane rotations to be applied.
X
X is DOUBLE PRECISION array,
dimension (1+(N-1)*INCX)
The vector x.
Y
Y is DOUBLE PRECISION array,
dimension (1+(N-1)*INCX)
The vector y.
Z
Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlarf (character SIDE, integer M, integer N, double precision, dimension( * ) V,
integer INCV, double precision TAU, double precision, dimension( ldc, * ) C, integer LDC,
double precision, dimension( * ) WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Purpose:
DLARF applies a real elementary reflector H to a real m by n matrix
C, from either the left or the right. H is represented in the form
H = I - tau * v * v**T
where tau is a real scalar and v is a real vector.
If tau = 0, then H is taken to be the unit matrix.
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 DOUBLE PRECISION 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 DOUBLE PRECISION
The value tau in the representation of H.
C
C is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlarfb (character SIDE, character TRANS, character DIRECT, character STOREV,
integer M, integer N, integer K, double precision, dimension( ldv, * ) V, integer LDV,
double precision, dimension( ldt, * ) T, integer LDT, double precision, dimension( ldc, *
) C, integer LDC, double precision, dimension( ldwork, * ) WORK, integer LDWORK)
DLARFB applies a block reflector or its transpose to a general rectangular matrix.
Purpose:
DLARFB applies a real block reflector H or its transpose H**T to a
real m by n matrix C, from either the left or the right.
Parameters
SIDE
SIDE is CHARACTER*1
= 'L': apply H or H**T from the Left
= 'R': apply H or H**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'T': apply H**T (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 DOUBLE PRECISION array, dimension
(LDV,K) if STOREV = 'C'
(LDV,M) if STOREV = 'R' and SIDE = 'L'
(LDV,N) if STOREV = 'R' and SIDE = 'R'
The matrix V. 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDC,N)
On entry, the m by n matrix C.
On exit, C is overwritten by H*C or H**T*C or C*H or C*H**T.
LDC
LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).
WORK
WORK is DOUBLE PRECISION 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 dlarfb_gett (character IDENT, integer M, integer N, integer K, double precision,
dimension( ldt, * ) T, integer LDT, double precision, dimension( lda, * ) A, integer LDA,
double precision, dimension( ldb, * ) B, integer LDB, double precision, dimension( ldwork,
* ) WORK, integer LDWORK)
DLARFB_GETT
Purpose:
DLARFB_GETT applies a real Householder block reflector H from the
left to a real (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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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**T ) * ( A_in ) =
( B_out ) ( B_in ) ( B_in )
= ( I - ( V1 ) * T * ( V1**T, V2**T ) ) * ( 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**T ) * ( A1_in )
( B1_out ) ( 0 ) ( 0 )
( A2_out ) := H * ( A2_in ) = ( I - V * T * V**T ) * ( A2_in )
( B2_out ) ( B2_in ) ( B2_in )
If IDENT != 'I':
The computation for column block 1:
A1_out: = A1_in - V1*T*(V1**T)*A1_in
B1_out: = - V2*T*(V1**T)*A1_in
The computation for column block 2, which exists if N > K:
A2_out: = A2_in - V1*T*( (V1**T)*A2_in + (V2**T)*B2_in )
B2_out: = B2_in - V2*T*( (V1**T)*A2_in + (V2**T)*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**T)*B2_in )
B2_out: = B2_in - V2*T*( A2_in + (V2**T)*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**T) * W2 = (unit_lower_tr_of_(A1)**T) * W2
col2_(3) W2: = W2 + (V2**T) * B2 = W2 + (B1**T) * 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**T) * W1 = (unit_lower_tr_of_(A1)**T) * 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**T) * B2 = W2 + (B1**T) * 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**T) * W2
= (unit_lower_tr_of_(A1)**T) * W2
end if
col2_(3) W2: = W2 + (V2**T) * B2 = W2 + (B1**T) * 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**T) * W1
= (unit_lower_tr_of_(A1)**T) * 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 dlarfg (integer N, double precision ALPHA, double precision, dimension( * ) X,
integer INCX, double precision TAU)
DLARFG generates an elementary reflector (Householder matrix).
Purpose:
DLARFG generates a real elementary reflector H of order n, such
that
H * ( alpha ) = ( beta ), H**T * H = I.
( x ) ( 0 )
where alpha and beta are scalars, and x is an (n-1)-element real
vector. H is represented in the form
H = I - tau * ( 1 ) * ( 1 v**T ) ,
( v )
where tau is a real scalar and v is a real (n-1)-element
vector.
If the elements of x are all zero, then tau = 0 and H is taken to be
the unit matrix.
Otherwise 1 <= tau <= 2.
Parameters
N
N is INTEGER
The order of the elementary reflector.
ALPHA
ALPHA is DOUBLE PRECISION
On entry, the value alpha.
On exit, it is overwritten with the value beta.
X
X is DOUBLE PRECISION 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 DOUBLE PRECISION
The value tau.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlarfgp (integer N, double precision ALPHA, double precision, dimension( * ) X,
integer INCX, double precision TAU)
DLARFGP generates an elementary reflector (Householder matrix) with non-negative beta.
Purpose:
DLARFGP generates a real elementary reflector H of order n, such
that
H * ( alpha ) = ( beta ), H**T * H = I.
( x ) ( 0 )
where alpha and beta are scalars, beta is non-negative, and x is
an (n-1)-element real vector. H is represented in the form
H = I - tau * ( 1 ) * ( 1 v**T ) ,
( v )
where tau is a real scalar and v is a real (n-1)-element
vector.
If the elements of x are all zero, 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 DOUBLE PRECISION
On entry, the value alpha.
On exit, it is overwritten with the value beta.
X
X is DOUBLE PRECISION 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 DOUBLE PRECISION
The value tau.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlarft (character DIRECT, character STOREV, integer N, integer K, double precision,
dimension( ldv, * ) V, integer LDV, double precision, dimension( * ) TAU, double
precision, dimension( ldt, * ) T, integer LDT)
DLARFT forms the triangular factor T of a block reflector H = I - vtvH
Purpose:
DLARFT forms the triangular factor T of a real 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**T
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**T * 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (K)
TAU(i) must contain the scalar factor of the elementary
reflector H(i).
T
T is DOUBLE PRECISION 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 dlarfx (character SIDE, integer M, integer N, double precision, dimension( * ) V,
double precision TAU, double precision, dimension( ldc, * ) C, integer LDC, double
precision, dimension( * ) WORK)
DLARFX applies an elementary reflector to a general rectangular matrix, with loop
unrolling when the reflector has order ≤ 10.
Purpose:
DLARFX applies a real elementary reflector H to a real m by n
matrix C, from either the left or the right. H is represented in the
form
H = I - tau * v * v**T
where tau is a real scalar and v is a real 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 DOUBLE PRECISION array, dimension (M) if SIDE = 'L'
or (N) if SIDE = 'R'
The vector v in the representation of H.
TAU
TAU is DOUBLE PRECISION
The value tau in the representation of H.
C
C is DOUBLE PRECISION 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 >= (1,M).
WORK
WORK is DOUBLE PRECISION 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 dlarfy (character UPLO, integer N, double precision, dimension( * ) V, integer
INCV, double precision TAU, double precision, dimension( ldc, * ) C, integer LDC, double
precision, dimension( * ) WORK)
DLARFY
Purpose:
DLARFY applies an elementary reflector, or Householder matrix, H,
to an n x n symmetric 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
symmetric 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 DOUBLE PRECISION 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 DOUBLE PRECISION
The value tau as described above.
C
C is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dlargv (integer N, double precision, dimension( * ) X, integer INCX, double
precision, dimension( * ) Y, integer INCY, double precision, dimension( * ) C, integer
INCC)
DLARGV generates a vector of plane rotations with real cosines and real sines.
Purpose:
DLARGV generates a vector of real plane rotations, determined by
elements of the real vectors x and y. For i = 1,2,...,n
( c(i) s(i) ) ( x(i) ) = ( a(i) )
( -s(i) c(i) ) ( y(i) ) = ( 0 )
Parameters
N
N is INTEGER
The number of plane rotations to be generated.
X
X is DOUBLE PRECISION array,
dimension (1+(N-1)*INCX)
On entry, the vector x.
On exit, x(i) is overwritten by a(i), for i = 1,...,n.
INCX
INCX is INTEGER
The increment between elements of X. INCX > 0.
Y
Y is DOUBLE PRECISION 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.
subroutine dlarrv (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, double
precision, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, double
precision, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO)
DLARRV computes the eigenvectors of the tridiagonal matrix T = L D LT given L, D and the
eigenvalues of L D LT.
Purpose:
DLARRV 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.
Note: VU is currently not used by this implementation of DLARRV, VU is
passed to DLARRV because it could be used compute gaps on the right end
of the extremal eigenvalues. However, with not much initial accuracy in
LAMBDA and VU, the formula can lead to an overestimation of the right gap
and thus to inadequately early RQI 'convergence'. This is currently
prevented this by forcing a small right gap. And so it turns out that VU
is currently not used by this implementation of DLARRV.
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 DOUBLE PRECISION 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 DLARRV.
< 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 dlartv (integer N, double precision, dimension( * ) X, integer INCX, double
precision, dimension( * ) Y, integer INCY, double precision, dimension( * ) C, double
precision, dimension( * ) S, integer INCC)
DLARTV applies a vector of plane rotations with real cosines and real sines to the
elements of a pair of vectors.
Purpose:
DLARTV applies a vector of real plane rotations to elements of the
real vectors x and y. For i = 1,2,...,n
( x(i) ) := ( c(i) s(i) ) ( x(i) )
( y(i) ) ( -s(i) c(i) ) ( y(i) )
Parameters
N
N is INTEGER
The number of plane rotations to be applied.
X
X is DOUBLE PRECISION array,
dimension (1+(N-1)*INCX)
The vector x.
INCX
INCX is INTEGER
The increment between elements of X. INCX > 0.
Y
Y is DOUBLE PRECISION 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 DOUBLE PRECISION 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 dlaswp (integer N, double precision, dimension( lda, * ) A, integer LDA, integer
K1, integer K2, integer, dimension( * ) IPIV, integer INCX)
DLASWP performs a series of row interchanges on a general rectangular matrix.
Purpose:
DLASWP 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 DOUBLE PRECISION 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 dlat2s (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
LDA, real, dimension( ldsa, * ) SA, integer LDSA, integer INFO)
DLAT2S converts a double-precision triangular matrix to a single-precision triangular
matrix.
Purpose:
DLAT2S converts a DOUBLE PRECISION triangular matrix, SA, to a SINGLE
PRECISION triangular matrix, A.
RMAX is the overflow for the SINGLE PRECISION arithmetic
DLAS2S 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 DOUBLE PRECISION 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 REAL 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 dlatbs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, integer KD, double precision, dimension( ldab, * ) AB, integer LDAB, double precision,
dimension( * ) X, double precision SCALE, double precision, dimension( * ) CNORM, integer
INFO)
DLATBS solves a triangular banded system of equations.
Purpose:
DLATBS solves one of the triangular systems
A *x = s*b or A**T*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 DTBSV 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**T* x = s*b (Conjugate transpose = 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 or A**T* 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, DTBSV
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 DTBSV 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. The basic
algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 DTBSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
subroutine dlatdf (integer IJOB, integer N, double precision, dimension( ldz, * ) Z, integer
LDZ, double precision, dimension( * ) RHS, double precision RDSUM, double precision
RDSCAL, integer, dimension( * ) IPIV, integer, dimension( * ) JPIV)
DLATDF uses the LU factorization of the n-by-n matrix computed by sgetc2 and computes a
contribution to the reciprocal Dif-estimate.
Purpose:
DLATDF uses the LU factorization of the n-by-n matrix Z computed by
DGETC2 and computes a contribution to the reciprocal Dif-estimate
by solving Z * x = b for x, and choosing the r.h.s. b such that
the norm of x is as large as possible. On entry RHS = b holds the
contribution from earlier solved sub-systems, and on return RHS = x.
The factorization of Z returned by DGETC2 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 DGECON, 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 DOUBLE PRECISION array, dimension (LDZ, N)
On entry, the LU part of the factorization of the n-by-n
matrix Z computed by DGETC2: Z = P * L * U * Q
LDZ
LDZ is INTEGER
The leading dimension of the array Z. LDA >= max(1, N).
RHS
RHS is DOUBLE PRECISION 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 DTGSYL, 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 DTGSY2 is called by STGSYL.
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 DTGSY2 is called by
DTGSYL.
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 IMINF-95.05, Departement of
Computing Science, Umea University, S-901 87 Umea, Sweden, 1995.
subroutine dlatps (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, double precision, dimension( * ) AP, double precision, dimension( * ) X, double
precision SCALE, double precision, dimension( * ) CNORM, integer INFO)
DLATPS solves a triangular system of equations with the matrix held in packed storage.
Purpose:
DLATPS solves one of the triangular systems
A *x = s*b or A**T*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, 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
DTPSV 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**T* x = s*b (Conjugate transpose = 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 or A**T* 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, DTPSV
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 DTPSV 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. The basic
algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 DTPSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
subroutine dlatrd (character UPLO, integer N, integer NB, double precision, dimension( lda, *
) A, integer LDA, double precision, dimension( * ) E, double precision, dimension( * )
TAU, double precision, dimension( ldw, * ) W, integer LDW)
DLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
tridiagonal form by an orthogonal similarity transformation.
Purpose:
DLATRD reduces NB rows and columns of a real symmetric matrix A to
symmetric tridiagonal form by an orthogonal similarity
transformation Q**T * 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', DLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', DLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by DSYTRD.
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.
NB
NB is INTEGER
The number of rows and columns to be reduced.
A
A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the symmetric 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 orthogonal 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 orthogonal matrix Q as a
product of elementary reflectors.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= (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 DOUBLE PRECISION 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 DOUBLE PRECISION 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**T
where tau is a real scalar, and v is a real 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**T
where tau is a real scalar, and v is a real 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 symmetric rank-2k update of the form:
A := A - V*W**T - W*V**T.
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 dlatrs (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, double precision, dimension( lda, * ) A, integer LDA, double precision, dimension( * )
X, double precision SCALE, double precision, dimension( * ) CNORM, integer INFO)
DLATRS solves a triangular system of equations with the scale factor set to prevent
overflow.
Purpose:
DLATRS solves one of the triangular systems
A *x = s*b or A**T *x = s*b
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, 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 DTRSV 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**T* x = s*b (Conjugate transpose = 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 or A**T* 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, DTRSV
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 DTRSV 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. The basic
algorithm for A upper triangular is
for j = 1, ..., n
x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 DTRSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
subroutine dlatrs3 (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, integer NRHS, double precision, dimension( lda, * ) A, integer LDA, double precision,
dimension( ldx, * ) X, integer LDX, double precision, dimension( * ) SCALE, double
precision, dimension( * ) CNORM, double precision, dimension( * ) WORK, integer LWORK,
integer INFO)
DLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
Purpose:
DLATRS3 solves one of the triangular systems
A * X = B * diag(scale) or A**T * X = B * diag(scale)
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A. X and B are
n by nrhs matrices and scale is an nrhs element vector of scaling
factors. A scaling factor scale(j) is usually less than or equal
to 1, chosen such that X(:,j) is less than the overflow threshold.
If the matrix A is singular (A(j,j) = 0 for some j), then
a non-trivial solution to A*X = 0 is returned. If the system is
so badly scaled that the solution cannot be represented as
(1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
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**T* x = s*b (Conjugate transpose = 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.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is DOUBLE PRECISION array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
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.
WORK
WORK is DOUBLE PRECISION array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N
+ NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal
dimensions of the WORK array, returns this value as the first entry of the WORK array, and
no error message related to LWORK is issued by XERBLA.
Parameters
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:
subroutine dlauu2 (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
LDA, integer INFO)
DLAUU2 computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (unblocked algorithm).
Purpose:
DLAUU2 computes the product U * U**T or L**T * 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 DOUBLE PRECISION 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**T;
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L**T * 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 dlauum (character UPLO, integer N, double precision, dimension( lda, * ) A, integer
LDA, integer INFO)
DLAUUM computes the product UUH or LHL, where U and L are upper or lower triangular
matrices (blocked algorithm).
Purpose:
DLAUUM computes the product U * U**T or L**T * 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 DOUBLE PRECISION 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**T;
if UPLO = 'L', the lower triangle of A is overwritten with
the lower triangle of the product L**T * 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 drscl (integer N, double precision SA, double precision, dimension( * ) SX, integer
INCX)
DRSCL multiplies a vector by the reciprocal of a real scalar.
Purpose:
DRSCL multiplies an n-element real 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 DOUBLE PRECISION 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 dtprfb (character SIDE, character TRANS, character DIRECT, character STOREV,
integer M, integer N, integer K, integer L, double precision, dimension( ldv, * ) V,
integer LDV, double precision, dimension( ldt, * ) T, integer LDT, double precision,
dimension( lda, * ) A, integer LDA, double precision, dimension( ldb, * ) B, integer LDB,
double precision, dimension( ldwork, * ) WORK, integer LDWORK)
DTPRFB applies a real 'triangular-pentagonal' block reflector to a real matrix, which is
composed of two blocks.
Purpose:
DTPRFB applies a real 'triangular-pentagonal' block reflector H or its
transpose H**T to a real 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**T from the Left
= 'R': apply H or H**T from the Right
TRANS
TRANS is CHARACTER*1
= 'N': apply H (No transpose)
= 'T': apply H**T (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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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**T*C or C*H or C*H**T. 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 DOUBLE PRECISION 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**T*C or C*H or C*H**T. See Further Details.
LDB
LDB is INTEGER
The leading dimension of the array B.
LDB >= max(1,M).
WORK
WORK is DOUBLE PRECISION 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.
subroutine slatrd (character UPLO, integer N, integer NB, real, dimension( lda, * ) A, integer
LDA, real, dimension( * ) E, real, dimension( * ) TAU, real, dimension( ldw, * ) W,
integer LDW)
SLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real
tridiagonal form by an orthogonal similarity transformation.
Purpose:
SLATRD reduces NB rows and columns of a real symmetric matrix A to
symmetric tridiagonal form by an orthogonal similarity
transformation Q**T * 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', SLATRD reduces the last NB rows and columns of a
matrix, of which the upper triangle is supplied;
if UPLO = 'L', SLATRD reduces the first NB rows and columns of a
matrix, of which the lower triangle is supplied.
This is an auxiliary routine called by SSYTRD.
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.
NB
NB is INTEGER
The number of rows and columns to be reduced.
A
A is REAL array, dimension (LDA,N)
On entry, the symmetric 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 orthogonal 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 orthogonal matrix Q as a
product of elementary reflectors.
See Further Details.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= (1,N).
E
E is REAL 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 REAL 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 REAL 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**T
where tau is a real scalar, and v is a real 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**T
where tau is a real scalar, and v is a real 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 symmetric rank-2k update of the form:
A := A - V*W**T - W*V**T.
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 slatrs3 (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldx, * ) X,
integer LDX, real, dimension( * ) SCALE, real, dimension( * ) CNORM, real, dimension( * )
WORK, integer LWORK, integer INFO)
SLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
Purpose:
SLATRS3 solves one of the triangular systems
A * X = B * diag(scale) or A**T * X = B * diag(scale)
with scaling to prevent overflow. Here A is an upper or lower
triangular matrix, A**T denotes the transpose of A. X and B are
n by nrhs matrices and scale is an nrhs element vector of scaling
factors. A scaling factor scale(j) is usually less than or equal
to 1, chosen such that X(:,j) is less than the overflow threshold.
If the matrix A is singular (A(j,j) = 0 for some j), then
a non-trivial solution to A*X = 0 is returned. If the system is
so badly scaled that the solution cannot be represented as
(1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
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**T* x = s*b (Conjugate transpose = 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.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 0.
A
A is REAL 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 REAL array, dimension (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is REAL array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
CNORM
CNORM is REAL 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.
WORK
WORK is REAL array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N
+ NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal
dimensions of the WORK array, returns this value as the first entry of the WORK array, and
no error message related to LWORK is issued by XERBLA.
Parameters
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:
subroutine zlatrs3 (character UPLO, character TRANS, character DIAG, character NORMIN, integer
N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension(
ldx, * ) X, integer LDX, double precision, dimension( * ) SCALE, double precision,
dimension( * ) CNORM, double precision, dimension( * ) WORK, integer LWORK, integer INFO)
ZLATRS3 solves a triangular system of equations with the scale factors set to prevent
overflow.
Purpose:
ZLATRS3 solves one of the triangular systems
A * X = B * diag(scale), A**T * X = B * diag(scale), or
A**H * X = B * diag(scale)
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-by-nrhs matrices and scale
is an nrhs-element vector of scaling factors. A scaling factor scale(j)
is usually less than or equal to 1, chosen such that X(:,j) is less
than the overflow threshold. If the matrix A is singular (A(j,j) = 0
for some j), then a non-trivial solution to A*X = 0 is returned. If
the system is so badly scaled that the solution cannot be represented
as (1/scale(k))*X(:,k), then x(:,k) = 0 and scale(k) is returned.
This is a BLAS-3 version of LATRS for solving several right
hand sides simultaneously.
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**T* 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.
NRHS
NRHS is INTEGER
The number of columns of X. NRHS >= 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 (LDX,NRHS)
On entry, the right hand side B of the triangular system.
On exit, X is overwritten by the solution matrix X.
LDX
LDX is INTEGER
The leading dimension of the array X. LDX >= max (1,N).
SCALE
SCALE is DOUBLE PRECISION array, dimension (NRHS)
The scaling factor s(k) is for the triangular system
A * x(:,k) = s(k)*b(:,k) or A**T* x(:,k) = s(k)*b(:,k).
If SCALE = 0, the matrix A is singular or badly scaled.
If A(j,j) = 0 is encountered, a non-trivial vector x(:,k)
that is an exact or approximate solution to A*x(:,k) = 0
is returned. If the system so badly scaled that solution
cannot be presented as x(:,k) * 1/s(k), then x(:,k) = 0
is returned.
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.
WORK
WORK is DOUBLE PRECISION array, dimension (LWORK).
On exit, if INFO = 0, WORK(1) returns the optimal size of
WORK.
LWORK LWORK is INTEGER LWORK >= MAX(1, 2*NBA * MAX(NBA, MIN(NRHS, 32)), where NBA = (N
+ NB - 1)/NB and NB is the optimal block size.
If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal
dimensions of the WORK array, returns this value as the first entry of the WORK array, and
no error message related to LWORK is issued by XERBLA.
Parameters
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:
Author
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