ZGESVD(3F) ZGESVD(3F)
ZGESVD - compute the singular value decomposition (SVD) of a complex Mby-N
matrix A, optionally computing the left and/or right singular
vectors
SUBROUTINE ZGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
LWORK, RWORK, INFO )
CHARACTER JOBU, JOBVT
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
DOUBLE PRECISION RWORK( * ), S( * )
COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
ZGESVD computes the singular value decomposition (SVD) of a complex Mby-N
matrix A, optionally computing the left and/or right singular
vectors. The SVD is written
A = U * SIGMA * conjugate-transpose(V)
where SIGMA is an M-by-N matrix which is zero except for its min(m,n)
diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N
unitary matrix. The diagonal elements of SIGMA are the singular values
of A; they are real and non-negative, and are returned in descending
order. The first min(m,n) columns of U and V are the left and right
singular vectors of A.
Note that the routine returns V**H, not V.
JOBU (input) CHARACTER*1
Specifies options for computing all or part of the matrix U:
= 'A': all M columns of U are returned in array U:
= 'S': the first min(m,n) columns of U (the left singular
vectors) are returned in the array U; = 'O': the first min(m,n)
columns of U (the left singular vectors) are overwritten on the
array A; = 'N': no columns of U (no left singular vectors) are
computed.
JOBVT (input) CHARACTER*1
Specifies options for computing all or part of the matrix V**H:
= 'A': all N rows of V**H are returned in the array VT;
= 'S': the first min(m,n) rows of V**H (the right singular
vectors) are returned in the array VT; = 'O': the first min(m,n)
rows of V**H (the right singular vectors) are overwritten on the
array A; = 'N': no rows of V**H (no right singular vectors) are
computed.
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ZGESVD(3F) ZGESVD(3F)
JOBVT and JOBU cannot both be 'O'.
M (input) INTEGER
The number of rows of the input matrix A. M >= 0.
N (input) INTEGER
The number of columns of the input matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if JOBU = 'O', A is
overwritten with the first min(m,n) columns of U (the left
singular vectors, stored columnwise); if JOBVT = 'O', A is
overwritten with the first min(m,n) rows of V**H (the right
singular vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT
.ne. 'O', the contents of A are destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
S (output) DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >= S(i+1).
U (output) COMPLEX*16 array, dimension (LDU,UCOL)
(LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU =
'A', U contains the M-by-M unitary matrix U; if JOBU = 'S', U
contains the first min(m,n) columns of U (the left singular
vectors, stored columnwise); if JOBU = 'N' or 'O', U is not
referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= 1; if JOBU = 'S' or
'A', LDU >= M.
VT (output) COMPLEX*16 array, dimension (LDVT,N)
If JOBVT = 'A', VT contains the N-by-N unitary matrix V**H; if
JOBVT = 'S', VT contains the first min(m,n) rows of V**H (the
right singular vectors, stored rowwise); if JOBVT = 'N' or 'O',
VT is not referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1; if JOBVT =
'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. LWORK >=
2*MIN(M,N)+MAX(M,N). For good performance, LWORK should
generally be larger.
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ZGESVD(3F) ZGESVD(3F)
RWORK (workspace) DOUBLE PRECISION array, dimension
(max(3*min(M,N),5*min(M,N)-4)) On exit, if INFO > 0,
RWORK(1:MIN(M,N)-1) contains the unconverged superdiagonal
elements of an upper bidiagonal matrix B whose diagonal is in S
(not necessarily sorted). B satisfies A = U * B * VT, so it has
the same singular values as A, and singular vectors related by U
and VT.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if ZBDSQR did not converge, INFO specifies how many
superdiagonals of an intermediate bidiagonal form B did not
converge to zero. See the description of RWORK above for details.
ZGESVD(3F) ZGESVD(3F)
ZGESVD - compute the singular value decomposition (SVD) of a complex Mby-N
matrix A, optionally computing the left and/or right singular
vectors
SUBROUTINE ZGESVD( JOBU, JOBVT, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
LWORK, RWORK, INFO )
CHARACTER JOBU, JOBVT
INTEGER INFO, LDA, LDU, LDVT, LWORK, M, N
DOUBLE PRECISION RWORK( * ), S( * )
COMPLEX*16 A( LDA, * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
ZGESVD computes the singular value decomposition (SVD) of a complex Mby-N
matrix A, optionally computing the left and/or right singular
vectors. The SVD is written
A = U * SIGMA * conjugate-transpose(V)
where SIGMA is an M-by-N matrix which is zero except for its min(m,n)
diagonal elements, U is an M-by-M unitary matrix, and V is an N-by-N
unitary matrix. The diagonal elements of SIGMA are the singular values
of A; they are real and non-negative, and are returned in descending
order. The first min(m,n) columns of U and V are the left and right
singular vectors of A.
Note that the routine returns V**H, not V.
JOBU (input) CHARACTER*1
Specifies options for computing all or part of the matrix U:
= 'A': all M columns of U are returned in array U:
= 'S': the first min(m,n) columns of U (the left singular
vectors) are returned in the array U; = 'O': the first min(m,n)
columns of U (the left singular vectors) are overwritten on the
array A; = 'N': no columns of U (no left singular vectors) are
computed.
JOBVT (input) CHARACTER*1
Specifies options for computing all or part of the matrix V**H:
= 'A': all N rows of V**H are returned in the array VT;
= 'S': the first min(m,n) rows of V**H (the right singular
vectors) are returned in the array VT; = 'O': the first min(m,n)
rows of V**H (the right singular vectors) are overwritten on the
array A; = 'N': no rows of V**H (no right singular vectors) are
computed.
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ZGESVD(3F) ZGESVD(3F)
JOBVT and JOBU cannot both be 'O'.
M (input) INTEGER
The number of rows of the input matrix A. M >= 0.
N (input) INTEGER
The number of columns of the input matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A. On exit, if JOBU = 'O', A is
overwritten with the first min(m,n) columns of U (the left
singular vectors, stored columnwise); if JOBVT = 'O', A is
overwritten with the first min(m,n) rows of V**H (the right
singular vectors, stored rowwise); if JOBU .ne. 'O' and JOBVT
.ne. 'O', the contents of A are destroyed.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
S (output) DOUBLE PRECISION array, dimension (min(M,N))
The singular values of A, sorted so that S(i) >= S(i+1).
U (output) COMPLEX*16 array, dimension (LDU,UCOL)
(LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'. If JOBU =
'A', U contains the M-by-M unitary matrix U; if JOBU = 'S', U
contains the first min(m,n) columns of U (the left singular
vectors, stored columnwise); if JOBU = 'N' or 'O', U is not
referenced.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= 1; if JOBU = 'S' or
'A', LDU >= M.
VT (output) COMPLEX*16 array, dimension (LDVT,N)
If JOBVT = 'A', VT contains the N-by-N unitary matrix V**H; if
JOBVT = 'S', VT contains the first min(m,n) rows of V**H (the
right singular vectors, stored rowwise); if JOBVT = 'N' or 'O',
VT is not referenced.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= 1; if JOBVT =
'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
WORK (workspace/output) COMPLEX*16 array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK (input) INTEGER
The dimension of the array WORK. LWORK >= 1. LWORK >=
2*MIN(M,N)+MAX(M,N). For good performance, LWORK should
generally be larger.
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ZGESVD(3F) ZGESVD(3F)
RWORK (workspace) DOUBLE PRECISION array, dimension
(max(3*min(M,N),5*min(M,N)-4)) On exit, if INFO > 0,
RWORK(1:MIN(M,N)-1) contains the unconverged superdiagonal
elements of an upper bidiagonal matrix B whose diagonal is in S
(not necessarily sorted). B satisfies A = U * B * VT, so it has
the same singular values as A, and singular vectors related by U
and VT.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if ZBDSQR did not converge, INFO specifies how many
superdiagonals of an intermediate bidiagonal form B did not
converge to zero. See the description of RWORK above for details.
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