CBDSQR(3F) CBDSQR(3F)
CBDSQR - compute the singular value decomposition (SVD) of a real N-by-N
(upper or lower) bidiagonal matrix B
SUBROUTINE CBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
LDC, RWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
REAL D( * ), E( * ), RWORK( * )
COMPLEX C( LDC, * ), U( LDU, * ), VT( LDVT, * )
CBDSQR computes the singular value decomposition (SVD) of a real N-by-N
(upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the
transpose of P), where S is a diagonal matrix with non-negative diagonal
elements (the singular values of B), and Q and P are orthogonal matrices.
The routine computes S, and optionally computes U * Q, P' * VT, or Q' *
C, for given complex input matrices U, VT, and C.
See "Computing Small Singular Values of Bidiagonal Matrices With
Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK
Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp.
873-912, Sept 1990) and
"Accurate singular values and differential qd algorithms," by B. Parlett
and V. Fernando, Technical Report CPAM-554, Mathematics Department,
University of California at Berkeley, July 1992 for a detailed
description of the algorithm.
UPLO (input) CHARACTER*1
= 'U': B is upper bidiagonal;
= 'L': B is lower bidiagonal.
N (input) INTEGER
The order of the matrix B. N >= 0.
NCVT (input) INTEGER
The number of columns of the matrix VT. NCVT >= 0.
NRU (input) INTEGER
The number of rows of the matrix U. NRU >= 0.
NCC (input) INTEGER
The number of columns of the matrix C. NCC >= 0.
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CBDSQR(3F) CBDSQR(3F)
D (input/output) REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B. On
exit, if INFO=0, the singular values of B in decreasing order.
E (input/output) REAL array, dimension (N)
On entry, the elements of E contain the offdiagonal elements of
of the bidiagonal matrix whose SVD is desired. On normal exit
(INFO = 0), E is destroyed. If the algorithm does not converge
(INFO > 0), D and E will contain the diagonal and superdiagonal
elements of a bidiagonal matrix orthogonally equivalent to the
one given as input. E(N) is used for workspace.
VT (input/output) COMPLEX array, dimension (LDVT, NCVT)
On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by
P' * VT. VT is not referenced if NCVT = 0.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= max(1,N) if NCVT
> 0; LDVT >= 1 if NCVT = 0.
U (input/output) COMPLEX array, dimension (LDU, N)
On entry, an NRU-by-N matrix U. On exit, U is overwritten by U *
Q. U is not referenced if NRU = 0.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,NRU).
C (input/output) COMPLEX array, dimension (LDC, NCC)
On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q'
* C. C is not referenced if NCC = 0.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1,N) if NCC >
0; LDC >=1 if NCC = 0.
RWORK (workspace) REAL array, dimension
2*N if only singular values wanted (NCVT = NRU = NCC = 0) max(
1, 4*N-4 ) otherwise
INFO (output) INTEGER
= 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
> 0: the algorithm did not converge; D and E contain the
elements of a bidiagonal matrix which is orthogonally similar to
the input matrix B; if INFO = i, i elements of E have not
converged to zero.
TOLMUL REAL, default = max(10,min(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop. If it
is positive, TOLMUL*EPS is the desired relative precision in the
computed singular values. If it is negative,
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CBDSQR(3F) CBDSQR(3F)
abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the
computed singular values (corresponds to relative accuracy
abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL)
should be between 1 and 1/EPS, and preferably between 10 (for
fast convergence) and .1/EPS (for there to be some accuracy in
the results). Default is to lose at either one eighth or 2 of
the available decimal digits in each computed singular value
(whichever is smaller).
MAXITR INTEGER, default = 6
MAXITR controls the maximum number of passes of the algorithm
through its inner loop. The algorithms stops (and so fails to
converge) if the number of passes through the inner loop exceeds
MAXITR*N**2.
CBDSQR(3F) CBDSQR(3F)
CBDSQR - compute the singular value decomposition (SVD) of a real N-by-N
(upper or lower) bidiagonal matrix B
SUBROUTINE CBDSQR( UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
LDC, RWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDC, LDU, LDVT, N, NCC, NCVT, NRU
REAL D( * ), E( * ), RWORK( * )
COMPLEX C( LDC, * ), U( LDU, * ), VT( LDVT, * )
CBDSQR computes the singular value decomposition (SVD) of a real N-by-N
(upper or lower) bidiagonal matrix B: B = Q * S * P' (P' denotes the
transpose of P), where S is a diagonal matrix with non-negative diagonal
elements (the singular values of B), and Q and P are orthogonal matrices.
The routine computes S, and optionally computes U * Q, P' * VT, or Q' *
C, for given complex input matrices U, VT, and C.
See "Computing Small Singular Values of Bidiagonal Matrices With
Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, LAPACK
Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, no. 5, pp.
873-912, Sept 1990) and
"Accurate singular values and differential qd algorithms," by B. Parlett
and V. Fernando, Technical Report CPAM-554, Mathematics Department,
University of California at Berkeley, July 1992 for a detailed
description of the algorithm.
UPLO (input) CHARACTER*1
= 'U': B is upper bidiagonal;
= 'L': B is lower bidiagonal.
N (input) INTEGER
The order of the matrix B. N >= 0.
NCVT (input) INTEGER
The number of columns of the matrix VT. NCVT >= 0.
NRU (input) INTEGER
The number of rows of the matrix U. NRU >= 0.
NCC (input) INTEGER
The number of columns of the matrix C. NCC >= 0.
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CBDSQR(3F) CBDSQR(3F)
D (input/output) REAL array, dimension (N)
On entry, the n diagonal elements of the bidiagonal matrix B. On
exit, if INFO=0, the singular values of B in decreasing order.
E (input/output) REAL array, dimension (N)
On entry, the elements of E contain the offdiagonal elements of
of the bidiagonal matrix whose SVD is desired. On normal exit
(INFO = 0), E is destroyed. If the algorithm does not converge
(INFO > 0), D and E will contain the diagonal and superdiagonal
elements of a bidiagonal matrix orthogonally equivalent to the
one given as input. E(N) is used for workspace.
VT (input/output) COMPLEX array, dimension (LDVT, NCVT)
On entry, an N-by-NCVT matrix VT. On exit, VT is overwritten by
P' * VT. VT is not referenced if NCVT = 0.
LDVT (input) INTEGER
The leading dimension of the array VT. LDVT >= max(1,N) if NCVT
> 0; LDVT >= 1 if NCVT = 0.
U (input/output) COMPLEX array, dimension (LDU, N)
On entry, an NRU-by-N matrix U. On exit, U is overwritten by U *
Q. U is not referenced if NRU = 0.
LDU (input) INTEGER
The leading dimension of the array U. LDU >= max(1,NRU).
C (input/output) COMPLEX array, dimension (LDC, NCC)
On entry, an N-by-NCC matrix C. On exit, C is overwritten by Q'
* C. C is not referenced if NCC = 0.
LDC (input) INTEGER
The leading dimension of the array C. LDC >= max(1,N) if NCC >
0; LDC >=1 if NCC = 0.
RWORK (workspace) REAL array, dimension
2*N if only singular values wanted (NCVT = NRU = NCC = 0) max(
1, 4*N-4 ) otherwise
INFO (output) INTEGER
= 0: successful exit
< 0: If INFO = -i, the i-th argument had an illegal value
> 0: the algorithm did not converge; D and E contain the
elements of a bidiagonal matrix which is orthogonally similar to
the input matrix B; if INFO = i, i elements of E have not
converged to zero.
TOLMUL REAL, default = max(10,min(100,EPS**(-1/8)))
TOLMUL controls the convergence criterion of the QR loop. If it
is positive, TOLMUL*EPS is the desired relative precision in the
computed singular values. If it is negative,
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CBDSQR(3F) CBDSQR(3F)
abs(TOLMUL*EPS*sigma_max) is the desired absolute accuracy in the
computed singular values (corresponds to relative accuracy
abs(TOLMUL*EPS) in the largest singular value. abs(TOLMUL)
should be between 1 and 1/EPS, and preferably between 10 (for
fast convergence) and .1/EPS (for there to be some accuracy in
the results). Default is to lose at either one eighth or 2 of
the available decimal digits in each computed singular value
(whichever is smaller).
MAXITR INTEGER, default = 6
MAXITR controls the maximum number of passes of the algorithm
through its inner loop. The algorithms stops (and so fails to
converge) if the number of passes through the inner loop exceeds
MAXITR*N**2.
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