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| gesvdq(3) | LAPACK | gesvdq(3) |
NAME¶
gesvdq - gesvdq: SVD, QR with pivoting
SYNOPSIS¶
Functions¶
subroutine cgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a,
lda, s, u, ldu, v, ldv, numrank, iwork, liwork, cwork, lcwork, rwork,
lrwork, info)
CGESVDQ computes the singular value decomposition (SVD) with a
QR-Preconditioned QR SVD Method for GE matrices subroutine
dgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v,
ldv, numrank, iwork, liwork, work, lwork, rwork, lrwork, info)
DGESVDQ computes the singular value decomposition (SVD) with a
QR-Preconditioned QR SVD Method for GE matrices subroutine
sgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v,
ldv, numrank, iwork, liwork, work, lwork, rwork, lrwork, info)
SGESVDQ computes the singular value decomposition (SVD) with a
QR-Preconditioned QR SVD Method for GE matrices subroutine
zgesvdq (joba, jobp, jobr, jobu, jobv, m, n, a, lda, s, u, ldu, v,
ldv, numrank, iwork, liwork, cwork, lcwork, rwork, lrwork, info)
ZGESVDQ computes the singular value decomposition (SVD) with a
QR-Preconditioned QR SVD Method for GE matrices
Detailed Description¶
Function Documentation¶
subroutine cgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, complex, dimension( lda, * ) a, integer lda, real, dimension( * ) s, complex, dimension( ldu, * ) u, integer ldu, complex, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, complex, dimension( * ) cwork, integer lcwork, real, dimension( * ) rwork, integer lrwork, integer info)¶
CGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
Purpose:
CGESVDQ computes the singular value decomposition (SVD) of a complex
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N unitary matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
Parameters
JOBA
JOBA is CHARACTER*1
Specifies the level of accuracy in the computed SVD
= 'A' The requested accuracy corresponds to having the backward
error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
where EPS = SLAMCH('Epsilon'). This authorises CGESVDQ to
truncate the computed triangular factor in a rank revealing
QR factorization whenever the truncated part is below the
threshold of the order of EPS * ||A||_F. This is aggressive
truncation level.
= 'M' Similarly as with 'A', but the truncation is more gentle: it
is allowed only when there is a drop on the diagonal of the
triangular factor in the QR factorization. This is medium
truncation level.
= 'H' High accuracy requested. No numerical rank determination based
on the rank revealing QR factorization is attempted.
= 'E' Same as 'H', and in addition the condition number of column
scaled A is estimated and returned in RWORK(1).
N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)
JOBP
JOBP is CHARACTER*1
= 'P' The rows of A are ordered in decreasing order with respect to
||A(i,:)||_\infty. This enhances numerical accuracy at the cost
of extra data movement. Recommended for numerical robustness.
= 'N' No row pivoting.
JOBR
JOBR is CHARACTER*1
= 'T' After the initial pivoted QR factorization, CGESVD is applied to
the adjoint R**H of the computed triangular factor R. This involves
some extra data movement (matrix transpositions). Useful for
experiments, research and development.
= 'N' The triangular factor R is given as input to CGESVD. This may be
preferred as it involves less data movement.
JOBU
JOBU is CHARACTER*1
= 'A' All M left singular vectors are computed and returned in the
matrix U. See the description of U.
= 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
in the matrix U. See the description of U.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
vectors are computed and returned in the matrix U.
= 'F' The N left singular vectors are returned in factored form as the
product of the Q factor from the initial QR factorization and the
N left singular vectors of (R**H , 0)**H. If row pivoting is used,
then the necessary information on the row pivoting is stored in
IWORK(N+1:N+M-1).
= 'N' The left singular vectors are not computed.
JOBV
JOBV is CHARACTER*1
= 'A', 'V' All N right singular vectors are computed and returned in
the matrix V.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
vectors are computed and returned in the matrix V. This option is
allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
= 'N' The right singular vectors are not computed.
M
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N
N is INTEGER
The number of columns of the input matrix A. M >= N >= 0.
A
A is COMPLEX array of dimensions LDA x N
On entry, the input matrix A.
On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
the Householder vectors as stored by CGEQP3. If JOBU = 'F', these Householder
vectors together with CWORK(1:N) can be used to restore the Q factors from
the initial pivoted QR factorization of A. See the description of U.
LDA
LDA is INTEGER.
The leading dimension of the array A. LDA >= max(1,M).
S
S is REAL array of dimension N.
The singular values of A, ordered so that S(i) >= S(i+1).
U
U is COMPLEX array, dimension
LDU x M if JOBU = 'A'; see the description of LDU. In this case,
on exit, U contains the M left singular vectors.
LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
case, U contains the leading N or the leading NUMRANK left singular vectors.
LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
contains N x N unitary matrix that can be used to form the left
singular vectors.
If JOBU = 'N', U is not referenced.
LDU
LDU is INTEGER.
The leading dimension of the array U.
If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
If JOBU = 'F', LDU >= max(1,N).
Otherwise, LDU >= 1.
V
V is COMPLEX array, dimension
LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
If JOBV = 'A', or 'V', V contains the N-by-N unitary matrix V**H;
If JOBV = 'R', V contains the first NUMRANK rows of V**H (the right
singular vectors, stored rowwise, of the NUMRANK largest singular values).
If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
If JOBV = 'N', and JOBA.NE.'E', V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V.
If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
Otherwise, LDV >= 1.
NUMRANK
NUMRANK is INTEGER
NUMRANK is the numerical rank first determined after the rank
revealing QR factorization, following the strategy specified by the
value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
leading singular values and vectors are then requested in the call
of CGESVD. The final value of NUMRANK might be further reduced if
some singular values are computed as zeros.
IWORK
IWORK is INTEGER array, dimension (max(1, LIWORK)).
On exit, IWORK(1:N) contains column pivoting permutation of the
rank revealing QR factorization.
If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
of row swaps used in row pivoting. These can be used to restore the
left singular vectors in the case JOBU = 'F'.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
IWORK(1) returns the minimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
LIWORK >= N + M - 1, if JOBP = 'P';
LIWORK >= N if JOBP = 'N'.
If LIWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.
CWORK
CWORK is COMPLEX array, dimension (max(2, LCWORK)), used as a workspace.
On exit, if, on entry, LCWORK.NE.-1, CWORK(1:N) contains parameters
needed to recover the Q factor from the QR factorization computed by
CGEQP3.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
CWORK(1) returns the optimal LCWORK, and
CWORK(2) returns the minimal LCWORK.
LCWORK
LCWORK is INTEGER
The dimension of the array CWORK. It is determined as follows:
Let LWQP3 = N+1, LWCON = 2*N, and let
LWUNQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U'
{ MAX( M, 1 ), if JOBU = 'A'
LWSVD = MAX( 3*N, 1 )
LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
Then the minimal value of LCWORK is:
= MAX( N + LWQP3, LWSVD ) if only the singular values are needed;
= MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
and a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD, LWUNQ ) if the singular values and the left
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the singular values and the left
singular vectors are requested, and also
a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD ) if the singular values and the right
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
singular vectors are requested, and also
a scaled condition etimate requested;
= N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
independent of JOBR;
= N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
JOBV = 'R' and, also a scaled condition
estimate requested; independent of JOBR;
= MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
if the full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N', and also a scaled condition number estimate
requested.
= MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
if the full SVD is requested with JOBV = 'A', 'V' and
JOBR ='T', and also a scaled condition number estimate
requested.
Finally, LCWORK must be at least two: LCWORK = MAX( 2, LCWORK ).
If LCWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.
RWORK
RWORK is REAL array, dimension (max(1, LRWORK)).
On exit,
1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
number of column scaled A. If A = C * D where D is diagonal and C
has unit columns in the Euclidean norm, then, assuming full column rank,
N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
Otherwise, RWORK(1) = -1.
2. RWORK(2) contains the number of singular values computed as
exact zeros in CGESVD applied to the upper triangular or trapezoidal
R (from the initial QR factorization). In case of early exit (no call to
CGESVD, such as in the case of zero matrix) RWORK(2) = -1.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
RWORK(1) returns the minimal LRWORK.
LRWORK
LRWORK is INTEGER.
The dimension of the array RWORK.
If JOBP ='P', then LRWORK >= MAX(2, M, 5*N);
Otherwise, LRWORK >= MAX(2, 5*N).
If LRWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if CBDSQR did not converge, INFO specifies how many superdiagonals
of an intermediate bidiagonal form B (computed in CGESVD) did not
converge to zero.
Further Details:
1. The data movement (matrix transpose) is coded using simple nested
DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
Those DO-loops are easily identified in this source code - by the CONTINUE
statements labeled with 11**. In an optimized version of this code, the
nested DO loops should be replaced with calls to an optimized subroutine.
2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
column norm overflow. This is the minial precaution and it is left to the
SVD routine (CGESVD) to do its own preemptive scaling if potential over-
or underflows are detected. To avoid repeated scanning of the array A,
an optimal implementation would do all necessary scaling before calling
CGESVD and the scaling in CGESVD can be switched off.
3. Other comments related to code optimization are given in comments in the
code, enclosed in [[double brackets]].
Bugs, examples and comments
Please report all bugs and send interesting examples and/or comments to
drmac@math.hr. Thank you.
References
[1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
44(1): 11:1-11:30 (2017)
SIGMA library, xGESVDQ section updated February 2016.
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Contributors:
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine dgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, double precision, dimension( ldu, * ) u, integer ldu, double precision, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, double precision, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer info)¶
DGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
Purpose:
DGESVDQ computes the singular value decomposition (SVD) of a real
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
Parameters
JOBA
JOBA is CHARACTER*1
Specifies the level of accuracy in the computed SVD
= 'A' The requested accuracy corresponds to having the backward
error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
where EPS = DLAMCH('Epsilon'). This authorises DGESVDQ to
truncate the computed triangular factor in a rank revealing
QR factorization whenever the truncated part is below the
threshold of the order of EPS * ||A||_F. This is aggressive
truncation level.
= 'M' Similarly as with 'A', but the truncation is more gentle: it
is allowed only when there is a drop on the diagonal of the
triangular factor in the QR factorization. This is medium
truncation level.
= 'H' High accuracy requested. No numerical rank determination based
on the rank revealing QR factorization is attempted.
= 'E' Same as 'H', and in addition the condition number of column
scaled A is estimated and returned in RWORK(1).
N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)
JOBP
JOBP is CHARACTER*1
= 'P' The rows of A are ordered in decreasing order with respect to
||A(i,:)||_\infty. This enhances numerical accuracy at the cost
of extra data movement. Recommended for numerical robustness.
= 'N' No row pivoting.
JOBR
JOBR is CHARACTER*1
= 'T' After the initial pivoted QR factorization, DGESVD is applied to
the transposed R**T of the computed triangular factor R. This involves
some extra data movement (matrix transpositions). Useful for
experiments, research and development.
= 'N' The triangular factor R is given as input to DGESVD. This may be
preferred as it involves less data movement.
JOBU
JOBU is CHARACTER*1
= 'A' All M left singular vectors are computed and returned in the
matrix U. See the description of U.
= 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
in the matrix U. See the description of U.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
vectors are computed and returned in the matrix U.
= 'F' The N left singular vectors are returned in factored form as the
product of the Q factor from the initial QR factorization and the
N left singular vectors of (R**T , 0)**T. If row pivoting is used,
then the necessary information on the row pivoting is stored in
IWORK(N+1:N+M-1).
= 'N' The left singular vectors are not computed.
JOBV
JOBV is CHARACTER*1
= 'A', 'V' All N right singular vectors are computed and returned in
the matrix V.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
vectors are computed and returned in the matrix V. This option is
allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
= 'N' The right singular vectors are not computed.
M
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N
N is INTEGER
The number of columns of the input matrix A. M >= N >= 0.
A
A is DOUBLE PRECISION array of dimensions LDA x N
On entry, the input matrix A.
On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
the Householder vectors as stored by DGEQP3. If JOBU = 'F', these Householder
vectors together with WORK(1:N) can be used to restore the Q factors from
the initial pivoted QR factorization of A. See the description of U.
LDA
LDA is INTEGER.
The leading dimension of the array A. LDA >= max(1,M).
S
S is DOUBLE PRECISION array of dimension N.
The singular values of A, ordered so that S(i) >= S(i+1).
U
U is DOUBLE PRECISION array, dimension
LDU x M if JOBU = 'A'; see the description of LDU. In this case,
on exit, U contains the M left singular vectors.
LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
case, U contains the leading N or the leading NUMRANK left singular vectors.
LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
contains N x N orthogonal matrix that can be used to form the left
singular vectors.
If JOBU = 'N', U is not referenced.
LDU
LDU is INTEGER.
The leading dimension of the array U.
If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
If JOBU = 'F', LDU >= max(1,N).
Otherwise, LDU >= 1.
V
V is DOUBLE PRECISION array, dimension
LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
If JOBV = 'A', or 'V', V contains the N-by-N orthogonal matrix V**T;
If JOBV = 'R', V contains the first NUMRANK rows of V**T (the right
singular vectors, stored rowwise, of the NUMRANK largest singular values).
If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
If JOBV = 'N', and JOBA.NE.'E', V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V.
If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
Otherwise, LDV >= 1.
NUMRANK
NUMRANK is INTEGER
NUMRANK is the numerical rank first determined after the rank
revealing QR factorization, following the strategy specified by the
value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
leading singular values and vectors are then requested in the call
of DGESVD. The final value of NUMRANK might be further reduced if
some singular values are computed as zeros.
IWORK
IWORK is INTEGER array, dimension (max(1, LIWORK)).
On exit, IWORK(1:N) contains column pivoting permutation of the
rank revealing QR factorization.
If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
of row swaps used in row pivoting. These can be used to restore the
left singular vectors in the case JOBU = 'F'.
If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
IWORK(1) returns the minimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
LIWORK >= N + M - 1, if JOBP = 'P' and JOBA .NE. 'E';
LIWORK >= N if JOBP = 'N' and JOBA .NE. 'E';
LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E';
LIWORK >= N + N if JOBP = 'N' and JOBA = 'E'.
If LIWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.
WORK
WORK is DOUBLE PRECISION array, dimension (max(2, LWORK)), used as a workspace.
On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters
needed to recover the Q factor from the QR factorization computed by
DGEQP3.
If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
WORK(1) returns the optimal LWORK, and
WORK(2) returns the minimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. It is determined as follows:
Let LWQP3 = 3*N+1, LWCON = 3*N, and let
LWORQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U'
{ MAX( M, 1 ), if JOBU = 'A'
LWSVD = MAX( 5*N, 1 )
LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ),
LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 )
Then the minimal value of LWORK is:
= MAX( N + LWQP3, LWSVD ) if only the singular values are needed;
= MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
and a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD, LWORQ ) if the singular values and the left
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the singular values and the left
singular vectors are requested, and also
a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD ) if the singular values and the right
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
singular vectors are requested, and also
a scaled condition etimate requested;
= N + MAX( LWQP3, LWSVD, LWORQ ) if the full SVD is requested with JOBV = 'R';
independent of JOBR;
= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested,
JOBV = 'R' and, also a scaled condition
estimate requested; independent of JOBR;
= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the
full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) )
if the full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N', and also a scaled condition number estimate
requested.
= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the
full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) )
if the full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='T', and also a scaled condition number estimate
requested.
Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.
RWORK
RWORK is DOUBLE PRECISION array, dimension (max(1, LRWORK)).
On exit,
1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
number of column scaled A. If A = C * D where D is diagonal and C
has unit columns in the Euclidean norm, then, assuming full column rank,
N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
Otherwise, RWORK(1) = -1.
2. RWORK(2) contains the number of singular values computed as
exact zeros in DGESVD applied to the upper triangular or trapezoidal
R (from the initial QR factorization). In case of early exit (no call to
DGESVD, such as in the case of zero matrix) RWORK(2) = -1.
If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
RWORK(1) returns the minimal LRWORK.
LRWORK
LRWORK is INTEGER.
The dimension of the array RWORK.
If JOBP ='P', then LRWORK >= MAX(2, M).
Otherwise, LRWORK >= 2
If LRWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if DBDSQR did not converge, INFO specifies how many superdiagonals
of an intermediate bidiagonal form B (computed in DGESVD) did not
converge to zero.
Further Details:
1. The data movement (matrix transpose) is coded using simple nested
DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
Those DO-loops are easily identified in this source code - by the CONTINUE
statements labeled with 11**. In an optimized version of this code, the
nested DO loops should be replaced with calls to an optimized subroutine.
2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
column norm overflow. This is the minial precaution and it is left to the
SVD routine (CGESVD) to do its own preemptive scaling if potential over-
or underflows are detected. To avoid repeated scanning of the array A,
an optimal implementation would do all necessary scaling before calling
CGESVD and the scaling in CGESVD can be switched off.
3. Other comments related to code optimization are given in comments in the
code, enclosed in [[double brackets]].
Bugs, examples and comments
Please report all bugs and send interesting examples and/or comments to
drmac@math.hr. Thank you.
References
[1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
44(1): 11:1-11:30 (2017)
SIGMA library, xGESVDQ section updated February 2016.
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Contributors:
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine sgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( * ) s, real, dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, real, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer info)¶
SGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
Purpose:
SGESVDQ computes the singular value decomposition (SVD) of a real
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N orthogonal matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
Parameters
JOBA
JOBA is CHARACTER*1
Specifies the level of accuracy in the computed SVD
= 'A' The requested accuracy corresponds to having the backward
error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
where EPS = SLAMCH('Epsilon'). This authorises CGESVDQ to
truncate the computed triangular factor in a rank revealing
QR factorization whenever the truncated part is below the
threshold of the order of EPS * ||A||_F. This is aggressive
truncation level.
= 'M' Similarly as with 'A', but the truncation is more gentle: it
is allowed only when there is a drop on the diagonal of the
triangular factor in the QR factorization. This is medium
truncation level.
= 'H' High accuracy requested. No numerical rank determination based
on the rank revealing QR factorization is attempted.
= 'E' Same as 'H', and in addition the condition number of column
scaled A is estimated and returned in RWORK(1).
N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)
JOBP
JOBP is CHARACTER*1
= 'P' The rows of A are ordered in decreasing order with respect to
||A(i,:)||_\infty. This enhances numerical accuracy at the cost
of extra data movement. Recommended for numerical robustness.
= 'N' No row pivoting.
JOBR
JOBR is CHARACTER*1
= 'T' After the initial pivoted QR factorization, SGESVD is applied to
the transposed R**T of the computed triangular factor R. This involves
some extra data movement (matrix transpositions). Useful for
experiments, research and development.
= 'N' The triangular factor R is given as input to SGESVD. This may be
preferred as it involves less data movement.
JOBU
JOBU is CHARACTER*1
= 'A' All M left singular vectors are computed and returned in the
matrix U. See the description of U.
= 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
in the matrix U. See the description of U.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
vectors are computed and returned in the matrix U.
= 'F' The N left singular vectors are returned in factored form as the
product of the Q factor from the initial QR factorization and the
N left singular vectors of (R**T , 0)**T. If row pivoting is used,
then the necessary information on the row pivoting is stored in
IWORK(N+1:N+M-1).
= 'N' The left singular vectors are not computed.
JOBV
JOBV is CHARACTER*1
= 'A', 'V' All N right singular vectors are computed and returned in
the matrix V.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
vectors are computed and returned in the matrix V. This option is
allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
= 'N' The right singular vectors are not computed.
M
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N
N is INTEGER
The number of columns of the input matrix A. M >= N >= 0.
A
A is REAL array of dimensions LDA x N
On entry, the input matrix A.
On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
the Householder vectors as stored by SGEQP3. If JOBU = 'F', these Householder
vectors together with WORK(1:N) can be used to restore the Q factors from
the initial pivoted QR factorization of A. See the description of U.
LDA
LDA is INTEGER.
The leading dimension of the array A. LDA >= max(1,M).
S
S is REAL array of dimension N.
The singular values of A, ordered so that S(i) >= S(i+1).
U
U is REAL array, dimension
LDU x M if JOBU = 'A'; see the description of LDU. In this case,
on exit, U contains the M left singular vectors.
LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
case, U contains the leading N or the leading NUMRANK left singular vectors.
LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
contains N x N orthogonal matrix that can be used to form the left
singular vectors.
If JOBU = 'N', U is not referenced.
LDU
LDU is INTEGER.
The leading dimension of the array U.
If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
If JOBU = 'F', LDU >= max(1,N).
Otherwise, LDU >= 1.
V
V is REAL array, dimension
LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
If JOBV = 'A', or 'V', V contains the N-by-N orthogonal matrix V**T;
If JOBV = 'R', V contains the first NUMRANK rows of V**T (the right
singular vectors, stored rowwise, of the NUMRANK largest singular values).
If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
If JOBV = 'N', and JOBA.NE.'E', V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V.
If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
Otherwise, LDV >= 1.
NUMRANK
NUMRANK is INTEGER
NUMRANK is the numerical rank first determined after the rank
revealing QR factorization, following the strategy specified by the
value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
leading singular values and vectors are then requested in the call
of SGESVD. The final value of NUMRANK might be further reduced if
some singular values are computed as zeros.
IWORK
IWORK is INTEGER array, dimension (max(1, LIWORK)).
On exit, IWORK(1:N) contains column pivoting permutation of the
rank revealing QR factorization.
If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
of row swaps used in row pivoting. These can be used to restore the
left singular vectors in the case JOBU = 'F'.
If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
IWORK(1) returns the minimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
LIWORK >= N + M - 1, if JOBP = 'P' and JOBA .NE. 'E';
LIWORK >= N if JOBP = 'N' and JOBA .NE. 'E';
LIWORK >= N + M - 1 + N, if JOBP = 'P' and JOBA = 'E';
LIWORK >= N + N if JOBP = 'N' and JOBA = 'E'.
If LIWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.
WORK
WORK is REAL array, dimension (max(2, LWORK)), used as a workspace.
On exit, if, on entry, LWORK.NE.-1, WORK(1:N) contains parameters
needed to recover the Q factor from the QR factorization computed by
SGEQP3.
If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
WORK(1) returns the optimal LWORK, and
WORK(2) returns the minimal LWORK.
LWORK
LWORK is INTEGER
The dimension of the array WORK. It is determined as follows:
Let LWQP3 = 3*N+1, LWCON = 3*N, and let
LWORQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U'
{ MAX( M, 1 ), if JOBU = 'A'
LWSVD = MAX( 5*N, 1 )
LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 5*(N/2), 1 ), LWORLQ = MAX( N, 1 ),
LWQRF = MAX( N/2, 1 ), LWORQ2 = MAX( N, 1 )
Then the minimal value of LWORK is:
= MAX( N + LWQP3, LWSVD ) if only the singular values are needed;
= MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
and a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD, LWORQ ) if the singular values and the left
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the singular values and the left
singular vectors are requested, and also
a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD ) if the singular values and the right
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
singular vectors are requested, and also
a scaled condition etimate requested;
= N + MAX( LWQP3, LWSVD, LWORQ ) if the full SVD is requested with JOBV = 'R';
independent of JOBR;
= N + MAX( LWQP3, LWCON, LWSVD, LWORQ ) if the full SVD is requested,
JOBV = 'R' and, also a scaled condition
estimate requested; independent of JOBR;
= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ) ) if the
full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWORLQ, LWORQ ) )
if the full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N', and also a scaled condition number estimate
requested.
= MAX( N + MAX( LWQP3, LWSVD, LWORQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) ) if the
full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWORQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWORQ2, LWORQ ) )
if the full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='T', and also a scaled condition number estimate
requested.
Finally, LWORK must be at least two: LWORK = MAX( 2, LWORK ).
If LWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.
RWORK
RWORK is REAL array, dimension (max(1, LRWORK)).
On exit,
1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
number of column scaled A. If A = C * D where D is diagonal and C
has unit columns in the Euclidean norm, then, assuming full column rank,
N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
Otherwise, RWORK(1) = -1.
2. RWORK(2) contains the number of singular values computed as
exact zeros in SGESVD applied to the upper triangular or trapezoidal
R (from the initial QR factorization). In case of early exit (no call to
SGESVD, such as in the case of zero matrix) RWORK(2) = -1.
If LIWORK, LWORK, or LRWORK = -1, then on exit, if INFO = 0,
RWORK(1) returns the minimal LRWORK.
LRWORK
LRWORK is INTEGER.
The dimension of the array RWORK.
If JOBP ='P', then LRWORK >= MAX(2, M).
Otherwise, LRWORK >= 2
If LRWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the WORK, IWORK, and RWORK arrays, and no error
message related to LWORK is issued by XERBLA.
INFO
INFO is INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if SBDSQR did not converge, INFO specifies how many superdiagonals
of an intermediate bidiagonal form B (computed in SGESVD) did not
converge to zero.
Further Details:
1. The data movement (matrix transpose) is coded using simple nested
DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
Those DO-loops are easily identified in this source code - by the CONTINUE
statements labeled with 11**. In an optimized version of this code, the
nested DO loops should be replaced with calls to an optimized subroutine.
2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
column norm overflow. This is the minial precaution and it is left to the
SVD routine (CGESVD) to do its own preemptive scaling if potential over-
or underflows are detected. To avoid repeated scanning of the array A,
an optimal implementation would do all necessary scaling before calling
CGESVD and the scaling in CGESVD can be switched off.
3. Other comments related to code optimization are given in comments in the
code, enclosed in [[double brackets]].
Bugs, examples and comments
Please report all bugs and send interesting examples and/or comments to
drmac@math.hr. Thank you.
References
[1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
44(1): 11:1-11:30 (2017)
SIGMA library, xGESVDQ section updated February 2016.
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Contributors:
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
subroutine zgesvdq (character joba, character jobp, character jobr, character jobu, character jobv, integer m, integer n, complex*16, dimension( lda, * ) a, integer lda, double precision, dimension( * ) s, complex*16, dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v, integer ldv, integer numrank, integer, dimension( * ) iwork, integer liwork, complex*16, dimension( * ) cwork, integer lcwork, double precision, dimension( * ) rwork, integer lrwork, integer info)¶
ZGESVDQ computes the singular value decomposition (SVD) with a QR-Preconditioned QR SVD Method for GE matrices
Purpose:
ZCGESVDQ computes the singular value decomposition (SVD) of a complex
M-by-N matrix A, where M >= N. The SVD of A is written as
[++] [xx] [x0] [xx]
A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
[++] [xx]
where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
matrix, and V is an N-by-N unitary matrix. The diagonal elements
of SIGMA are the singular values of A. The columns of U and V are the
left and the right singular vectors of A, respectively.
Parameters
JOBA
JOBA is CHARACTER*1
Specifies the level of accuracy in the computed SVD
= 'A' The requested accuracy corresponds to having the backward
error bounded by || delta A ||_F <= f(m,n) * EPS * || A ||_F,
where EPS = DLAMCH('Epsilon'). This authorises ZGESVDQ to
truncate the computed triangular factor in a rank revealing
QR factorization whenever the truncated part is below the
threshold of the order of EPS * ||A||_F. This is aggressive
truncation level.
= 'M' Similarly as with 'A', but the truncation is more gentle: it
is allowed only when there is a drop on the diagonal of the
triangular factor in the QR factorization. This is medium
truncation level.
= 'H' High accuracy requested. No numerical rank determination based
on the rank revealing QR factorization is attempted.
= 'E' Same as 'H', and in addition the condition number of column
scaled A is estimated and returned in RWORK(1).
N^(-1/4)*RWORK(1) <= ||pinv(A_scaled)||_2 <= N^(1/4)*RWORK(1)
JOBP
JOBP is CHARACTER*1
= 'P' The rows of A are ordered in decreasing order with respect to
||A(i,:)||_\infty. This enhances numerical accuracy at the cost
of extra data movement. Recommended for numerical robustness.
= 'N' No row pivoting.
JOBR
JOBR is CHARACTER*1
= 'T' After the initial pivoted QR factorization, ZGESVD is applied to
the adjoint R**H of the computed triangular factor R. This involves
some extra data movement (matrix transpositions). Useful for
experiments, research and development.
= 'N' The triangular factor R is given as input to CGESVD. This may be
preferred as it involves less data movement.
JOBU
JOBU is CHARACTER*1
= 'A' All M left singular vectors are computed and returned in the
matrix U. See the description of U.
= 'S' or 'U' N = min(M,N) left singular vectors are computed and returned
in the matrix U. See the description of U.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK left singular
vectors are computed and returned in the matrix U.
= 'F' The N left singular vectors are returned in factored form as the
product of the Q factor from the initial QR factorization and the
N left singular vectors of (R**H , 0)**H. If row pivoting is used,
then the necessary information on the row pivoting is stored in
IWORK(N+1:N+M-1).
= 'N' The left singular vectors are not computed.
JOBV
JOBV is CHARACTER*1
= 'A', 'V' All N right singular vectors are computed and returned in
the matrix V.
= 'R' Numerical rank NUMRANK is determined and only NUMRANK right singular
vectors are computed and returned in the matrix V. This option is
allowed only if JOBU = 'R' or JOBU = 'N'; otherwise it is illegal.
= 'N' The right singular vectors are not computed.
M
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N
N is INTEGER
The number of columns of the input matrix A. M >= N >= 0.
A
A is COMPLEX*16 array of dimensions LDA x N
On entry, the input matrix A.
On exit, if JOBU .NE. 'N' or JOBV .NE. 'N', the lower triangle of A contains
the Householder vectors as stored by ZGEQP3. If JOBU = 'F', these Householder
vectors together with CWORK(1:N) can be used to restore the Q factors from
the initial pivoted QR factorization of A. See the description of U.
LDA
LDA is INTEGER.
The leading dimension of the array A. LDA >= max(1,M).
S
S is DOUBLE PRECISION array of dimension N.
The singular values of A, ordered so that S(i) >= S(i+1).
U
U is COMPLEX*16 array, dimension
LDU x M if JOBU = 'A'; see the description of LDU. In this case,
on exit, U contains the M left singular vectors.
LDU x N if JOBU = 'S', 'U', 'R' ; see the description of LDU. In this
case, U contains the leading N or the leading NUMRANK left singular vectors.
LDU x N if JOBU = 'F' ; see the description of LDU. In this case U
contains N x N unitary matrix that can be used to form the left
singular vectors.
If JOBU = 'N', U is not referenced.
LDU
LDU is INTEGER.
The leading dimension of the array U.
If JOBU = 'A', 'S', 'U', 'R', LDU >= max(1,M).
If JOBU = 'F', LDU >= max(1,N).
Otherwise, LDU >= 1.
V
V is COMPLEX*16 array, dimension
LDV x N if JOBV = 'A', 'V', 'R' or if JOBA = 'E' .
If JOBV = 'A', or 'V', V contains the N-by-N unitary matrix V**H;
If JOBV = 'R', V contains the first NUMRANK rows of V**H (the right
singular vectors, stored rowwise, of the NUMRANK largest singular values).
If JOBV = 'N' and JOBA = 'E', V is used as a workspace.
If JOBV = 'N', and JOBA.NE.'E', V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V.
If JOBV = 'A', 'V', 'R', or JOBA = 'E', LDV >= max(1,N).
Otherwise, LDV >= 1.
NUMRANK
NUMRANK is INTEGER
NUMRANK is the numerical rank first determined after the rank
revealing QR factorization, following the strategy specified by the
value of JOBA. If JOBV = 'R' and JOBU = 'R', only NUMRANK
leading singular values and vectors are then requested in the call
of CGESVD. The final value of NUMRANK might be further reduced if
some singular values are computed as zeros.
IWORK
IWORK is INTEGER array, dimension (max(1, LIWORK)).
On exit, IWORK(1:N) contains column pivoting permutation of the
rank revealing QR factorization.
If JOBP = 'P', IWORK(N+1:N+M-1) contains the indices of the sequence
of row swaps used in row pivoting. These can be used to restore the
left singular vectors in the case JOBU = 'F'.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
IWORK(1) returns the minimal LIWORK.
LIWORK
LIWORK is INTEGER
The dimension of the array IWORK.
LIWORK >= N + M - 1, if JOBP = 'P';
LIWORK >= N if JOBP = 'N'.
If LIWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.
CWORK
CWORK is COMPLEX*12 array, dimension (max(2, LCWORK)), used as a workspace.
On exit, if, on entry, LCWORK.NE.-1, CWORK(1:N) contains parameters
needed to recover the Q factor from the QR factorization computed by
ZGEQP3.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
CWORK(1) returns the optimal LCWORK, and
CWORK(2) returns the minimal LCWORK.
LCWORK
LCWORK is INTEGER
The dimension of the array CWORK. It is determined as follows:
Let LWQP3 = N+1, LWCON = 2*N, and let
LWUNQ = { MAX( N, 1 ), if JOBU = 'R', 'S', or 'U'
{ MAX( M, 1 ), if JOBU = 'A'
LWSVD = MAX( 3*N, 1 )
LWLQF = MAX( N/2, 1 ), LWSVD2 = MAX( 3*(N/2), 1 ), LWUNLQ = MAX( N, 1 ),
LWQRF = MAX( N/2, 1 ), LWUNQ2 = MAX( N, 1 )
Then the minimal value of LCWORK is:
= MAX( N + LWQP3, LWSVD ) if only the singular values are needed;
= MAX( N + LWQP3, LWCON, LWSVD ) if only the singular values are needed,
and a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD, LWUNQ ) if the singular values and the left
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the singular values and the left
singular vectors are requested, and also
a scaled condition estimate requested;
= N + MAX( LWQP3, LWSVD ) if the singular values and the right
singular vectors are requested;
= N + MAX( LWQP3, LWCON, LWSVD ) if the singular values and the right
singular vectors are requested, and also
a scaled condition etimate requested;
= N + MAX( LWQP3, LWSVD, LWUNQ ) if the full SVD is requested with JOBV = 'R';
independent of JOBR;
= N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ) if the full SVD is requested,
JOBV = 'R' and, also a scaled condition
estimate requested; independent of JOBR;
= MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ) ) if the
full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWLQF, N/2+LWSVD2, N/2+LWUNLQ, LWUNQ ) )
if the full SVD is requested with JOBV = 'A' or 'V', and
JOBR ='N', and also a scaled condition number estimate
requested.
= MAX( N + MAX( LWQP3, LWSVD, LWUNQ ),
N + MAX( LWQP3, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) ) if the
full SVD is requested with JOBV = 'A', 'V', and JOBR ='T'
= MAX( N + MAX( LWQP3, LWCON, LWSVD, LWUNQ ),
N + MAX( LWQP3, LWCON, N/2+LWQRF, N/2+LWSVD2, N/2+LWUNQ2, LWUNQ ) )
if the full SVD is requested with JOBV = 'A', 'V' and
JOBR ='T', and also a scaled condition number estimate
requested.
Finally, LCWORK must be at least two: LCWORK = MAX( 2, LCWORK ).
If LCWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.
RWORK
RWORK is DOUBLE PRECISION array, dimension (max(1, LRWORK)).
On exit,
1. If JOBA = 'E', RWORK(1) contains an estimate of the condition
number of column scaled A. If A = C * D where D is diagonal and C
has unit columns in the Euclidean norm, then, assuming full column rank,
N^(-1/4) * RWORK(1) <= ||pinv(C)||_2 <= N^(1/4) * RWORK(1).
Otherwise, RWORK(1) = -1.
2. RWORK(2) contains the number of singular values computed as
exact zeros in ZGESVD applied to the upper triangular or trapezoidal
R (from the initial QR factorization). In case of early exit (no call to
ZGESVD, such as in the case of zero matrix) RWORK(2) = -1.
If LIWORK, LCWORK, or LRWORK = -1, then on exit, if INFO = 0,
RWORK(1) returns the minimal LRWORK.
LRWORK
LRWORK is INTEGER.
The dimension of the array RWORK.
If JOBP ='P', then LRWORK >= MAX(2, M, 5*N);
Otherwise, LRWORK >= MAX(2, 5*N).
If LRWORK = -1, then a workspace query is assumed; the routine
only calculates and returns the optimal and minimal sizes
for the CWORK, IWORK, and RWORK arrays, and no error
message related to LCWORK is issued by XERBLA.
INFO
INFO is 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 (computed in ZGESVD) did not
converge to zero.
Further Details:
1. The data movement (matrix transpose) is coded using simple nested
DO-loops because BLAS and LAPACK do not provide corresponding subroutines.
Those DO-loops are easily identified in this source code - by the CONTINUE
statements labeled with 11**. In an optimized version of this code, the
nested DO loops should be replaced with calls to an optimized subroutine.
2. This code scales A by 1/SQRT(M) if the largest ABS(A(i,j)) could cause
column norm overflow. This is the minial precaution and it is left to the
SVD routine (CGESVD) to do its own preemptive scaling if potential over-
or underflows are detected. To avoid repeated scanning of the array A,
an optimal implementation would do all necessary scaling before calling
CGESVD and the scaling in CGESVD can be switched off.
3. Other comments related to code optimization are given in comments in the
code, enclosed in [[double brackets]].
Bugs, examples and comments
Please report all bugs and send interesting examples and/or comments to
drmac@math.hr. Thank you.
References
[1] Zlatko Drmac, Algorithm 977: A QR-Preconditioned QR SVD Method for
Computing the SVD with High Accuracy. ACM Trans. Math. Softw.
44(1): 11:1-11:30 (2017)
SIGMA library, xGESVDQ section updated February 2016.
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Contributors:
Developed and coded by Zlatko Drmac, Department of Mathematics
University of Zagreb, Croatia, drmac@math.hr
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Author¶
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