sgejsv - N matrix A, where M >= N
SUBROUTINE SGEJSV(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO) CHARACTER*1 JOBA, JOBU, JOBV, JOBR, JOBT, JOBP INTEGER INFO, LDA, LDU, LDV, LWORK, M, N REAL A(LDA,*), SVA(N), U(LDU,*), V(LDV,*), WORK(LWORK) INTEGER IWORK(*) SUBROUTINE SGEJSV_64(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO) CHARACTER*1 JOBA, JOBU, JOBV, JOBR, JOBT, JOBP INTEGER*8 INFO, LDA, LDU, LDV, LWORK, M, N REAL A(LDA,*), SVA(N), U(LDU,*), V(LDV,*), WORK(LWORK) INTEGER*8 IWORK(*) F95 INTERFACE SUBROUTINE GEJSV(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO) REAL, DIMENSION(:,:) :: A, U, V INTEGER :: M, N, LDA, LDU, LDV, LWORK, INFO CHARACTER(LEN=1) :: JOBA, JOBU, JOBV, JOBR, JOBT, JOBP INTEGER, DIMENSION(:) :: IWORK REAL, DIMENSION(:) :: SVA, WORK SUBROUTINE GEJSV_64(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO) REAL, DIMENSION(:,:) :: A, U, V INTEGER(8) :: M, N, LDA, LDU, LDV, LWORK, INFO CHARACTER(LEN=1) :: JOBA, JOBU, JOBV, JOBR, JOBT, JOBP INTEGER(8), DIMENSION(:) :: IWORK REAL, DIMENSION(:) :: SVA, WORK C INTERFACE #include <sunperf.h> void sgejsv (char joba, char jobu, char jobv, char jobr, char jobt, char jobp, int m, int n, float *a, int lda, float *sva, float *u, int ldu, float *v, int ldv, int lwork, int *info); void sgejsv_64 (char joba, char jobu, char jobv, char jobr, char jobt, char jobp, long m, long n, float *a, long lda, float *sva, float *u, long ldu, float *v, long ldv, long lwork, long *info);
Oracle Solaris Studio Performance Library sgejsv(3P)
NAME
sgejsv - compute the singular value decomposition (SVD) of a real M-by-
N matrix A, where M >= N
SYNOPSIS
SUBROUTINE SGEJSV(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA,
SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
CHARACTER*1 JOBA, JOBU, JOBV, JOBR, JOBT, JOBP
INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
REAL A(LDA,*), SVA(N), U(LDU,*), V(LDV,*), WORK(LWORK)
INTEGER IWORK(*)
SUBROUTINE SGEJSV_64(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA,
SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
CHARACTER*1 JOBA, JOBU, JOBV, JOBR, JOBT, JOBP
INTEGER*8 INFO, LDA, LDU, LDV, LWORK, M, N
REAL A(LDA,*), SVA(N), U(LDU,*), V(LDV,*), WORK(LWORK)
INTEGER*8 IWORK(*)
F95 INTERFACE
SUBROUTINE GEJSV(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA,
U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
REAL, DIMENSION(:,:) :: A, U, V
INTEGER :: M, N, LDA, LDU, LDV, LWORK, INFO
CHARACTER(LEN=1) :: JOBA, JOBU, JOBV, JOBR, JOBT, JOBP
INTEGER, DIMENSION(:) :: IWORK
REAL, DIMENSION(:) :: SVA, WORK
SUBROUTINE GEJSV_64(JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA,
SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, INFO)
REAL, DIMENSION(:,:) :: A, U, V
INTEGER(8) :: M, N, LDA, LDU, LDV, LWORK, INFO
CHARACTER(LEN=1) :: JOBA, JOBU, JOBV, JOBR, JOBT, JOBP
INTEGER(8), DIMENSION(:) :: IWORK
REAL, DIMENSION(:) :: SVA, WORK
C INTERFACE
#include <sunperf.h>
void sgejsv (char joba, char jobu, char jobv, char jobr, char jobt,
char jobp, int m, int n, float *a, int lda, float *sva, float
*u, int ldu, float *v, int ldv, int lwork, int *info);
void sgejsv_64 (char joba, char jobu, char jobv, char jobr, char jobt,
char jobp, long m, long n, float *a, long lda, float *sva,
float *u, long ldu, float *v, long ldv, long lwork, long
*info);
PURPOSE
sgejsv computes the singular value decomposition (SVD) of a real M-by-N
matrix [A], where M >= N. The SVD of [A] is written as
[A] = [U] * [SIGMA] * [V]^t,
where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its
N diagonal elements, [U] is an M-by-N (or M-by-M) 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. The
matrices [U] and [V] are computed and stored in the arrays U and V,
respectively. The diagonal of [SIGMA] is computed and stored in the
array SVA.
ARGUMENTS
JOBA (input)
JOBA is CHARACTER*1
Specifies the level of accuracy:
= 'C': This option works well (high relative accuracy) if
A=B*D, with well-conditioned B and arbitrary diagonal matrix
D. The accuracy cannot be spoiled by COLUMN scaling. The
accuracy of the computed output depends on the condition of
B, and the procedure aims at the best theoretical accuracy.
The relative error max_{i=1:N}|d sigma_i| / sigma_i is
bounded by f(M,N)*epsilon* cond(B), independent of D. The
input matrix is preprocessed with the QRF with column pivot-
ing. This initial preprocessing and preconditioning by a rank
revealing QR factorization is common for all values of JOBA.
Additional actions are specified as follows:
= 'E': Computation as with 'C' with an additional estimate of
the condition number of B. It provides a realistic error
bound.
= 'F': If A = D1 * C * D2 with ill-conditioned diagonal scal-
ings D1, D2, and well-conditioned matrix C, this option gives
higher accuracy than the 'C' option. If the structure of the
input matrix is not known, and relative accuracy is desir-
able, then this option is advisable. The input matrix A is
preprocessed with QR factorization with FULL (row and column)
pivoting.
= 'G' Computation as with 'F' with an additional estimate of
the condition number of B, where A=D*B. If A has heavily
weighted rows, then using this condition number gives too
pessimistic error bound.
= 'A': Small singular values are the noise and the matrix is
treated as numerically rank defficient. The error in the com-
puted singular values is bounded by f(m,n)*epsilon*||A||.
The computed SVD A = U * S * V^t restores A up to
f(m,n)*epsilon*||A||.
This gives the procedure the licence to discard (set to zero)
all singular values below N*epsilon*||A||.
= 'R': Similar as in 'A'. Rank revealing property of the ini-
tial QR factorization is used do reveal (using triangular
factor) a gap sigma_{r+1} < epsilon * sigma_r in which case
the numerical RANK is declared to be r. The SVD is computed
with absolute error bounds, but more accurately than with
'A'.
JOBU (input)
JOBU is CHARACTER*1
Specifies whether to compute the columns of U:
= 'U': N columns of U are returned in the array U.
= 'F': full set of M left sing. vectors is returned in the
array U.
= 'W': U may be used as workspace of length M*N. See the
description of U.
= 'N': U is not computed.
JOBV (input)
JOBV is CHARACTER*1
Specifies whether to compute the matrix V:
= 'V': N columns of V are returned in the array V; Jacobi
rotations are not explicitly accumulated.
= 'J': N columns of V are returned in the array V, but they
are computed as the product of Jacobi rotations. This option
is allowed only if JOBU .NE. 'N', i.e. in computing the full
SVD.
= 'W': V may be used as workspace of length N*N. See the
description of V.
= 'N': V is not computed.
JOBR (input)
JOBR is CHARACTER*1
Specifies the RANGE for the singular values. Issues the
licence to set to zero small positive singular values if they
are outside specified range. If A .NE. 0 is scaled so that
the largest singular value of c*A is around SQRT(BIG),
BIG=SLAMCH('O'), then JOBR issues the licence to kill columns
of A whose norm in c*A is less than SQRT(SFMIN) (for
JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN, where
SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
= 'N': Do not kill small columns of c*A. This option assumes
that BLAS and QR factorizations and triangular solvers are
implemented to work in that range. If the condition of A is
greater than BIG, use SGESVJ.
= 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN),
SQRT(BIG)] (roughly, as described above). This option is rec-
ommended.
For computing the singular values in the FULL range
[SFMIN,BIG] use SGESVJ.
JOBT (input)
JOBT is CHARACTER*1
If the matrix is square then the procedure may determine to
use transposed A if A^t seems to be better with respect to
convergence. If the matrix is not square, JOBT is ignored.
This is subject to changes in the future. The decision is
based on two values of entropy over the adjoint orbit of A^t
* A. See the descriptions of WORK(6) and WORK(7).
= 'T': transpose if entropy test indicates possibly faster
convergence of Jacobi process if A^t is taken as input. If A
is replaced with A^t, then the row pivoting is included auto-
matically.
= 'N': do not speculate.
This option can be used to compute only the singular values,
or the full SVD (U, SIGMA and V). For only one set of singu-
lar vectors (U or V), the caller should provide both U and V,
as one of the matrices is used as workspace if the matrix A
is transposed. The implementer can easily remove this con-
straint and make the code more complicated. See the descrip-
tions of U and V.
JOBP (input)
JOBP is CHARACTER*1
Issues the licence to introduce structured perturbations to
drown denormalized numbers. This licence should be active if
the denormals are poorly implemented, causing slow computa-
tion, especially in cases of fast convergence (!). For
details see [1,2].
For the sake of simplicity, this perturbations are included
only when the full SVD or only the singular values are
requested. The implementer/user can easily add the perturba-
tion for the cases of computing one set of singular vectors.
= 'P': introduce perturbation
= 'N': do not perturb
M (input)
M is INTEGER
The number of rows of the input matrix A. M >= 0.
N (input)
N is INTEGER
The number of columns of the input matrix A. M >= N >= 0.
A (input/output)
A is REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
LDA (input)
LDA is INTEGER
The leading dimension of the array A.
LDA >= max(1,M).
SVA (output)
SVA is REAL array, dimension (N)
On exit,
- For WORK(1)/WORK(2) = ONE: The singular values of A. During
the computation SVA contains Euclidean column norms of the
iterated matrices in the array A.
- For WORK(1) .NE. WORK(2): The singular values of A are
(WORK(1)/WORK(2))*SVA(1:N).This factored form is used if
sigma_max(A) overflows or if small singular values have been
saved from underflow by scaling the input matrix A.
- If JOBR='R' then some of the singular values may be
returned as exact zeros obtained by "set to zero" because
they are below the numerical rank threshold or are denormal-
ized numbers.
U (output)
U is REAL array, dimension ( LDU, N )
If JOBU = 'U', then U contains on exit the M-by-N matrix of
the left singular vectors.
If JOBU = 'F', then U contains on exit the M-by-M matrix of
the left singular vectors, including an ONB of the orthogonal
complement of the Range(A).
If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND.
M.EQ.N), then U is used as workspace if the procedure
replaces A with A^t. In that case, [V] is computed in U as
left singular vectors of A^t and then copied back to the V
array. This 'W' option is just a reminder to the caller that
in this case U is reserved as workspace of length N*N.
If JOBU = 'N' U is not referenced.
LDU (input)
LDU is INTEGER
The leading dimension of the array U, LDU >= 1.
IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
V (output)
V is REAL array, dimension ( LDV, N )
If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix
of the right singular vectors;
If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
then V is used as workspace if the procedure replaces A with
A^t. In that case, [U] is computed in V as right singular
vectors of A^t and then copied back to the U array. This 'W'
option is just a reminder to the caller that in this case V
is reserved as workspace of length N*N.
If JOBV = 'N' V is not referenced.
LDV (input)
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = 'V' or 'J' or 'W', then LDV >= N.
WORK (output)
WORK is REAL array, dimension at least LWORK.
On exit,
WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor
such that SCALE*SVA(1:N) are the computed singular values of
A. (See the description of SVA().)
WORK(2) = See the description of WORK(1).
WORK(3) = SCONDA is an estimate for the condition number of
column equilibrated A. (If JOBA .EQ. 'E' or 'G') SCONDA is an
estimate of SQRT(||(R^t * R)^(-1)||_1). It is computed using
SPOCON. It holds
N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
where R is the triangular factor from the QRF of A. However,
if R is truncated and the numerical rank is determined to be
strictly smaller than N, SCONDA is returned as -1, thus indi-
cating that the smallest singular values might be lost.
If full SVD is needed, the following two condition numbers
are useful for the analysis of the algorithm. They are
provied for a developer/implementer who is familiar with the
details of the method.
WORK(4) = an estimate of the scaled condition number of the
triangular factor in the first QR factorization.
WORK(5) = an estimate of the scaled condition number of the
triangular factor in the second QR factorization. The fol-
lowing two parameters are computed if JOBT .EQ. 'T'. They
are provided for a developer/implementer who is familiar with
the details of the method.
WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
of diag(A^t*A) / Trace(A^t*A) taken as point in the probabil-
ity simplex.
WORK(7) = the entropy of A*A^t.
LWORK (input)
LWORK is INTEGER
Length of WORK to confirm proper allocation of work space.
LWORK depends on the job:
If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
-> .. no scaled condition estimate required (JOBE.EQ.'N'):
LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
block size for SGEQP3 and SGEQRF.
In general, optimal LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SGEQRF), 7).
-> .. an estimate of the scaled condition number of A is
required (JOBA='E', 'G'). In this case, LWORK is the maximum
of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SGEQRF),
N+N*N+LWORK(SPOCON),7).
If SIGMA and the right singular vectors are needed
(JOBV.EQ.'V'),
-> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
-> For optimal performance, LWORK >=
max(2*M+N,3*N+(N+1)*NB,7), where NB is the optimal block size
for SGEQP3, SGEQRF, SGELQ, SORMLQ. In general, the optimal
length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(SGEQP3), N+LWORK(SPOCON),
N+LWORK(SGELQ), 2*N+LWORK(SGEQRF), N+LWORK(SORMLQ)).
If SIGMA and the left singular vectors are needed
-> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
-> For optimal performance:
if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
where NB is the optimal block size for SGEQP3, SGEQRF, SOR-
MQR.
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(SGEQP3),N+LWORK(SPOCON),
2*N+LWORK(SGEQRF), N+LWORK(SORMQR)).
Here LWORK(SORMQR) equals N*NB (for JOBU.EQ.'U') or M*NB (for
JOBU.EQ.'F').
If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
-> if JOBV.EQ.'V' the minimal requirement is LWORK >=
max(2*M+N,6*N+2*N*N).
-> if JOBV.EQ.'J' the minimal requirement is
LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
-> For optimal performance, LWORK should be additionally
larger than N+M*NB, where NB is the optimal block size for
SORMQR.
IWORK (output)
IWORK is INTEGER array, dimension M+3*N.
On exit,
IWORK(1) = the numerical rank determined after the initial QR
factorization with pivoting. See the descriptions of JOBA and
JOBR.
IWORK(2) = the number of the computed nonzero singular values
IWORK(3) = if nonzero, a warning message:
If IWORK(3).EQ.1 then some of the column norms of A were
denormalized floats. The requested high accuracy is not war-
ranted by the data.
INFO (output)
INFO is INTEGER
< 0 : if INFO = -i, then the i-th argument had an illegal
value.
= 0 : successfull exit;
> 0 : SGEJSV did not converge in the maximal allowed number
of sweeps. The computed values may be inaccurate.
FURTHER DETAILS
SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses
SGEQP3, SGEQRF, and SGELQF as preprocessors and preconditioners.
Optionally, an additional row pivoting can be used as a preprocessor,
which in some cases results in much higher accuracy. An example is
matrix A with the structure A = D1 * C * D2, where D1, D2 are arbitrar-
ily ill-conditioned diagonal matrices and C is well-conditioned matrix.
In that case, complete pivoting in the first QR factorizations provides
accuracy dependent on the condition number of C, and independent of D1,
D2. Such higher accuracy is not completely understood theoretically,
but it works well in practice. Further, if A can be written as A =
B*D, with well-conditioned B and some diagonal D, then the high accu-
racy is guaranteed, both theoretically and in software, independent of
D. For more details see [1], [2]. The computational range for the sin-
gular values can be the full range ( UNDERFLOW,OVERFLOW ), provided
that the machine arithmetic and the BLAS LAPACK routines called by SGE-
JSV are implemented to work in that range. If that is not the case,
then the restriction for safe computation with the singular values in
the range of normalized IEEE numbers is that the spectral condition
number kappa(A)=sigma_max(A)/sigma_min(A) does not overflow. This code
(SGEJSV) is best used in this restricted range, meaning that singular
values of magnitude below ||A||_2 / SLAMCH('O') are returned as zeros.
See JOBR for details on this. Further, this implementation is somewhat
slower than the one described in [1,2] due to replacement of some non-
LAPACK components, and because the choice of some tuning parameters in
the iterative part (SGESVJ) is left to the implementer on a particular
machine. The rank revealing QR factorization (in this code: SGEQP3)
should be implemented as in [3]. We have a new version of SGEQP3 under
development that is more robust than the current one in LAPACK, with a
cleaner cut in rank defficient cases. It will be available in the SIGMA
library [4]. If M is much larger than N, it is obvious that the inital
QRF with column pivoting can be preprocessed by the QRF without pivot-
ing. That well known trick is not used in SGEJSV because in some cases
heavy row weighting can be treated with complete pivoting. The overhead
in cases M much larger than N is then only due to pivoting, but the
benefits in terms of accuracy have prevailed. The implementer/user can
incorporate this extra QRF step easily. The implementer can also
improve data movement (matrix transpose, matrix copy, matrix transposed
copy) - this implementation of SGEJSV uses only the simplest, naive
data movement.
[1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm
I.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
[2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm
II.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
[3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
factorization software - a case study.
ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
LAPACK Working note 176.
[4] Z. Drmac: SIGMA - mathematical software library for accurate SVD,
PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008.
7 Nov 2015 sgejsv(3P)