SGEJSV(1LAPACK routine (version 3.2)SGEJSV(1)NAME
SGEJSV - [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 )
IMPLICIT NONE
INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
REAL A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
& WORK( LWORK )
INTEGER IWORK( * )
CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
PURPOSE
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) 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 pivoting. This initial
preprocessing and preconditioning by a rank revealing QR factor‐
ization 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 scalings
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 desirable, 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 computed 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 initial
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) 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) CHARACTER*1
Specifies whether to compute the matrix V:
= 'V': N columns of V are returned in the array V; Jacobi rota‐
tions 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) CHARACTER*1
Specifies the RANGE for the singular values. Issues the licence
to set to zero small positive singular values if they are out‐
side 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 recommended. ===========================
For computing the singular values in the FULL range [SFMIN,BIG]
use SGESVJ.
JOBT (input) 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 conver‐
gence. 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 automatically. = '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
singular 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 descriptions
of U and V.
JOBP (input) 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 computation,
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 perturbation
for the cases of computing one set of singular vectors. = 'P':
introduce perturbation
= 'N': do not perturb
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. M >= N >= 0.
A (input/workspace) REAL array, dimension (LDA,N)
On entry, the M-by-N matrix A.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SVA (workspace/output) 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 denormalized numbers.
U (workspace/output) 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.
The leading dimension of the array U, LDU >= 1. IF JOBU =
'U' or 'F' or 'W', then LDU >= M.
V (workspace/output) 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
pprocedure 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) INTEGER
The leading dimension of the array V, LDV >= 1. If JOBV = 'V'
or 'J' or 'W', then LDV >= N.
WORK (workspace/output) 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 val‐
ues 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 tri‐
angular 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 indicating 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 fac‐
tor in the first QR factorization. WORK(5) = an estimate of
the scaled condition number of the triangular factor in the
second QR factorization. The following two parameters are com‐
puted if JOBT .EQ. 'T'. They are provided for a devel‐
oper/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
probability simplex. WORK(7) = the entropy of A*A^t.
LWORK (input) 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 xGEQP3/xGEQRF. -> .. 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+4N,7). If SIGMA and the right singular vec‐
tors are needed (JOBV.EQ.'V'), -> the minimal requirement is
LWORK >= max(2*N+M,7). -> For optimal performance, LWORK >=
max(2*N+M,2*N+N*NB,7), where NB is the optimal block size. If
SIGMA and the left singular vectors are needed -> the minimal
requirement is LWORK >= max(2*N+M,7). -> For optimal perfor‐
mance, LWORK >= max(2*N+M,2*N+N*NB,7), where NB is the optimal
block size. If full SVD is needed ( JOBU.EQ.'U' or 'F',
JOBV.EQ.'V' ) and -> .. the singular vectors are computed with‐
out explicit accumulation of the Jacobi rotations, LWORK >=
6*N+2*N*N -> .. in the iterative part, the Jacobi rotations are
explicitly accumulated (option, see the description of JOBV),
then the minimal requirement is LWORK >= max(M+3*N+N*N,7). For
better performance, if NB is the optimal block size, LWORK >=
max(3*N+N*N+M,3*N+N*N+N*NB,7).
IWORK (workspace/output) 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 denormal‐
ized floats. The requested high accuracy is not warranted by
the data.
INFO (output) 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 singular values can be the full
range ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and
the BLAS & LAPACK routines called by SGEJSV 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 (SGE‐
JSV) is best used in this restricted range, meaning that singular val‐
ues 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 develop‐
ment 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 pivoting.
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 bene‐
fits 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. Contributors
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
References
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
LAPACK Working note 169.
SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
LAPACK Working note 170.
factorization software - a case study.
ACM Trans. math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
LAPACK Working note 176.
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008. Bugs, exam‐
ples and comments
Please report all bugs and send interesting examples and/or comments to
drmac@math.hr. Thank you.
LAPACK routine (version 3.2) November 2008 SGEJSV(1)
