subroutine zgesvj (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO)
ZGESVJ
ZGESVJ
Purpose:
ZGESVJ 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 is CHARACTER*1
Specifies the structure of A.
= 'L': The input matrix A is lower triangular;
= 'U': The input matrix A is upper triangular;
= 'G': The input matrix A is general M-by-N matrix, M >= N.
JOBU
JOBU is CHARACTER*1
Specifies whether to compute the left singular vectors
(columns of U):
= 'U' or 'F': The left singular vectors corresponding to the nonzero
singular values are computed and returned in the leading
columns of A. See more details in the description of A.
The default numerical orthogonality threshold is set to
approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=DLAMCH('E').
= 'C': Analogous to JOBU='U', except that user can control the
level of numerical orthogonality of the computed left
singular vectors. TOL can be set to TOL = CTOL*EPS, where
CTOL is given on input in the array WORK.
No CTOL smaller than ONE is allowed. CTOL greater
than 1 / EPS is meaningless. The option 'C'
can be used if M*EPS is satisfactory orthogonality
of the computed left singular vectors, so CTOL=M could
save few sweeps of Jacobi rotations.
See the descriptions of A and WORK(1).
= 'N': The matrix U is not computed. However, see the
description of A.
JOBV
JOBV is CHARACTER*1
Specifies whether to compute the right singular vectors, that
is, the matrix V:
= 'V' or 'J': the matrix V is computed and returned in the array V
= 'A' : the Jacobi rotations are applied to the MV-by-N
array V. In other words, the right singular vector
matrix V is not computed explicitly; instead it is
applied to an MV-by-N matrix initially stored in the
first MV rows of V.
= 'N' : the matrix V is not computed and the array V is not
referenced
M
M is INTEGER
The number of rows of the input matrix A. 1/DLAMCH('E') > M >= 0.
N
N is INTEGER
The number of columns of the input matrix A.
M >= N >= 0.
A
A is COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit,
If JOBU .EQ. 'U' .OR. JOBU .EQ. 'C':
If INFO .EQ. 0 :
RANKA orthonormal columns of U are returned in the
leading RANKA columns of the array A. Here RANKA <= N
is the number of computed singular values of A that are
above the underflow threshold DLAMCH('S'). The singular
vectors corresponding to underflowed or zero singular
values are not computed. The value of RANKA is returned
in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
descriptions of SVA and RWORK. The computed columns of U
are mutually numerically orthogonal up to approximately
TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU.EQ.'C'),
see the description of JOBU.
If INFO .GT. 0,
the procedure ZGESVJ did not converge in the given number
of iterations (sweeps). In that case, the computed
columns of U may not be orthogonal up to TOL. The output
U (stored in A), SIGMA (given by the computed singular
values in SVA(1:N)) and V is still a decomposition of the
input matrix A in the sense that the residual
|| A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
If JOBU .EQ. 'N':
If INFO .EQ. 0 :
Note that the left singular vectors are 'for free' in the
one-sided Jacobi SVD algorithm. However, if only the
singular values are needed, the level of numerical
orthogonality of U is not an issue and iterations are
stopped when the columns of the iterated matrix are
numerically orthogonal up to approximately M*EPS. Thus,
on exit, A contains the columns of U scaled with the
corresponding singular values.
If INFO .GT. 0 :
the procedure ZGESVJ did not converge in the given number
of iterations (sweeps).
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SVA
SVA is DOUBLE PRECISION array, dimension (N)
On exit,
If INFO .EQ. 0 :
depending on the value SCALE = RWORK(1), we have:
If SCALE .EQ. ONE:
SVA(1:N) contains the computed singular values of A.
During the computation SVA contains the Euclidean column
norms of the iterated matrices in the array A.
If SCALE .NE. ONE:
The singular values of A are SCALE*SVA(1:N), and this
factored representation is due to the fact that some of the
singular values of A might underflow or overflow.
If INFO .GT. 0 :
the procedure ZGESVJ did not converge in the given number of
iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
MV
MV is INTEGER
If JOBV .EQ. 'A', then the product of Jacobi rotations in ZGESVJ
is applied to the first MV rows of V. See the description of JOBV.
V
V is COMPLEX*16 array, dimension (LDV,N)
If JOBV = 'V', then V contains on exit the N-by-N matrix of
the right singular vectors;
If JOBV = 'A', then V contains the product of the computed right
singular vector matrix and the initial matrix in
the array V.
If JOBV = 'N', then V is not referenced.
LDV
LDV is INTEGER
The leading dimension of the array V, LDV .GE. 1.
If JOBV .EQ. 'V', then LDV .GE. max(1,N).
If JOBV .EQ. 'A', then LDV .GE. max(1,MV) .
CWORK
CWORK is COMPLEX*16 array, dimension (max(1,LWORK))
Used as workspace.
If on entry LWORK .EQ. -1, then a workspace query is assumed and
no computation is done; CWORK(1) is set to the minial (and optimal)
length of CWORK.
LWORK
LWORK is INTEGER.
Length of CWORK, LWORK >= M+N.
RWORK
RWORK is DOUBLE PRECISION array, dimension (max(6,LRWORK))
On entry,
If JOBU .EQ. 'C' :
RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
The process stops if all columns of A are mutually
orthogonal up to CTOL*EPS, EPS=DLAMCH('E').
It is required that CTOL >= ONE, i.e. it is not
allowed to force the routine to obtain orthogonality
below EPSILON.
On exit,
RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
are the computed singular values of A.
(See description of SVA().)
RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
singular values.
RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
values that are larger than the underflow threshold.
RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
rotations needed for numerical convergence.
RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
This is useful information in cases when ZGESVJ did
not converge, as it can be used to estimate whether
the output is stil useful and for post festum analysis.
RWORK(6) = the largest absolute value over all sines of the
Jacobi rotation angles in the last sweep. It can be
useful for a post festum analysis.
If on entry LRWORK .EQ. -1, then a workspace query is assumed and
no computation is done; RWORK(1) is set to the minial (and optimal)
length of RWORK.
LRWORK
LRWORK is INTEGER
Length of RWORK, LRWORK >= MAX(6,N).
INFO
INFO is INTEGER
= 0 : successful exit.
< 0 : if INFO = -i, then the i-th argument had an illegal value
> 0 : ZGESVJ did not converge in the maximal allowed number
(NSWEEP=30) of sweeps. The output may still be useful.
See the description of RWORK.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
rotations. In the case of underflow of the tangent of the Jacobi angle, a
modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
column interchanges of de Rijk [1]. The relative accuracy of the computed
singular values and the accuracy of the computed singular vectors (in
angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
The condition number that determines the accuracy in the full rank case
is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
spectral condition number. The best performance of this Jacobi SVD
procedure is achieved if used in an accelerated version of Drmac and
Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
Some tunning parameters (marked with [TP]) are available for the
implementer.
The computational range for the nonzero singular values is the machine
number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
denormalized singular values can be computed with the corresponding
gradual loss of accurate digits.
Contributor:
============ Zlatko Drmac (Zagreb, Croatia)
References:
[1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
singular value decomposition on a vector computer.
SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
[2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
[3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
value computation in floating point arithmetic.
SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
[4] 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.
[5] 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.
[6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
QSVD, (H,K)-SVD computations.
Department of Mathematics, University of Zagreb, 2008, 2015.
Bugs, examples and comments:
=========================== Please report all bugs and send interesting test examples and comments to drmac@math.hr. Thank you.
Definition at line 353 of file zgesvj.f.
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