subroutine cgejsv (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA, SVA, U, LDU, V, LDV, CWORK, LWORK, RWORK, LRWORK, IWORK, INFO)
CGEJSV
CGEJSV
Purpose:
CGEJSV computes the singular value decomposition (SVD) of a complex M-by-N
matrix [A], where M >= N. The SVD of [A] is written as
[A] = [U] * [SIGMA] * [V]^*,
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) unitary 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. 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.
Parameters:
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
pivoting. 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 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=B*D. If A has heavily weighted
rows, then using this condition number gives too pessimistic
error bound.
= 'A': Small singular values are not well determined by the data
and are considered as noisy; 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^* 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
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
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, if JOBT .EQ. 'N'.
= 'W': V may be used as workspace of length N*N. See the description
of V.
= 'N': V is not computed.
JOBR
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 CGESVJ.
= '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 CGESVJ.
JOBT
JOBT is CHARACTER*1
If the matrix is square then the procedure may determine to use
transposed A if A^* seems to be better with respect to convergence.
If the matrix is not square, JOBT is ignored.
The decision is based on two values of entropy over the adjoint
orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
= 'T': transpose if entropy test indicates possibly faster
convergence of Jacobi process if A^* is taken as input. If A is
replaced with A^*, then the row pivoting is included automatically.
= 'N': do not speculate.
The option 'T' 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 constraint and make the
code more complicated. See the descriptions of U and V.
In general, this option is considered experimental, and 'N'; should
be preferred. This is subject to changes in the future.
JOBP
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 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
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, dimension (LDA,N)
On entry, the M-by-N matrix A.
LDA
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
SVA
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 denormalized numbers.
U
U is COMPLEX array, dimension ( LDU, N ) or ( LDU, M )
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^*. In that case, [V] is computed
in U as left singular vectors of A^* 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, unless JOBT='T'.
LDU
LDU is INTEGER
The leading dimension of the array U, LDU >= 1.
IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
V
V is COMPLEX 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^*. In that case, [U] is computed
in V as right singular vectors of A^* 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, unless JOBT='T'.
LDV
LDV is INTEGER
The leading dimension of the array V, LDV >= 1.
If JOBV = 'V' or 'J' or 'W', then LDV >= N.
CWORK
CWORK is COMPLEX array, dimension (MAX(2,LWORK))
If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
LRWORK=-1), then on exit CWORK(1) contains the required length of
CWORK for the job parameters used in the call.
LWORK
LWORK is INTEGER
Length of CWORK to confirm proper allocation of workspace.
LWORK depends on the job:
1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
1.1 .. no scaled condition estimate required (JOBA.NE.'E'.AND.JOBA.NE.'G'):
LWORK >= 2*N+1. This is the minimal requirement.
->> For optimal performance (blocked code) the optimal value
is LWORK >= N + (N+1)*NB. Here NB is the optimal
block size for CGEQP3 and CGEQRF.
In general, optimal LWORK is computed as
LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ)).
1.2. .. an estimate of the scaled condition number of A is
required (JOBA='E', or 'G'). In this case, LWORK the minimal
requirement is LWORK >= N*N + 2*N.
->> For optimal performance (blocked code) the optimal value
is LWORK >= max(N+(N+1)*NB, N*N+2*N)=N**2+2*N.
In general, the optimal length LWORK is computed as
LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CGEQRF), LWORK(CGESVJ),
N*N+LWORK(CPOCON)).
2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
(JOBU.EQ.'N')
2.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
-> the minimal requirement is LWORK >= 3*N.
-> For optimal performance,
LWORK >= max(N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
CUNMLQ. In general, the optimal length LWORK is computed as
LWORK >= max(N+LWORK(CGEQP3), N+LWORK(CGESVJ),
N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
2.2 .. an estimate of the scaled condition number of A is
required (JOBA='E', or 'G').
-> the minimal requirement is LWORK >= 3*N.
-> For optimal performance,
LWORK >= max(N+(N+1)*NB, 2*N,2*N+N*NB)=2*N+N*NB,
where NB is the optimal block size for CGEQP3, CGEQRF, CGELQ,
CUNMLQ. In general, the optimal length LWORK is computed as
LWORK >= max(N+LWORK(CGEQP3), LWORK(CPOCON), N+LWORK(CGESVJ),
N+LWORK(CGELQF), 2*N+LWORK(CGEQRF), N+LWORK(CUNMLQ)).
3. If SIGMA and the left singular vectors are needed
3.1 .. no scaled condition estimate requested (JOBE.EQ.'N'):
-> the minimal requirement is LWORK >= 3*N.
-> For optimal performance:
if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
In general, the optimal length LWORK is computed as
LWORK >= max(N+LWORK(CGEQP3), 2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
3.2 .. an estimate of the scaled condition number of A is
required (JOBA='E', or 'G').
-> the minimal requirement is LWORK >= 3*N.
-> For optimal performance:
if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB)=2*N+N*NB,
where NB is the optimal block size for CGEQP3, CGEQRF, CUNMQR.
In general, the optimal length LWORK is computed as
LWORK >= max(N+LWORK(CGEQP3),N+LWORK(CPOCON),
2*N+LWORK(CGEQRF), N+LWORK(CUNMQR)).
4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
4.1. if JOBV.EQ.'V'
the minimal requirement is LWORK >= 5*N+2*N*N.
4.2. if JOBV.EQ.'J' the minimal requirement is
LWORK >= 4*N+N*N.
In both cases, the allocated CWORK can accommodate blocked runs
of CGEQP3, CGEQRF, CGELQF, CUNMQR, CUNMLQ.
If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
LRWORK=-1), then on exit CWORK(1) contains the optimal and CWORK(2) contains the
minimal length of CWORK for the job parameters used in the call.
RWORK
RWORK is REAL array, dimension (MAX(7,LWORK))
On exit,
RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
such that SCALE*SVA(1:N) are the computed singular values
of A. (See the description of SVA().)
RWORK(2) = See the description of RWORK(1).
RWORK(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^* * 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 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.
RWORK(4) = an estimate of the scaled condition number of the
triangular factor in the first QR factorization.
RWORK(5) = an estimate of the scaled condition number of the
triangular factor in the second QR factorization.
The following two parameters are computed if JOBT .EQ. 'T'.
They are provided for a developer/implementer who is familiar
with the details of the method.
RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
of diag(A^* * A) / Trace(A^* * A) taken as point in the
probability simplex.
RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
If the call to CGEJSV is a workspace query (indicated by LWORK=-1 or
LRWORK=-1), then on exit RWORK(1) contains the required length of
RWORK for the job parameters used in the call.
LRWORK
LRWORK is INTEGER
Length of RWORK to confirm proper allocation of workspace.
LRWORK depends on the job:
1. If only the singular values are requested i.e. if
LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
then:
1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
then: LRWORK = max( 7, 2 * M ).
1.2. Otherwise, LRWORK = max( 7, N ).
2. If singular values with the right singular vectors are requested
i.e. if
(LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
.NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
then:
2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
then LRWORK = max( 7, 2 * M ).
2.2. Otherwise, LRWORK = max( 7, N ).
3. If singular values with the left singular vectors are requested, i.e. if
(LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
.NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
then:
3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
then LRWORK = max( 7, 2 * M ).
3.2. Otherwise, LRWORK = max( 7, N ).
4. If singular values with both the left and the right singular vectors
are requested, i.e. if
(LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
then:
4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
then LRWORK = max( 7, 2 * M ).
4.2. Otherwise, LRWORK = max( 7, N ).
If, on entry, LRWORK = -1 or LWORK=-1, a workspace query is assumed and
the length of RWORK is returned in RWORK(1).
IWORK
IWORK is INTEGER array, of dimension at least 4, that further depends
on the job:
1. If only the singular values are requested then:
If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
then the length of IWORK is N+M; otherwise the length of IWORK is N.
2. If the singular values and the right singular vectors are requested then:
If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
then the length of IWORK is N+M; otherwise the length of IWORK is N.
3. If the singular values and the left singular vectors are requested then:
If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
then the length of IWORK is N+M; otherwise the length of IWORK is N.
4. If the singular values with both the left and the right singular vectors
are requested, then:
4.1. If LSAME(JOBV,'J') the length of IWORK is determined as follows:
If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
then the length of IWORK is N+M; otherwise the length of IWORK is N.
4.2. If LSAME(JOBV,'V') the length of IWORK is determined as follows:
If ( LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G') )
then the length of IWORK is 2*N+M; otherwise the length of IWORK is 2*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 warranted by the data.
IWORK(4) = 1 or -1. If IWORK(4) .EQ. 1, then the procedure used A^* to
do the job as specified by the JOB parameters.
If the call to CGEJSV is a workspace query (indicated by LWORK .EQ. -1 and
LRWORK .EQ. -1), then on exit IWORK(1) contains the required length of
IWORK for the job parameters used in the call.
INFO
INFO is INTEGER
< 0 : if INFO = -i, then the i-th argument had an illegal value.
= 0 : successful exit;
> 0 : CGEJSV did not converge in the maximal allowed number
of sweeps. The computed values may be inaccurate.
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
Further Details:
CGEJSV implements a preconditioned Jacobi SVD algorithm. It uses CGEQP3,
CGEQRF, and CGELQF 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 arbitrarily 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 accuracy 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 CGEJSV 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 (CGEJSV) 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 (CGESVJ) is
left to the implementer on a particular machine.
The rank revealing QR factorization (in this code: CGEQP3) should be
implemented as in [3]. We have a new version of CGEQP3 under development
that is more robust than the current one in LAPACK, with a cleaner cut in
rank deficient cases. It will be available in the SIGMA library [4].
If M is much larger than N, it is obvious that the initial QRF with
column pivoting can be preprocessed by the QRF without pivoting. That
well known trick is not used in CGEJSV 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 CGEJSV uses only the simplest, naive data movement.
Contributor:
References:
[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, 2016.
Bugs, examples and comments:
Definition at line 570 of file cgejsv.f.
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