SGEJSV(3) LAPACK routine of NEC Numeric Library Collection SGEJSV(3)
NAME
SGEJSV
SYNOPSIS
SUBROUTINE SGEJSV (JOBA, JOBU, JOBV, JOBR, JOBT, JOBP, M, N, A, LDA,
SVA, U, LDU, V, LDV, WORK, LWORK, IWORK, 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
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=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)
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 recommended.
===========================
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 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 constraint and make the
code more complicated. See the descriptions 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 computation,
especially in cases of fast convergence (!).
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)
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 denormalized 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 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)
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 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 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 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.
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)
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 DGEQP3 and DGEQRF.
In general, optimal LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 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(DGEQP3),N+LWORK(DGEQRF),
N+N*N+LWORK(DPOCON),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 DGEQP3, DGEQRF, DGELQ,
DORMLQ. In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
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 DGEQP3, DGEQRF, DORMQR.
In general, the optimal length LWORK is computed as
LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
Here LWORK(DORMQR) 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 DORMQR.
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 warranted 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 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.
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 (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.
LAPACK routine 31 October 2017 SGEJSV(3)