SYNOPSIS

SUBROUTINE DGEJSV(

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

    

DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),

    

& WORK( LWORK )

    

INTEGER IWORK( * )

    

CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV

PURPOSE

DGEJSV 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

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'. 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. 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. 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 DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues the licence to kill columns of A whose norm in c*A is less than DSQRT(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 DGESVJ. = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)] (roughly, as described above). This option is recommended. ~~~~~~~~~~~~~~~~~~~~~~~~~~~ For computing the singular values in the FULL range [SFMIN,BIG] use DGESVJ. 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. 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 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 DSQRT(||(R^t * R)^(-1)||_1). It is computed using DPOCON. 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) 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 vectors 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 performance, 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 without 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 denormalized 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 : DGEJSV did not converge in the maximal allowed number of sweeps. The computed values may be inaccurate.

FURTHER DETAILS

DGEJSV 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. 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 DGEJSV 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 (DGEJSV) 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 (DGESVJ) 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 pivoting. That well known trick is not used in DGEJSV 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 DGEJSV 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, examples and comments
Please report all bugs and send interesting examples and/or comments to [email protected]. Thank you.