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.


 SGEJSV can sometimes compute tiny singular values and their singular vectors much


 more accurately than other SVD routines, see below under Further Details.

JOBA

          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 deficient. 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

          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. 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

          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

          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

          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 REAL 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 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, 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 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, 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.

WORK

          WORK is REAL array, dimension (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

          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

          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

          INFO is INTEGER


           < 0  : if INFO = -i, then the i-th argument had an illegal value.


           = 0 :  successful exit;


           > 0 :  SGEJSV  did not converge in the maximal allowed number


                  of sweeps. The computed values may be inaccurate.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

  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. 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 (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.


     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 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 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 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 SGEJSV uses only the simplest, naive data movement.

Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)

 [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.

Please report all bugs and send interesting examples and/or comments to drmac@math.hr. Thank you.