SGESVJ computes the singular value decomposition (SVD) of a real


 M-by-N matrix A, where M >= N. The SVD of A is written as


                                    [++]   [xx]   [x0]   [xx]


              A = U * SIGMA * V^t,  [++] = [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 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.


 SGESVJ 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 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': 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=SLAMCH('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' : 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/SLAMCH('E') > 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.


          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 SLAMCH('S'). The singular


                 vectors corresponding to underflowed or zero singular


                 values are not computed. The value of RANKA is returned


                 in the array WORK as RANKA=NINT(WORK(2)). Also see the


                 descriptions of SVA and WORK. 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 SGESVJ 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^T||_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 SGESVJ 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 REAL array, dimension (N)


          On exit,


          If INFO .EQ. 0 :


          depending on the value SCALE = WORK(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 SGESVJ 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 SGESVJ


          is applied to the first MV rows of V. See the description of JOBV.

V

          V is REAL 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) .

WORK

          WORK is REAL array, dimension (LWORK)


          On entry,


          If JOBU .EQ. 'C' :


          WORK(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=SLAMCH('E').


                    It is required that CTOL >= ONE, i.e. it is not


                    allowed to force the routine to obtain orthogonality


                    below EPSILON.


          On exit,


          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)


                    are the computed singular vcalues of A.


                    (See description of SVA().)


          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero


                    singular values.


          WORK(3) = NINT(WORK(3)) is the number of the computed singular


                    values that are larger than the underflow threshold.


          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi


                    rotations needed for numerical convergence.


          WORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.


                    This is useful information in cases when SGESVJ did


                    not converge, as it can be used to estimate whether


                    the output is stil useful and for post festum analysis.


          WORK(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.

LWORK

          LWORK is INTEGER


         length of WORK, WORK >= MAX(6,M+N)

INFO

          INFO is INTEGER


          = 0 : successful exit.


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


          > 0 : SGESVJ did not converge in the maximal allowed number (30)


                of sweeps. The output may still be useful. See the


                description of WORK.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2017

The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane rotations. The rotations are implemented as fast scaled rotations of Anda and Park [1]. In the case of underflow of the Jacobi angle, a modified Jacobi transformation of Drmac [4] is used. Pivot strategy uses column interchanges of de Rijk [2]. 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 [3]. 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 [5,6], and it is the kernel routine in the SIGMA library [7]. 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.

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

[1] A. A. Anda and H. Park: Fast plane rotations with dynamic scaling.
SIAM J. matrix Anal. Appl., Vol. 15 (1994), pp. 162-174.

[2] 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.

[3] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
[4] 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.

[5] 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.

[6] 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.

[7] 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 test examples and comments to drmac@math.hr. Thank you.