SGESVJ(3)      LAPACK routine of NEC Numeric Library Collection      SGESVJ(3)



NAME
       SGESVJ

SYNOPSIS
       SUBROUTINE SGESVJ (JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V, LDV,
           WORK, LWORK, INFO)



PURPOSE
            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.




ARGUMENTS
           JOBA      (input)
                     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      (input)
                     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      (input)
                     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         (input)
                     M is INTEGER
                     The number of rows of the input matrix A. 1/SLAMCH('E') > 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.
                     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       (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,
                     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        (input)
                     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         (input/output)
                     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       (input)
                     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      (input/output)
                     WORK is REAL array, dimension max(4,M+N).
                     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     (input)
                     LWORK is INTEGER
                    length of WORK, WORK >= MAX(6,M+N)

           INFO      (output)
                     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.






FURTHER DETAILS
           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.



LAPACK routine                  31 October 2017                      SGESVJ(3)