!>
!> 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 = 'U' .OR. JOBU = 'C':
!>                 If INFO = 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 = 'C'),
!>                 see the description of JOBU.
!>                 If INFO > 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 = 'N':
!>                 If INFO = 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 > 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 = 0 :
!>          depending on the value SCALE = WORK(1), we have:
!>                 If SCALE = 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 > 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 = '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 >= 1.
!>          If JOBV = 'V', then LDV >= max(1,N).
!>          If JOBV = 'A', then LDV >= max(1,MV) .
!> 

WORK

!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On entry,
!>          If JOBU = '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 still 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.
!>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(6,M+N), otherwise.
!>
!>          If on entry LWORK = -1, then a workspace query is assumed and
!>          no computation is done; WORK(1) is set to the minial (and optimal)
!>          length of WORK.
!> 

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.

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 tuning 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.