path: root/SRC/slasd1.f

*> \brief \b SLASD1 computes the SVD of an upper bidiagonal matrix B of the specified size. Used by sbdsdc.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
*                          IDXQ, IWORK, WORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDU, LDVT, NL, NR, SQRE
*       REAL               ALPHA, BETA
*       ..
*       .. Array Arguments ..
*       INTEGER            IDXQ( * ), IWORK( * )
*       REAL               D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SLASD1 computes the SVD of an upper bidiagonal N-by-M matrix B,
*> where N = NL + NR + 1 and M = N + SQRE. SLASD1 is called from SLASD0.
*>
*> A related subroutine SLASD7 handles the case in which the singular
*> values (and the singular vectors in factored form) are desired.
*>
*> SLASD1 computes the SVD as follows:
*>
*>               ( D1(in)    0    0       0 )
*>   B = U(in) * (   Z1**T   a   Z2**T    b ) * VT(in)
*>               (   0       0   D2(in)   0 )
*>
*>     = U(out) * ( D(out) 0) * VT(out)
*>
*> where Z**T = (Z1**T a Z2**T b) = u**T VT**T, and u is a vector of dimension M
*> with ALPHA and BETA in the NL+1 and NL+2 th entries and zeros
*> elsewhere; and the entry b is empty if SQRE = 0.
*>
*> The left singular vectors of the original matrix are stored in U, and
*> the transpose of the right singular vectors are stored in VT, and the
*> singular values are in D.  The algorithm consists of three stages:
*>
*>    The first stage consists of deflating the size of the problem
*>    when there are multiple singular values or when there are zeros in
*>    the Z vector.  For each such occurrence the dimension of the
*>    secular equation problem is reduced by one.  This stage is
*>    performed by the routine SLASD2.
*>
*>    The second stage consists of calculating the updated
*>    singular values. This is done by finding the square roots of the
*>    roots of the secular equation via the routine SLASD4 (as called
*>    by SLASD3). This routine also calculates the singular vectors of
*>    the current problem.
*>
*>    The final stage consists of computing the updated singular vectors
*>    directly using the updated singular values.  The singular vectors
*>    for the current problem are multiplied with the singular vectors
*>    from the overall problem.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] NL
*> \verbatim
*>          NL is INTEGER
*>         The row dimension of the upper block.  NL >= 1.
*> \endverbatim
*>
*> \param[in] NR
*> \verbatim
*>          NR is INTEGER
*>         The row dimension of the lower block.  NR >= 1.
*> \endverbatim
*>
*> \param[in] SQRE
*> \verbatim
*>          SQRE is INTEGER
*>         = 0: the lower block is an NR-by-NR square matrix.
*>         = 1: the lower block is an NR-by-(NR+1) rectangular matrix.
*>
*>         The bidiagonal matrix has row dimension N = NL + NR + 1,
*>         and column dimension M = N + SQRE.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is REAL array, dimension (NL+NR+1).
*>         N = NL+NR+1
*>         On entry D(1:NL,1:NL) contains the singular values of the
*>         upper block; and D(NL+2:N) contains the singular values of
*>         the lower block. On exit D(1:N) contains the singular values
*>         of the modified matrix.
*> \endverbatim
*>
*> \param[in,out] ALPHA
*> \verbatim
*>          ALPHA is REAL
*>         Contains the diagonal element associated with the added row.
*> \endverbatim
*>
*> \param[in,out] BETA
*> \verbatim
*>          BETA is REAL
*>         Contains the off-diagonal element associated with the added
*>         row.
*> \endverbatim
*>
*> \param[in,out] U
*> \verbatim
*>          U is REAL array, dimension (LDU,N)
*>         On entry U(1:NL, 1:NL) contains the left singular vectors of
*>         the upper block; U(NL+2:N, NL+2:N) contains the left singular
*>         vectors of the lower block. On exit U contains the left
*>         singular vectors of the bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>         The leading dimension of the array U.  LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[in,out] VT
*> \verbatim
*>          VT is REAL array, dimension (LDVT,M)
*>         where M = N + SQRE.
*>         On entry VT(1:NL+1, 1:NL+1)**T contains the right singular
*>         vectors of the upper block; VT(NL+2:M, NL+2:M)**T contains
*>         the right singular vectors of the lower block. On exit
*>         VT**T contains the right singular vectors of the
*>         bidiagonal matrix.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>         The leading dimension of the array VT.  LDVT >= max( 1, M ).
*> \endverbatim
*>
*> \param[in,out] IDXQ
*> \verbatim
*>          IDXQ is INTEGER array, dimension (N)
*>         This contains the permutation which will reintegrate the
*>         subproblem just solved back into sorted order, i.e.
*>         D( IDXQ( I = 1, N ) ) will be in ascending order.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (4*N)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (3*M**2+2*M)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  if INFO = 1, a singular value did not converge
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup OTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE SLASD1( NL, NR, SQRE, D, ALPHA, BETA, U, LDU, VT, LDVT,
     $                   IDXQ, IWORK, WORK, INFO )
*
*  -- LAPACK auxiliary routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDU, LDVT, NL, NR, SQRE
      REAL               ALPHA, BETA
*     ..
*     .. Array Arguments ..
      INTEGER            IDXQ( * ), IWORK( * )
      REAL               D( * ), U( LDU, * ), VT( LDVT, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
*
      REAL               ONE, ZERO
      PARAMETER          ( ONE = 1.0E+0, ZERO = 0.0E+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            COLTYP, I, IDX, IDXC, IDXP, IQ, ISIGMA, IU2,
     $                   IVT2, IZ, K, LDQ, LDU2, LDVT2, M, N, N1, N2
      REAL               ORGNRM
*     ..
*     .. External Subroutines ..
      EXTERNAL           SLAMRG, SLASCL, SLASD2, SLASD3, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IF( NL.LT.1 ) THEN
         INFO = -1
      ELSE IF( NR.LT.1 ) THEN
         INFO = -2
      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
         INFO = -3
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SLASD1', -INFO )
         RETURN
      END IF
*
      N = NL + NR + 1
      M = N + SQRE
*
*     The following values are for bookkeeping purposes only.  They are
*     integer pointers which indicate the portion of the workspace
*     used by a particular array in SLASD2 and SLASD3.
*
      LDU2 = N
      LDVT2 = M
*
      IZ = 1
      ISIGMA = IZ + M
      IU2 = ISIGMA + N
      IVT2 = IU2 + LDU2*N
      IQ = IVT2 + LDVT2*M
*
      IDX = 1
      IDXC = IDX + N
      COLTYP = IDXC + N
      IDXP = COLTYP + N
*
*     Scale.
*
      ORGNRM = MAX( ABS( ALPHA ), ABS( BETA ) )
      D( NL+1 ) = ZERO
      DO 10 I = 1, N
         IF( ABS( D( I ) ).GT.ORGNRM ) THEN
            ORGNRM = ABS( D( I ) )
         END IF
   10 CONTINUE
      CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
      ALPHA = ALPHA / ORGNRM
      BETA = BETA / ORGNRM
*
*     Deflate singular values.
*
      CALL SLASD2( NL, NR, SQRE, K, D, WORK( IZ ), ALPHA, BETA, U, LDU,
     $             VT, LDVT, WORK( ISIGMA ), WORK( IU2 ), LDU2,
     $             WORK( IVT2 ), LDVT2, IWORK( IDXP ), IWORK( IDX ),
     $             IWORK( IDXC ), IDXQ, IWORK( COLTYP ), INFO )
*
*     Solve Secular Equation and update singular vectors.
*
      LDQ = K
      CALL SLASD3( NL, NR, SQRE, K, D, WORK( IQ ), LDQ, WORK( ISIGMA ),
     $             U, LDU, WORK( IU2 ), LDU2, VT, LDVT, WORK( IVT2 ),
     $             LDVT2, IWORK( IDXC ), IWORK( COLTYP ), WORK( IZ ),
     $             INFO )
*
*     Report the possible convergence failure.
*
      IF( INFO.NE.0 ) THEN
         RETURN
      END IF
*
*     Unscale.
*
      CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
*
*     Prepare the IDXQ sorting permutation.
*
      N1 = K
      N2 = N - K
      CALL SLAMRG( N1, N2, D, 1, -1, IDXQ )
*
      RETURN
*
*     End of SLASD1
*
      END