Chimie Théorique » OpenPath

      SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,

     $                   DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,

     $                   PERM, GIVNUM, C, S, WORK, IWORK, INFO )

*

*  -- LAPACK auxiliary routine (version 3.2.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     June 2010

*

*     .. Scalar Arguments ..

      INTEGER            ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE

*     ..

*     .. Array Arguments ..

      INTEGER            GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),

     $                   K( * ), PERM( LDGCOL, * )

      DOUBLE PRECISION   C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),

     $                   E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),

     $                   S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),

     $                   Z( LDU, * )

*     ..

*

*  Purpose

*  =======

*

*  Using a divide and conquer approach, DLASDA computes the singular

*  value decomposition (SVD) of a real upper bidiagonal N-by-M matrix

*  B with diagonal D and offdiagonal E, where M = N + SQRE. The

*  algorithm computes the singular values in the SVD B = U * S * VT.

*  The orthogonal matrices U and VT are optionally computed in

*  compact form.

*

*  A related subroutine, DLASD0, computes the singular values and

*  the singular vectors in explicit form.

*

*  Arguments

*  =========

*

*  ICOMPQ (input) INTEGER

*         Specifies whether singular vectors are to be computed

*         in compact form, as follows

*         = 0: Compute singular values only.

*         = 1: Compute singular vectors of upper bidiagonal

*              matrix in compact form.

*

*  SMLSIZ (input) INTEGER

*         The maximum size of the subproblems at the bottom of the

*         computation tree.

*

*  N      (input) INTEGER

*         The row dimension of the upper bidiagonal matrix. This is

*         also the dimension of the main diagonal array D.

*

*  SQRE   (input) INTEGER

*         Specifies the column dimension of the bidiagonal matrix.

*         = 0: The bidiagonal matrix has column dimension M = N;

*         = 1: The bidiagonal matrix has column dimension M = N + 1.

*

*  D      (input/output) DOUBLE PRECISION array, dimension ( N )

*         On entry D contains the main diagonal of the bidiagonal

*         matrix. On exit D, if INFO = 0, contains its singular values.

*

*  E      (input) DOUBLE PRECISION array, dimension ( M-1 )

*         Contains the subdiagonal entries of the bidiagonal matrix.

*         On exit, E has been destroyed.

*

*  U      (output) DOUBLE PRECISION array,

*         dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced

*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left

*         singular vector matrices of all subproblems at the bottom

*         level.

*

*  LDU    (input) INTEGER, LDU = > N.

*         The leading dimension of arrays U, VT, DIFL, DIFR, POLES,

*         GIVNUM, and Z.

*

*  VT     (output) DOUBLE PRECISION array,

*         dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced

*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT' contains the right

*         singular vector matrices of all subproblems at the bottom

*         level.

*

*  K      (output) INTEGER array,

*         dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.

*         If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th

*         secular equation on the computation tree.

*

*  DIFL   (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),

*         where NLVL = floor(log_2 (N/SMLSIZ))).

*

*  DIFR   (output) DOUBLE PRECISION array,

*                  dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and

*                  dimension ( N ) if ICOMPQ = 0.

*         If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)

*         record distances between singular values on the I-th

*         level and singular values on the (I -1)-th level, and

*         DIFR(1:N, 2 * I ) contains the normalizing factors for

*         the right singular vector matrix. See DLASD8 for details.

*

*  Z      (output) DOUBLE PRECISION array,

*                  dimension ( LDU, NLVL ) if ICOMPQ = 1 and

*                  dimension ( N ) if ICOMPQ = 0.

*         The first K elements of Z(1, I) contain the components of

*         the deflation-adjusted updating row vector for subproblems

*         on the I-th level.

*

*  POLES  (output) DOUBLE PRECISION array,

*         dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced

*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and

*         POLES(1, 2*I) contain  the new and old singular values

*         involved in the secular equations on the I-th level.

*

*  GIVPTR (output) INTEGER array,

*         dimension ( N ) if ICOMPQ = 1, and not referenced if

*         ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records

*         the number of Givens rotations performed on the I-th

*         problem on the computation tree.

*

*  GIVCOL (output) INTEGER array,

*         dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not

*         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,

*         GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations

*         of Givens rotations performed on the I-th level on the

*         computation tree.

*

*  LDGCOL (input) INTEGER, LDGCOL = > N.

*         The leading dimension of arrays GIVCOL and PERM.

*

*  PERM   (output) INTEGER array,

*         dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced

*         if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records

*         permutations done on the I-th level of the computation tree.

*

*  GIVNUM (output) DOUBLE PRECISION array,

*         dimension ( LDU,  2 * NLVL ) if ICOMPQ = 1, and not

*         referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,

*         GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-

*         values of Givens rotations performed on the I-th level on

*         the computation tree.

*

*  C      (output) DOUBLE PRECISION array,

*         dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.

*         If ICOMPQ = 1 and the I-th subproblem is not square, on exit,

*         C( I ) contains the C-value of a Givens rotation related to

*         the right null space of the I-th subproblem.

*

*  S      (output) DOUBLE PRECISION array, dimension ( N ) if

*         ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1

*         and the I-th subproblem is not square, on exit, S( I )

*         contains the S-value of a Givens rotation related to

*         the right null space of the I-th subproblem.

*

*  WORK   (workspace) DOUBLE PRECISION array, dimension

*         (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).

*

*  IWORK  (workspace) INTEGER array.

*         Dimension must be at least (7 * N).

*

*  INFO   (output) 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

*

*  Further Details

*  ===============

*

*  Based on contributions by

*     Ming Gu and Huan Ren, Computer Science Division, University of

*     California at Berkeley, USA

*

*  =====================================================================

*

*     .. Parameters ..

      DOUBLE PRECISION   ZERO, ONE

      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )

*     ..

*     .. Local Scalars ..

      INTEGER            I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,

     $                   J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,

     $                   NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,

     $                   NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI

      DOUBLE PRECISION   ALPHA, BETA

*     ..

*     .. External Subroutines ..

      EXTERNAL           DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA

*     ..

*     .. Executable Statements ..

*

*     Test the input parameters.

*

      INFO = 0

*

      IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN

         INFO = -1

      ELSE IF( SMLSIZ.LT.3 ) THEN

         INFO = -2

      ELSE IF( N.LT.0 ) THEN

         INFO = -3

      ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN

         INFO = -4

      ELSE IF( LDU.LT.( N+SQRE ) ) THEN

         INFO = -8

      ELSE IF( LDGCOL.LT.N ) THEN

         INFO = -17

      END IF

      IF( INFO.NE.0 ) THEN

         CALL XERBLA( 'DLASDA', -INFO )

         RETURN

      END IF

*

      M = N + SQRE

*

*     If the input matrix is too small, call DLASDQ to find the SVD.

*

      IF( N.LE.SMLSIZ ) THEN

         IF( ICOMPQ.EQ.0 ) THEN

            CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU,

     $                   U, LDU, WORK, INFO )

         ELSE

            CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU,

     $                   U, LDU, WORK, INFO )

         END IF

         RETURN

      END IF

*

*     Book-keeping and  set up the computation tree.

*

      INODE = 1

      NDIML = INODE + N

      NDIMR = NDIML + N

      IDXQ = NDIMR + N

      IWK = IDXQ + N

*

      NCC = 0

      NRU = 0

*

      SMLSZP = SMLSIZ + 1

      VF = 1

      VL = VF + M

      NWORK1 = VL + M

      NWORK2 = NWORK1 + SMLSZP*SMLSZP

*

      CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),

     $             IWORK( NDIMR ), SMLSIZ )

*

*     for the nodes on bottom level of the tree, solve

*     their subproblems by DLASDQ.

*

      NDB1 = ( ND+1 ) / 2

      DO 30 I = NDB1, ND

*

*        IC : center row of each node

*        NL : number of rows of left  subproblem

*        NR : number of rows of right subproblem

*        NLF: starting row of the left   subproblem

*        NRF: starting row of the right  subproblem

*

         I1 = I - 1

         IC = IWORK( INODE+I1 )

         NL = IWORK( NDIML+I1 )

         NLP1 = NL + 1

         NR = IWORK( NDIMR+I1 )

         NLF = IC - NL

         NRF = IC + 1

         IDXQI = IDXQ + NLF - 2

         VFI = VF + NLF - 1

         VLI = VL + NLF - 1

         SQREI = 1

         IF( ICOMPQ.EQ.0 ) THEN

            CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),

     $                   SMLSZP )

            CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),

     $                   E( NLF ), WORK( NWORK1 ), SMLSZP,

     $                   WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,

     $                   WORK( NWORK2 ), INFO )

            ITEMP = NWORK1 + NL*SMLSZP

            CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )

            CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )

         ELSE

            CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )

            CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU )

            CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),

     $                   E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,

     $                   U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )

            CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )

            CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )

         END IF

         IF( INFO.NE.0 ) THEN

            RETURN

         END IF

         DO 10 J = 1, NL

            IWORK( IDXQI+J ) = J

   10    CONTINUE

         IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN

            SQREI = 0

         ELSE

            SQREI = 1

         END IF

         IDXQI = IDXQI + NLP1

         VFI = VFI + NLP1

         VLI = VLI + NLP1

         NRP1 = NR + SQREI

         IF( ICOMPQ.EQ.0 ) THEN

            CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),

     $                   SMLSZP )

            CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),

     $                   E( NRF ), WORK( NWORK1 ), SMLSZP,

     $                   WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,

     $                   WORK( NWORK2 ), INFO )

            ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP

            CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )

            CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )

         ELSE

            CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )

            CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU )

            CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),

     $                   E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,

     $                   U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )

            CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )

            CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )

         END IF

         IF( INFO.NE.0 ) THEN

            RETURN

         END IF

         DO 20 J = 1, NR

            IWORK( IDXQI+J ) = J

   20    CONTINUE

   30 CONTINUE

*

*     Now conquer each subproblem bottom-up.

*

      J = 2**NLVL

      DO 50 LVL = NLVL, 1, -1

         LVL2 = LVL*2 - 1

*

*        Find the first node LF and last node LL on

*        the current level LVL.

*

         IF( LVL.EQ.1 ) THEN

            LF = 1

            LL = 1

         ELSE

            LF = 2**( LVL-1 )

            LL = 2*LF - 1

         END IF

         DO 40 I = LF, LL

            IM1 = I - 1

            IC = IWORK( INODE+IM1 )

            NL = IWORK( NDIML+IM1 )

            NR = IWORK( NDIMR+IM1 )

            NLF = IC - NL

            NRF = IC + 1

            IF( I.EQ.LL ) THEN

               SQREI = SQRE

            ELSE

               SQREI = 1

            END IF

            VFI = VF + NLF - 1

            VLI = VL + NLF - 1

            IDXQI = IDXQ + NLF - 1

            ALPHA = D( IC )

            BETA = E( IC )

            IF( ICOMPQ.EQ.0 ) THEN

               CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),

     $                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,

     $                      IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,

     $                      LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,

     $                      K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),

     $                      IWORK( IWK ), INFO )

            ELSE

               J = J - 1

               CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),

     $                      WORK( VFI ), WORK( VLI ), ALPHA, BETA,

     $                      IWORK( IDXQI ), PERM( NLF, LVL ),

     $                      GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,

     $                      GIVNUM( NLF, LVL2 ), LDU,

     $                      POLES( NLF, LVL2 ), DIFL( NLF, LVL ),

     $                      DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),

     $                      C( J ), S( J ), WORK( NWORK1 ),

     $                      IWORK( IWK ), INFO )

            END IF

            IF( INFO.NE.0 ) THEN

               RETURN

            END IF

   40    CONTINUE

   50 CONTINUE

*

      RETURN

*

*     End of DLASDA

*

      END