dbdsdc.f

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      SUBROUTINE DBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK routine (instrumented to count ops, version 3.0) --
*     Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
*     Courant Institute, Argonne National Lab, and Rice University
*     October 31, 1999
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, UPLO
      INTEGER            INFO, LDU, LDVT, N
*     ..
*     .. Array Arguments ..
      INTEGER            IQ( * ), IWORK( * )
      DOUBLE PRECISION   D( * ), E( * ), Q( * ), U( LDU, * ),
     $                   VT( LDVT, * ), WORK( * )
*     ..
*     .. Common blocks ..
      COMMON             / LATIME / OPS, ITCNT
*     ..
*     .. Scalars in Common ..
      DOUBLE PRECISION   ITCNT, OPS
*     ..
*
*  Purpose
*  =======
*
*  DBDSDC computes the singular value decomposition (SVD) of a real
*  N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT,
*  using a divide and conquer method, where S is a diagonal matrix
*  with non-negative diagonal elements (the singular values of B), and
*  U and VT are orthogonal matrices of left and right singular vectors,
*  respectively. DBDSDC can be used to compute all singular values,
*  and optionally, singular vectors or singular vectors in compact form.
*
*  This code makes very mild assumptions about floating point
*  arithmetic. It will work on machines with a guard digit in
*  add/subtract, or on those binary machines without guard digits
*  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
*  It could conceivably fail on hexadecimal or decimal machines
*  without guard digits, but we know of none.  See DLASD3 for details.
*
*  The code currently call DLASDQ if singular values only are desired.
*  However, it can be slightly modified to compute singular values
*  using the divide and conquer method.
*
*  Arguments
*  =========
*
*  UPLO    (input) CHARACTER*1
*          = 'U':  B is upper bidiagonal.
*          = 'L':  B is lower bidiagonal.
*
*  COMPQ   (input) CHARACTER*1
*          Specifies whether singular vectors are to be computed
*          as follows:
*          = 'N':  Compute singular values only;
*          = 'P':  Compute singular values and compute singular
*                  vectors in compact form;
*          = 'I':  Compute singular values and singular vectors.
*
*  N       (input) INTEGER
*          The order of the matrix B.  N >= 0.
*
*  D       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the n diagonal elements of the bidiagonal matrix B.
*          On exit, if INFO=0, the singular values of B.
*
*  E       (input/output) DOUBLE PRECISION array, dimension (N)
*          On entry, the elements of E contain the offdiagonal
*          elements of the bidiagonal matrix whose SVD is desired.
*          On exit, E has been destroyed.
*
*  U       (output) DOUBLE PRECISION array, dimension (LDU,N)
*          If  COMPQ = 'I', then:
*             On exit, if INFO = 0, U contains the left singular vectors
*             of the bidiagonal matrix.
*          For other values of COMPQ, U is not referenced.
*
*  LDU     (input) INTEGER
*          The leading dimension of the array U.  LDU >= 1.
*          If singular vectors are desired, then LDU >= max( 1, N ).
*
*  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N)
*          If  COMPQ = 'I', then:
*             On exit, if INFO = 0, VT' contains the right singular
*             vectors of the bidiagonal matrix.
*          For other values of COMPQ, VT is not referenced.
*
*  LDVT    (input) INTEGER
*          The leading dimension of the array VT.  LDVT >= 1.
*          If singular vectors are desired, then LDVT >= max( 1, N ).
*
*  Q       (output) DOUBLE PRECISION array, dimension (LDQ)
*          If  COMPQ = 'P', then:
*             On exit, if INFO = 0, Q and IQ contain the left
*             and right singular vectors in a compact form,
*             requiring O(N log N) space instead of 2*N**2.
*             In particular, Q contains all the DOUBLE PRECISION data in
*             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1))))
*             words of memory, where SMLSIZ is returned by ILAENV and
*             is equal to the maximum size of the subproblems at the
*             bottom of the computation tree (usually about 25).
*          For other values of COMPQ, Q is not referenced.
*
*  IQ      (output) INTEGER array, dimension (LDIQ)
*          If  COMPQ = 'P', then:
*             On exit, if INFO = 0, Q and IQ contain the left
*             and right singular vectors in a compact form,
*             requiring O(N log N) space instead of 2*N**2.
*             In particular, IQ contains all INTEGER data in
*             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1))))
*             words of memory, where SMLSIZ is returned by ILAENV and
*             is equal to the maximum size of the subproblems at the
*             bottom of the computation tree (usually about 25).
*          For other values of COMPQ, IQ is not referenced.
*
*  WORK    (workspace) DOUBLE PRECISION array, dimension (LWORK)
*          If COMPQ = 'N' then LWORK >= (4 * N).
*          If COMPQ = 'P' then LWORK >= (6 * N).
*          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N).
*
*  IWORK   (workspace) INTEGER array, dimension (7*N)
*
*  INFO    (output) INTEGER
*          = 0:  successful exit.
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          > 0:  The algorithm failed to compute an singular value.
*                The update process of divide and conquer failed.
*
*  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, TWO
      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0, TWO = 2.0D+0 )
*     ..
*     .. Local Scalars ..
      INTEGER            DIFL, DIFR, GIVCOL, GIVNUM, GIVPTR, I, IC,
     $                   ICOMPQ, IERR, II, IS, IU, IUPLO, IVT, J, K, KK,
     $                   MLVL, NM1, NSIZE, PERM, POLES, QSTART, SMLSIZ,
     $                   SMLSZP, SQRE, START, WSTART, Z
      DOUBLE PRECISION   CS, EPS, ORGNRM, P, R, SN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANST
      EXTERNAL           LSAME, ILAENV, DLAMCH, DLANST
*     ..
*     .. External Subroutines ..
      EXTERNAL           DCOPY, DLARTG, DLASCL, DLASD0, DLASDA, DLASDQ,
     $                   DLASET, DLASR, DSWAP, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, DBLE, INT, LOG, SIGN
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      INFO = 0
*
      IUPLO = 0
      IF( LSAME( UPLO, 'U' ) )
     $   IUPLO = 1
      IF( LSAME( UPLO, 'L' ) )
     $   IUPLO = 2
      IF( LSAME( COMPQ, 'N' ) ) THEN
         ICOMPQ = 0
      ELSE IF( LSAME( COMPQ, 'P' ) ) THEN
         ICOMPQ = 1
      ELSE IF( LSAME( COMPQ, 'I' ) ) THEN
         ICOMPQ = 2
      ELSE
         ICOMPQ = -1
      END IF
      IF( IUPLO.EQ.0 ) THEN
         INFO = -1
      ELSE IF( ICOMPQ.LT.0 ) THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( ( LDU.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDU.LT.
     $         N ) ) ) THEN
         INFO = -7
      ELSE IF( ( LDVT.LT.1 ) .OR. ( ( ICOMPQ.EQ.2 ) .AND. ( LDVT.LT.
     $         N ) ) ) THEN
         INFO = -9
      END IF
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'DBDSDC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
      SMLSIZ = ILAENV( 9, 'DBDSDC', ' ', 0, 0, 0, 0 )
      IF( N.EQ.1 ) THEN
         IF( ICOMPQ.EQ.1 ) THEN
            Q( 1 ) = SIGN( ONE, D( 1 ) )
            Q( 1+SMLSIZ*N ) = ONE
         ELSE IF( ICOMPQ.EQ.2 ) THEN
            U( 1, 1 ) = SIGN( ONE, D( 1 ) )
            VT( 1, 1 ) = ONE
         END IF
         D( 1 ) = ABS( D( 1 ) )
         RETURN
      END IF
      NM1 = N - 1
*
*     If matrix lower bidiagonal, rotate to be upper bidiagonal
*     by applying Givens rotations on the left
*
      WSTART = 1
      QSTART = 3
      IF( ICOMPQ.EQ.1 ) THEN
         CALL DCOPY( N, D, 1, Q( 1 ), 1 )
         CALL DCOPY( N-1, E, 1, Q( N+1 ), 1 )
      END IF
      IF( IUPLO.EQ.2 ) THEN
         QSTART = 5
         WSTART = 2*N - 1
         OPS = OPS + DBLE( 8*( N-1 ) )
         DO 10 I = 1, N - 1
            CALL DLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
            IF( ICOMPQ.EQ.1 ) THEN
               Q( I+2*N ) = CS
               Q( I+3*N ) = SN
            ELSE IF( ICOMPQ.EQ.2 ) THEN
               WORK( I ) = CS
               WORK( NM1+I ) = -SN
            END IF
   10    CONTINUE
      END IF
*
*     If ICOMPQ = 0, use DLASDQ to compute the singular values.
*
      IF( ICOMPQ.EQ.0 ) THEN
         CALL DLASDQ( 'U', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
     $                LDU, WORK( WSTART ), INFO )
         GO TO 40
      END IF
*
*     If N is smaller than the minimum divide size SMLSIZ, then solve
*     the problem with another solver.
*
      IF( N.LE.SMLSIZ ) THEN
         IF( ICOMPQ.EQ.2 ) THEN
            CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
            CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
            CALL DLASDQ( 'U', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU, U,
     $                   LDU, WORK( WSTART ), INFO )
         ELSE IF( ICOMPQ.EQ.1 ) THEN
            IU = 1
            IVT = IU + N
            CALL DLASET( 'A', N, N, ZERO, ONE, Q( IU+( QSTART-1 )*N ),
     $                   N )
            CALL DLASET( 'A', N, N, ZERO, ONE, Q( IVT+( QSTART-1 )*N ),
     $                   N )
            CALL DLASDQ( 'U', 0, N, N, N, 0, D, E,
     $                   Q( IVT+( QSTART-1 )*N ), N,
     $                   Q( IU+( QSTART-1 )*N ), N,
     $                   Q( IU+( QSTART-1 )*N ), N, WORK( WSTART ),
     $                   INFO )
         END IF
         GO TO 40
      END IF
*
      IF( ICOMPQ.EQ.2 ) THEN
         CALL DLASET( 'A', N, N, ZERO, ONE, U, LDU )
         CALL DLASET( 'A', N, N, ZERO, ONE, VT, LDVT )
      END IF
*
*     Scale.
*
      ORGNRM = DLANST( 'M', N, D, E )
      IF( ORGNRM.EQ.ZERO )
     $   RETURN
      OPS = OPS + DBLE( N+NM1 )
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR )
      CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR )
*
      EPS = DLAMCH( 'Epsilon' )
*
      MLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
      SMLSZP = SMLSIZ + 1
*
      IF( ICOMPQ.EQ.1 ) THEN
         IU = 1
         IVT = 1 + SMLSIZ
         DIFL = IVT + SMLSZP
         DIFR = DIFL + MLVL
         Z = DIFR + MLVL*2
         IC = Z + MLVL
         IS = IC + 1
         POLES = IS + 1
         GIVNUM = POLES + 2*MLVL
*
         K = 1
         GIVPTR = 2
         PERM = 3
         GIVCOL = PERM + MLVL
      END IF
*
      DO 20 I = 1, N
         IF( ABS( D( I ) ).LT.EPS ) THEN
            D( I ) = SIGN( EPS, D( I ) )
         END IF
   20 CONTINUE
*
      START = 1
      SQRE = 0
*
      DO 30 I = 1, NM1
         IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
*
*        Subproblem found. First determine its size and then
*        apply divide and conquer on it.
*
            IF( I.LT.NM1 ) THEN
*
*        A subproblem with E(I) small for I < NM1.
*
               NSIZE = I - START + 1
            ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
*
*        A subproblem with E(NM1) not too small but I = NM1.
*
               NSIZE = N - START + 1
            ELSE
*
*        A subproblem with E(NM1) small. This implies an
*        1-by-1 subproblem at D(N). Solve this 1-by-1 problem
*        first.
*
               NSIZE = I - START + 1
               IF( ICOMPQ.EQ.2 ) THEN
                  U( N, N ) = SIGN( ONE, D( N ) )
                  VT( N, N ) = ONE
               ELSE IF( ICOMPQ.EQ.1 ) THEN
                  Q( N+( QSTART-1 )*N ) = SIGN( ONE, D( N ) )
                  Q( N+( SMLSIZ+QSTART-1 )*N ) = ONE
               END IF
               D( N ) = ABS( D( N ) )
            END IF
            IF( ICOMPQ.EQ.2 ) THEN
               CALL DLASD0( NSIZE, SQRE, D( START ), E( START ),
     $                      U( START, START ), LDU, VT( START, START ),
     $                      LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO )
            ELSE
               CALL DLASDA( ICOMPQ, SMLSIZ, NSIZE, SQRE, D( START ),
     $                      E( START ), Q( START+( IU+QSTART-2 )*N ), N,
     $                      Q( START+( IVT+QSTART-2 )*N ),
     $                      IQ( START+K*N ), Q( START+( DIFL+QSTART-2 )*
     $                      N ), Q( START+( DIFR+QSTART-2 )*N ),
     $                      Q( START+( Z+QSTART-2 )*N ),
     $                      Q( START+( POLES+QSTART-2 )*N ),
     $                      IQ( START+GIVPTR*N ), IQ( START+GIVCOL*N ),
     $                      N, IQ( START+PERM*N ),
     $                      Q( START+( GIVNUM+QSTART-2 )*N ),
     $                      Q( START+( IC+QSTART-2 )*N ),
     $                      Q( START+( IS+QSTART-2 )*N ),
     $                      WORK( WSTART ), IWORK, INFO )
               IF( INFO.NE.0 ) THEN
                  RETURN
               END IF
            END IF
            START = I + 1
         END IF
   30 CONTINUE
*
*     Unscale
*
      OPS = OPS + DBLE( N )
      CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, IERR )
   40 CONTINUE
*
*     Use Selection Sort to minimize swaps of singular vectors
*
      DO 60 II = 2, N
         I = II - 1
         KK = I
         P = D( I )
         DO 50 J = II, N
            IF( D( J ).GT.P ) THEN
               KK = J
               P = D( J )
            END IF
   50    CONTINUE
         IF( KK.NE.I ) THEN
            D( KK ) = D( I )
            D( I ) = P
            IF( ICOMPQ.EQ.1 ) THEN
               IQ( I ) = KK
            ELSE IF( ICOMPQ.EQ.2 ) THEN
               CALL DSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 )
               CALL DSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT )
            END IF
         ELSE IF( ICOMPQ.EQ.1 ) THEN
            IQ( I ) = I
         END IF
   60 CONTINUE
*
*     If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO
*
      IF( ICOMPQ.EQ.1 ) THEN
         IF( IUPLO.EQ.1 ) THEN
            IQ( N ) = 1
         ELSE
            IQ( N ) = 0
         END IF
      END IF
*
*     If B is lower bidiagonal, update U by those Givens rotations
*     which rotated B to be upper bidiagonal
*
      IF( ( IUPLO.EQ.2 ) .AND. ( ICOMPQ.EQ.2 ) ) THEN
         OPS = OPS + DBLE( 6*( N-1 )*N )
         CALL DLASR( 'L', 'V', 'B', N, N, WORK( 1 ), WORK( N ), U, LDU )
      END IF
*
      RETURN
*
*     End of DBDSDC
*
      END