sbdsdc.f

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*> \brief \b SBDSDC
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SBDSDC( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
*                          WORK, IWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          COMPQ, UPLO
*       INTEGER            INFO, LDU, LDVT, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IQ( * ), IWORK( * )
*       REAL               D( * ), E( * ), Q( * ), U( LDU, * ),
*      $                   VT( LDVT, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SBDSDC 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. SBDSDC 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 SLASD3 for details.
*>
*> The code currently calls SLASDQ if singular values only are desired.
*> However, it can be slightly modified to compute singular values
*> using the divide and conquer method.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  B is upper bidiagonal.
*>          = 'L':  B is lower bidiagonal.
*> \endverbatim
*>
*> \param[in] COMPQ
*> \verbatim
*>          COMPQ is 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.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix B.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>          On entry, the n diagonal elements of the bidiagonal matrix B.
*>          On exit, if INFO=0, the singular values of B.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is REAL array, dimension (N-1)
*>          On entry, the elements of E contain the offdiagonal
*>          elements of the bidiagonal matrix whose SVD is desired.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*>          U is REAL 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.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*>          LDU is INTEGER
*>          The leading dimension of the array U.  LDU >= 1.
*>          If singular vectors are desired, then LDU >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] VT
*> \verbatim
*>          VT is REAL array, dimension (LDVT,N)
*>          If  COMPQ = 'I', then:
*>             On exit, if INFO = 0, VT**T contains the right singular
*>             vectors of the bidiagonal matrix.
*>          For other values of COMPQ, VT is not referenced.
*> \endverbatim
*>
*> \param[in] LDVT
*> \verbatim
*>          LDVT is INTEGER
*>          The leading dimension of the array VT.  LDVT >= 1.
*>          If singular vectors are desired, then LDVT >= max( 1, N ).
*> \endverbatim
*>
*> \param[out] Q
*> \verbatim
*>          Q is REAL 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 REAL 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.
*> \endverbatim
*>
*> \param[out] IQ
*> \verbatim
*>          IQ is 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (MAX(1,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).
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          > 0:  The algorithm failed to compute a singular value.
*>                The update process of divide and conquer failed.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup auxOTHERcomputational
*
*> \par Contributors:
*  ==================
*>
*>     Ming Gu and Huan Ren, Computer Science Division, University of
*>     California at Berkeley, USA
*>
*  =====================================================================
      SUBROUTINE sbdsdc( UPLO, COMPQ, N, D, E, U, LDU, VT, LDVT, Q, IQ,
     $                   WORK, IWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          COMPQ, UPLO
      INTEGER            INFO, LDU, LDVT, N
*     ..
*     .. Array Arguments ..
      INTEGER            IQ( * ), IWORK( * )
      REAL               D( * ), E( * ), Q( * ), U( LDU, * ),
     $                   vt( ldvt, * ), work( * )
*     ..
*
*  =====================================================================
*  Changed dimension statement in comment describing E from (N) to
*  (N-1).  Sven, 17 Feb 05.
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE, TWO
      PARAMETER          ( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+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
      REAL               CS, EPS, ORGNRM, P, R, SN
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      REAL               SLAMCH, SLANST
      EXTERNAL           slamch, slanst, ilaenv, lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           scopy, slartg, slascl, slasd0, slasda, slasdq,
     $                   slaset, slasr, sswap, xerbla
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          real, abs, 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
.EQ.      IF( IUPLO0 ) THEN
         INFO = -1
.LT.      ELSE IF( ICOMPQ0 ) THEN
         INFO = -2
.LT.      ELSE IF( N0 ) THEN
         INFO = -3
.LT..OR..EQ..AND..LT.      ELSE IF( ( LDU1 )  ( ( ICOMPQ2 )  ( LDU
     $         N ) ) ) THEN
         INFO = -7
.LT..OR..EQ..AND..LT.      ELSE IF( ( LDVT1 )  ( ( ICOMPQ2 )  ( LDVT
     $         N ) ) ) THEN
         INFO = -9
      END IF
.NE.      IF( INFO0 ) THEN
         CALL XERBLA( 'sbdsdc', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
.EQ.      IF( N0 )
     $   RETURN
      SMLSIZ = ILAENV( 9, 'sbdsdc', ' ', 0, 0, 0, 0 )
.EQ.      IF( N1 ) THEN
.EQ.         IF( ICOMPQ1 ) THEN
            Q( 1 ) = SIGN( ONE, D( 1 ) )
            Q( 1+SMLSIZ*N ) = ONE
.EQ.         ELSE IF( ICOMPQ2 ) 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
.EQ.      IF( ICOMPQ1 ) THEN
         CALL SCOPY( N, D, 1, Q( 1 ), 1 )
         CALL SCOPY( N-1, E, 1, Q( N+1 ), 1 )
      END IF
.EQ.      IF( IUPLO2 ) THEN
         QSTART = 5
.EQ.         IF( ICOMPQ  2 ) WSTART = 2*N - 1
         DO 10 I = 1, N - 1
            CALL SLARTG( D( I ), E( I ), CS, SN, R )
            D( I ) = R
            E( I ) = SN*D( I+1 )
            D( I+1 ) = CS*D( I+1 )
.EQ.            IF( ICOMPQ1 ) THEN
               Q( I+2*N ) = CS
               Q( I+3*N ) = SN
.EQ.            ELSE IF( ICOMPQ2 ) THEN
               WORK( I ) = CS
               WORK( NM1+I ) = -SN
            END IF
   10    CONTINUE
      END IF
*
*     If ICOMPQ = 0, use SLASDQ to compute the singular values.
*
.EQ.      IF( ICOMPQ0 ) THEN
*        Ignore WSTART, instead using WORK( 1 ), since the two vectors
*        for CS and -SN above are added only if ICOMPQ == 2,
*        and adding them exceeds documented WORK size of 4*n.
         CALL SLASDQ( 'u', 0, N, 0, 0, 0, D, E, VT, LDVT, U, LDU, U,
     $                LDU, WORK( 1 ), INFO )
         GO TO 40
      END IF
*
*     If N is smaller than the minimum divide size SMLSIZ, then solve
*     the problem with another solver.
*
.LE.      IF( NSMLSIZ ) THEN
.EQ.         IF( ICOMPQ2 ) THEN
            CALL SLASET( 'a', N, N, ZERO, ONE, U, LDU )
            CALL SLASET( 'a', N, N, ZERO, ONE, VT, LDVT )
            CALL SLASDQ( 'u', 0, N, N, N, 0, D, E, VT, LDVT, U, LDU, U,
     $                   LDU, WORK( WSTART ), INFO )
.EQ.         ELSE IF( ICOMPQ1 ) THEN
            IU = 1
            IVT = IU + N
            CALL SLASET( 'a', N, N, ZERO, ONE, Q( IU+( QSTART-1 )*N ),
     $                   N )
            CALL SLASET( 'a', N, N, ZERO, ONE, Q( IVT+( QSTART-1 )*N ),
     $                   N )
            CALL SLASDQ( '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
*
.EQ.      IF( ICOMPQ2 ) THEN
         CALL SLASET( 'a', N, N, ZERO, ONE, U, LDU )
         CALL SLASET( 'a', N, N, ZERO, ONE, VT, LDVT )
      END IF
*
*     Scale.
*
      ORGNRM = SLANST( 'm', N, D, E )
.EQ.      IF( ORGNRMZERO )
     $   RETURN
      CALL SLASCL( 'g', 0, 0, ORGNRM, ONE, N, 1, D, N, IERR )
      CALL SLASCL( 'g', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, IERR )
*
      EPS = SLAMCH( 'epsilon' )
*
      MLVL = INT( LOG( REAL( N ) / REAL( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
      SMLSZP = SMLSIZ + 1
*
.EQ.      IF( ICOMPQ1 ) 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
.LT.         IF( ABS( D( I ) )EPS ) THEN
            D( I ) = SIGN( EPS, D( I ) )
         END IF
   20 CONTINUE
*
      START = 1
      SQRE = 0
*
      DO 30 I = 1, NM1
.LT..OR..EQ.         IF( ( ABS( E( I ) )EPS )  ( INM1 ) ) THEN
*
*        Subproblem found. First determine its size and then
*        apply divide and conquer on it.
*
.LT.            IF( INM1 ) THEN
*
*        A subproblem with E(I) small for I < NM1.
*
               NSIZE = I - START + 1
.GE.            ELSE IF( ABS( E( I ) )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
.EQ.               IF( ICOMPQ2 ) THEN
                  U( N, N ) = SIGN( ONE, D( N ) )
                  VT( N, N ) = ONE
.EQ.               ELSE IF( ICOMPQ1 ) 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
.EQ.            IF( ICOMPQ2 ) THEN
               CALL SLASD0( NSIZE, SQRE, D( START ), E( START ),
     $                      U( START, START ), LDU, VT( START, START ),
     $                      LDVT, SMLSIZ, IWORK, WORK( WSTART ), INFO )
            ELSE
               CALL SLASDA( 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 )
            END IF
.NE.            IF( INFO0 ) THEN
               RETURN
            END IF
            START = I + 1
         END IF
   30 CONTINUE
*
*     Unscale
*
      CALL SLASCL( '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
.GT.            IF( D( J )P ) THEN
               KK = J
               P = D( J )
            END IF
   50    CONTINUE
.NE.         IF( KKI ) THEN
            D( KK ) = D( I )
            D( I ) = P
.EQ.            IF( ICOMPQ1 ) THEN
               IQ( I ) = KK
.EQ.            ELSE IF( ICOMPQ2 ) THEN
               CALL SSWAP( N, U( 1, I ), 1, U( 1, KK ), 1 )
               CALL SSWAP( N, VT( I, 1 ), LDVT, VT( KK, 1 ), LDVT )
            END IF
.EQ.         ELSE IF( ICOMPQ1 ) THEN
            IQ( I ) = I
         END IF
   60 CONTINUE
*
*     If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO
*
.EQ.      IF( ICOMPQ1 ) THEN
.EQ.         IF( IUPLO1 ) 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
*
.EQ..AND..EQ.      IF( ( IUPLO2 )  ( ICOMPQ2 ) )
     $   CALL SLASR( 'l', 'v', 'b', N, N, WORK( 1 ), WORK( N ), U, LDU )
*
      RETURN
*
*     End of SBDSDC
*
      END