dlalsd.f

          SUBROUTINE DLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND,
         $                   RANK, WORK, IWORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, LDB, N, NRHS, RANK, SMLSIZ
          DOUBLE PRECISION   RCOND
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   B( LDB, * ), D( * ), E( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DLALSD uses the singular value decomposition of A to solve the least
    *  squares problem of finding X to minimize the Euclidean norm of each
    *  column of A*X-B, where A is N-by-N upper bidiagonal, and X and B
    *  are N-by-NRHS. The solution X overwrites B.
    *
    *  The singular values of A smaller than RCOND times the largest
    *  singular value are treated as zero in solving the least squares
    *  problem; in this case a minimum norm solution is returned.
    *  The actual singular values are returned in D in ascending order.
    *
    *  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 XMP, Cray YMP, Cray C 90, or Cray 2.
    *  It could conceivably fail on hexadecimal or decimal machines
    *  without guard digits, but we know of none.
    *
    *  Arguments
    *  =========
    *
    *  UPLO   (input) CHARACTER*1
    *         = 'U': D and E define an upper bidiagonal matrix.
    *         = 'L': D and E define a  lower bidiagonal matrix.
    *
    *  SMLSIZ (input) INTEGER
    *         The maximum size of the subproblems at the bottom of the
    *         computation tree.
    *
    *  N      (input) INTEGER
    *         The dimension of the  bidiagonal matrix.  N >= 0.
    *
    *  NRHS   (input) INTEGER
    *         The number of columns of B. NRHS must be at least 1.
    *
    *  D      (input/output) DOUBLE PRECISION array, dimension (N)
    *         On entry D contains the main diagonal of the bidiagonal
    *         matrix. On exit, if INFO = 0, D contains its singular values.
    *
    *  E      (input/output) DOUBLE PRECISION array, dimension (N-1)
    *         Contains the super-diagonal entries of the bidiagonal matrix.
    *         On exit, E has been destroyed.
    *
    *  B      (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
    *         On input, B contains the right hand sides of the least
    *         squares problem. On output, B contains the solution X.
    *
    *  LDB    (input) INTEGER
    *         The leading dimension of B in the calling subprogram.
    *         LDB must be at least max(1,N).
    *
    *  RCOND  (input) DOUBLE PRECISION
    *         The singular values of A less than or equal to RCOND times
    *         the largest singular value are treated as zero in solving
    *         the least squares problem. If RCOND is negative,
    *         machine precision is used instead.
    *         For example, if diag(S)*X=B were the least squares problem,
    *         where diag(S) is a diagonal matrix of singular values, the
    *         solution would be X(i) = B(i) / S(i) if S(i) is greater than
    *         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
    *         RCOND*max(S).
    *
    *  RANK   (output) INTEGER
    *         The number of singular values of A greater than RCOND times
    *         the largest singular value.
    *
    *  WORK   (workspace) DOUBLE PRECISION array, dimension at least
    *         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2),
    *         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
    *
    *  IWORK  (workspace) INTEGER array, dimension at least
    *         (3*N*NLVL + 11*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 while
    *               working on the submatrix lying in rows and columns
    *               INFO/(N+1) through MOD(INFO,N+1).
    *
    *  Further Details
    *  ===============
    *
    *  Based on contributions by
    *     Ming Gu and Ren-Cang Li, Computer Science Division, University of
    *       California at Berkeley, USA
    *     Osni Marques, LBNL/NERSC, USA
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ZERO, ONE, TWO
          PARAMETER          ( ZERO = 0.0D0, ONE = 1.0D0, TWO = 2.0D0 )
    *     ..
    *     .. Local Scalars ..
          INTEGER            BX, BXST, C, DIFL, DIFR, GIVCOL, GIVNUM,
         $                   GIVPTR, I, ICMPQ1, ICMPQ2, IWK, J, K, NLVL,
         $                   NM1, NSIZE, NSUB, NWORK, PERM, POLES, S, SIZEI,
         $                   SMLSZP, SQRE, ST, ST1, U, VT, Z
          DOUBLE PRECISION   CS, EPS, ORGNRM, R, RCND, SN, TOL
    *     ..
    *     .. External Functions ..
          INTEGER            IDAMAX
          DOUBLE PRECISION   DLAMCH, DLANST
          EXTERNAL           IDAMAX, DLAMCH, DLANST
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DCOPY, DGEMM, DLACPY, DLALSA, DLARTG, DLASCL,
         $                   DLASDA, DLASDQ, DLASET, DLASRT, DROT, XERBLA
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          ABS, DBLE, INT, LOG, SIGN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters.
    *
          INFO = 0
    *
          IF( N.LT.0 ) THEN
             INFO = -3
          ELSE IF( NRHS.LT.1 ) THEN
             INFO = -4
          ELSE IF( ( LDB.LT.1 ) .OR. ( LDB.LT.N ) ) THEN
             INFO = -8
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DLALSD', -INFO )
             RETURN
          END IF
    *
          EPS = DLAMCH( 'Epsilon' )
    *
    *     Set up the tolerance.
    *
          IF( ( RCOND.LE.ZERO ) .OR. ( RCOND.GE.ONE ) ) THEN
             RCND = EPS
          ELSE
             RCND = RCOND
          END IF
    *
          RANK = 0
    *
    *     Quick return if possible.
    *
          IF( N.EQ.0 ) THEN
             RETURN
          ELSE IF( N.EQ.1 ) THEN
             IF( D( 1 ).EQ.ZERO ) THEN
                CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B, LDB )
             ELSE
                RANK = 1
                CALL DLASCL( 'G', 0, 0, D( 1 ), ONE, 1, NRHS, B, LDB, INFO )
                D( 1 ) = ABS( D( 1 ) )
             END IF
             RETURN
          END IF
    *
    *     Rotate the matrix if it is lower bidiagonal.
    *
          IF( UPLO.EQ.'L' ) THEN
             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( NRHS.EQ.1 ) THEN
                   CALL DROT( 1, B( I, 1 ), 1, B( I+1, 1 ), 1, CS, SN )
                ELSE
                   WORK( I*2-1 ) = CS
                   WORK( I*2 ) = SN
                END IF
       10    CONTINUE
             IF( NRHS.GT.1 ) THEN
                DO 30 I = 1, NRHS
                   DO 20 J = 1, N - 1
                      CS = WORK( J*2-1 )
                      SN = WORK( J*2 )
                      CALL DROT( 1, B( J, I ), 1, B( J+1, I ), 1, CS, SN )
       20          CONTINUE
       30       CONTINUE
             END IF
          END IF
    *
    *     Scale.
    *
          NM1 = N - 1
          ORGNRM = DLANST( 'M', N, D, E )
          IF( ORGNRM.EQ.ZERO ) THEN
             CALL DLASET( 'A', N, NRHS, ZERO, ZERO, B, LDB )
             RETURN
          END IF
    *
          CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, 1, D, N, INFO )
          CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, NM1, 1, E, NM1, INFO )
    *
    *     If N is smaller than the minimum divide size SMLSIZ, then solve
    *     the problem with another solver.
    *
          IF( N.LE.SMLSIZ ) THEN
             NWORK = 1 + N*N
             CALL DLASET( 'A', N, N, ZERO, ONE, WORK, N )
             CALL DLASDQ( 'U', 0, N, N, 0, NRHS, D, E, WORK, N, WORK, N, B,
         $                LDB, WORK( NWORK ), INFO )
             IF( INFO.NE.0 ) THEN
                RETURN
             END IF
             TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
             DO 40 I = 1, N
                IF( D( I ).LE.TOL ) THEN
                   CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, B( I, 1 ), LDB )
                ELSE
                   CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS, B( I, 1 ),
         $                      LDB, INFO )
                   RANK = RANK + 1
                END IF
       40    CONTINUE
             CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO,
         $               WORK( NWORK ), N )
             CALL DLACPY( 'A', N, NRHS, WORK( NWORK ), N, B, LDB )
    *
    *        Unscale.
    *
             CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
             CALL DLASRT( 'D', N, D, INFO )
             CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
    *
             RETURN
          END IF
    *
    *     Book-keeping and setting up some constants.
    *
          NLVL = INT( LOG( DBLE( N ) / DBLE( SMLSIZ+1 ) ) / LOG( TWO ) ) + 1
    *
          SMLSZP = SMLSIZ + 1
    *
          U = 1
          VT = 1 + SMLSIZ*N
          DIFL = VT + SMLSZP*N
          DIFR = DIFL + NLVL*N
          Z = DIFR + NLVL*N*2
          C = Z + NLVL*N
          S = C + N
          POLES = S + N
          GIVNUM = POLES + 2*NLVL*N
          BX = GIVNUM + 2*NLVL*N
          NWORK = BX + N*NRHS
    *
          SIZEI = 1 + N
          K = SIZEI + N
          GIVPTR = K + N
          PERM = GIVPTR + N
          GIVCOL = PERM + NLVL*N
          IWK = GIVCOL + NLVL*N*2
    *
          ST = 1
          SQRE = 0
          ICMPQ1 = 1
          ICMPQ2 = 0
          NSUB = 0
    *
          DO 50 I = 1, N
             IF( ABS( D( I ) ).LT.EPS ) THEN
                D( I ) = SIGN( EPS, D( I ) )
             END IF
       50 CONTINUE
    *
          DO 60 I = 1, NM1
             IF( ( ABS( E( I ) ).LT.EPS ) .OR. ( I.EQ.NM1 ) ) THEN
                NSUB = NSUB + 1
                IWORK( NSUB ) = ST
    *
    *           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 - ST + 1
                   IWORK( SIZEI+NSUB-1 ) = NSIZE
                ELSE IF( ABS( E( I ) ).GE.EPS ) THEN
    *
    *              A subproblem with E(NM1) not too small but I = NM1.
    *
                   NSIZE = N - ST + 1
                   IWORK( SIZEI+NSUB-1 ) = NSIZE
                ELSE
    *
    *              A subproblem with E(NM1) small. This implies an
    *              1-by-1 subproblem at D(N), which is not solved
    *              explicitly.
    *
                   NSIZE = I - ST + 1
                   IWORK( SIZEI+NSUB-1 ) = NSIZE
                   NSUB = NSUB + 1
                   IWORK( NSUB ) = N
                   IWORK( SIZEI+NSUB-1 ) = 1
                   CALL DCOPY( NRHS, B( N, 1 ), LDB, WORK( BX+NM1 ), N )
                END IF
                ST1 = ST - 1
                IF( NSIZE.EQ.1 ) THEN
    *
    *              This is a 1-by-1 subproblem and is not solved
    *              explicitly.
    *
                   CALL DCOPY( NRHS, B( ST, 1 ), LDB, WORK( BX+ST1 ), N )
                ELSE IF( NSIZE.LE.SMLSIZ ) THEN
    *
    *              This is a small subproblem and is solved by DLASDQ.
    *
                   CALL DLASET( 'A', NSIZE, NSIZE, ZERO, ONE,
         $                      WORK( VT+ST1 ), N )
                   CALL DLASDQ( 'U', 0, NSIZE, NSIZE, 0, NRHS, D( ST ),
         $                      E( ST ), WORK( VT+ST1 ), N, WORK( NWORK ),
         $                      N, B( ST, 1 ), LDB, WORK( NWORK ), INFO )
                   IF( INFO.NE.0 ) THEN
                      RETURN
                   END IF
                   CALL DLACPY( 'A', NSIZE, NRHS, B( ST, 1 ), LDB,
         $                      WORK( BX+ST1 ), N )
                ELSE
    *
    *              A large problem. Solve it using divide and conquer.
    *
                   CALL DLASDA( ICMPQ1, SMLSIZ, NSIZE, SQRE, D( ST ),
         $                      E( ST ), WORK( U+ST1 ), N, WORK( VT+ST1 ),
         $                      IWORK( K+ST1 ), WORK( DIFL+ST1 ),
         $                      WORK( DIFR+ST1 ), WORK( Z+ST1 ),
         $                      WORK( POLES+ST1 ), IWORK( GIVPTR+ST1 ),
         $                      IWORK( GIVCOL+ST1 ), N, IWORK( PERM+ST1 ),
         $                      WORK( GIVNUM+ST1 ), WORK( C+ST1 ),
         $                      WORK( S+ST1 ), WORK( NWORK ), IWORK( IWK ),
         $                      INFO )
                   IF( INFO.NE.0 ) THEN
                      RETURN
                   END IF
                   BXST = BX + ST1
                   CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, B( ST, 1 ),
         $                      LDB, WORK( BXST ), N, WORK( U+ST1 ), N,
         $                      WORK( VT+ST1 ), IWORK( K+ST1 ),
         $                      WORK( DIFL+ST1 ), WORK( DIFR+ST1 ),
         $                      WORK( Z+ST1 ), WORK( POLES+ST1 ),
         $                      IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
         $                      IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ),
         $                      WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ),
         $                      IWORK( IWK ), INFO )
                   IF( INFO.NE.0 ) THEN
                      RETURN
                   END IF
                END IF
                ST = I + 1
             END IF
       60 CONTINUE
    *
    *     Apply the singular values and treat the tiny ones as zero.
    *
          TOL = RCND*ABS( D( IDAMAX( N, D, 1 ) ) )
    *
          DO 70 I = 1, N
    *
    *        Some of the elements in D can be negative because 1-by-1
    *        subproblems were not solved explicitly.
    *
             IF( ABS( D( I ) ).LE.TOL ) THEN
                CALL DLASET( 'A', 1, NRHS, ZERO, ZERO, WORK( BX+I-1 ), N )
             ELSE
                RANK = RANK + 1
                CALL DLASCL( 'G', 0, 0, D( I ), ONE, 1, NRHS,
         $                   WORK( BX+I-1 ), N, INFO )
             END IF
             D( I ) = ABS( D( I ) )
       70 CONTINUE
    *
    *     Now apply back the right singular vectors.
    *
          ICMPQ2 = 1
          DO 80 I = 1, NSUB
             ST = IWORK( I )
             ST1 = ST - 1
             NSIZE = IWORK( SIZEI+I-1 )
             BXST = BX + ST1
             IF( NSIZE.EQ.1 ) THEN
                CALL DCOPY( NRHS, WORK( BXST ), N, B( ST, 1 ), LDB )
             ELSE IF( NSIZE.LE.SMLSIZ ) THEN
                CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,
         $                  WORK( VT+ST1 ), N, WORK( BXST ), N, ZERO,
         $                  B( ST, 1 ), LDB )
             ELSE
                CALL DLALSA( ICMPQ2, SMLSIZ, NSIZE, NRHS, WORK( BXST ), N,
         $                   B( ST, 1 ), LDB, WORK( U+ST1 ), N,
         $                   WORK( VT+ST1 ), IWORK( K+ST1 ),
         $                   WORK( DIFL+ST1 ), WORK( DIFR+ST1 ),
         $                   WORK( Z+ST1 ), WORK( POLES+ST1 ),
         $                   IWORK( GIVPTR+ST1 ), IWORK( GIVCOL+ST1 ), N,
         $                   IWORK( PERM+ST1 ), WORK( GIVNUM+ST1 ),
         $                   WORK( C+ST1 ), WORK( S+ST1 ), WORK( NWORK ),
         $                   IWORK( IWK ), INFO )
                IF( INFO.NE.0 ) THEN
                   RETURN
                END IF
             END IF
       80 CONTINUE
    *
    *     Unscale and sort the singular values.
    *
          CALL DLASCL( 'G', 0, 0, ONE, ORGNRM, N, 1, D, N, INFO )
          CALL DLASRT( 'D', N, D, INFO )
          CALL DLASCL( 'G', 0, 0, ORGNRM, ONE, N, NRHS, B, LDB, INFO )
    *
          RETURN
    *
    *     End of DLALSD
    *
          END