dgelsd.f

          SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,
         $                   WORK, LWORK, IWORK, INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
          DOUBLE PRECISION   RCOND
    *     ..
    *     .. Array Arguments ..
          INTEGER            IWORK( * )
          DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DGELSD computes the minimum-norm solution to a real linear least
    *  squares problem:
    *      minimize 2-norm(| b - A*x |)
    *  using the singular value decomposition (SVD) of A. A is an M-by-N
    *  matrix which may be rank-deficient.
    *
    *  Several right hand side vectors b and solution vectors x can be
    *  handled in a single call; they are stored as the columns of the
    *  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
    *  matrix X.
    *
    *  The problem is solved in three steps:
    *  (1) Reduce the coefficient matrix A to bidiagonal form with
    *      Householder transformations, reducing the original problem
    *      into a "bidiagonal least squares problem" (BLS)
    *  (2) Solve the BLS using a divide and conquer approach.
    *  (3) Apply back all the Householder tranformations to solve
    *      the original least squares problem.
    *
    *  The effective rank of A is determined by treating as zero those
    *  singular values which are less than RCOND times the largest singular
    *  value.
    *
    *  The divide and conquer algorithm 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.
    *
    *  Arguments
    *  =========
    *
    *  M       (input) INTEGER
    *          The number of rows of A. M >= 0.
    *
    *  N       (input) INTEGER
    *          The number of columns of A. N >= 0.
    *
    *  NRHS    (input) INTEGER
    *          The number of right hand sides, i.e., the number of columns
    *          of the matrices B and X. NRHS >= 0.
    *
    *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the M-by-N matrix A.
    *          On exit, A has been destroyed.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,M).
    *
    *  B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
    *          On entry, the M-by-NRHS right hand side matrix B.
    *          On exit, B is overwritten by the N-by-NRHS solution
    *          matrix X.  If m >= n and RANK = n, the residual
    *          sum-of-squares for the solution in the i-th column is given
    *          by the sum of squares of elements n+1:m in that column.
    *
    *  LDB     (input) INTEGER
    *          The leading dimension of the array B. LDB >= max(1,max(M,N)).
    *
    *  S       (output) DOUBLE PRECISION array, dimension (min(M,N))
    *          The singular values of A in decreasing order.
    *          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
    *
    *  RCOND   (input) DOUBLE PRECISION
    *          RCOND is used to determine the effective rank of A.
    *          Singular values S(i) <= RCOND*S(1) are treated as zero.
    *          If RCOND < 0, machine precision is used instead.
    *
    *  RANK    (output) INTEGER
    *          The effective rank of A, i.e., the number of singular values
    *          which are greater than RCOND*S(1).
    *
    *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
    *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
    *
    *  LWORK   (input) INTEGER
    *          The dimension of the array WORK. LWORK must be at least 1.
    *          The exact minimum amount of workspace needed depends on M,
    *          N and NRHS. As long as LWORK is at least
    *              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
    *          if M is greater than or equal to N or
    *              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
    *          if M is less than N, the code will execute correctly.
    *          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), and
    *             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
    *          For good performance, LWORK should generally be larger.
    *
    *          If LWORK = -1, then a workspace query is assumed; the routine
    *          only calculates the optimal size of the WORK array, returns
    *          this value as the first entry of the WORK array, and no error
    *          message related to LWORK is issued by XERBLA.
    *
    *  IWORK   (workspace) INTEGER array, dimension (MAX(1,LIWORK))
    *          LIWORK >= 3 * MINMN * NLVL + 11 * MINMN,
    *          where MINMN = MIN( M,N ).
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *          > 0:  the algorithm for computing the SVD failed to converge;
    *                if INFO = i, i off-diagonal elements of an intermediate
    *                bidiagonal form did not converge to zero.
    *
    *  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 ..
          LOGICAL            LQUERY
          INTEGER            IASCL, IBSCL, IE, IL, ITAU, ITAUP, ITAUQ,
         $                   LDWORK, MAXMN, MAXWRK, MINMN, MINWRK, MM,
         $                   MNTHR, NLVL, NWORK, SMLSIZ, WLALSD
          DOUBLE PRECISION   ANRM, BIGNUM, BNRM, EPS, SFMIN, SMLNUM
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DGEBRD, DGELQF, DGEQRF, DLABAD, DLACPY, DLALSD,
         $                   DLASCL, DLASET, DORMBR, DORMLQ, DORMQR, XERBLA
    *     ..
    *     .. External Functions ..
          INTEGER            ILAENV
          DOUBLE PRECISION   DLAMCH, DLANGE
          EXTERNAL           ILAENV, DLAMCH, DLANGE
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          DBLE, INT, LOG, MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input arguments.
    *
          INFO = 0
          MINMN = MIN( M, N )
          MAXMN = MAX( M, N )
          MNTHR = ILAENV( 6, 'DGELSD', ' ', M, N, NRHS, -1 )
          LQUERY = ( LWORK.EQ.-1 )
          IF( M.LT.0 ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( NRHS.LT.0 ) THEN
             INFO = -3
          ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
             INFO = -5
          ELSE IF( LDB.LT.MAX( 1, MAXMN ) ) THEN
             INFO = -7
          END IF
    *
          SMLSIZ = ILAENV( 9, 'DGELSD', ' ', 0, 0, 0, 0 )
    *
    *     Compute workspace.
    *     (Note: Comments in the code beginning "Workspace:" describe the
    *     minimal amount of workspace needed at that point in the code,
    *     as well as the preferred amount for good performance.
    *     NB refers to the optimal block size for the immediately
    *     following subroutine, as returned by ILAENV.)
    *
          MINWRK = 1
          MINMN = MAX( 1, MINMN )
          NLVL = MAX( INT( LOG( DBLE( MINMN ) / DBLE( SMLSIZ+1 ) ) /
         $       LOG( TWO ) ) + 1, 0 )
    *
          IF( INFO.EQ.0 ) THEN
             MAXWRK = 0
             MM = M
             IF( M.GE.N .AND. M.GE.MNTHR ) THEN
    *
    *           Path 1a - overdetermined, with many more rows than columns.
    *
                MM = N
                MAXWRK = MAX( MAXWRK, N+N*ILAENV( 1, 'DGEQRF', ' ', M, N,
         $               -1, -1 ) )
                MAXWRK = MAX( MAXWRK, N+NRHS*
         $               ILAENV( 1, 'DORMQR', 'LT', M, NRHS, N, -1 ) )
             END IF
             IF( M.GE.N ) THEN
    *
    *           Path 1 - overdetermined or exactly determined.
    *
                MAXWRK = MAX( MAXWRK, 3*N+( MM+N )*
         $               ILAENV( 1, 'DGEBRD', ' ', MM, N, -1, -1 ) )
                MAXWRK = MAX( MAXWRK, 3*N+NRHS*
         $               ILAENV( 1, 'DORMBR', 'QLT', MM, NRHS, N, -1 ) )
                MAXWRK = MAX( MAXWRK, 3*N+( N-1 )*
         $               ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, N, -1 ) )
                WLALSD = 9*N+2*N*SMLSIZ+8*N*NLVL+N*NRHS+(SMLSIZ+1)**2
                MAXWRK = MAX( MAXWRK, 3*N+WLALSD )
                MINWRK = MAX( 3*N+MM, 3*N+NRHS, 3*N+WLALSD )
             END IF
             IF( N.GT.M ) THEN
                WLALSD = 9*M+2*M*SMLSIZ+8*M*NLVL+M*NRHS+(SMLSIZ+1)**2
                IF( N.GE.MNTHR ) THEN
    *
    *              Path 2a - underdetermined, with many more columns
    *              than rows.
    *
                   MAXWRK = M + M*ILAENV( 1, 'DGELQF', ' ', M, N, -1, -1 )
                   MAXWRK = MAX( MAXWRK, M*M+4*M+2*M*
         $                  ILAENV( 1, 'DGEBRD', ' ', M, M, -1, -1 ) )
                   MAXWRK = MAX( MAXWRK, M*M+4*M+NRHS*
         $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, M, -1 ) )
                   MAXWRK = MAX( MAXWRK, M*M+4*M+( M-1 )*
         $                  ILAENV( 1, 'DORMBR', 'PLN', M, NRHS, M, -1 ) )
                   IF( NRHS.GT.1 ) THEN
                      MAXWRK = MAX( MAXWRK, M*M+M+M*NRHS )
                   ELSE
                      MAXWRK = MAX( MAXWRK, M*M+2*M )
                   END IF
                   MAXWRK = MAX( MAXWRK, M+NRHS*
         $                  ILAENV( 1, 'DORMLQ', 'LT', N, NRHS, M, -1 ) )
                   MAXWRK = MAX( MAXWRK, M*M+4*M+WLALSD )
                ELSE
    *
    *              Path 2 - remaining underdetermined cases.
    *
                   MAXWRK = 3*M + ( N+M )*ILAENV( 1, 'DGEBRD', ' ', M, N,
         $                  -1, -1 )
                   MAXWRK = MAX( MAXWRK, 3*M+NRHS*
         $                  ILAENV( 1, 'DORMBR', 'QLT', M, NRHS, N, -1 ) )
                   MAXWRK = MAX( MAXWRK, 3*M+M*
         $                  ILAENV( 1, 'DORMBR', 'PLN', N, NRHS, M, -1 ) )
                   MAXWRK = MAX( MAXWRK, 3*M+WLALSD )
                END IF
                MINWRK = MAX( 3*M+NRHS, 3*M+M, 3*M+WLALSD )
             END IF
             MINWRK = MIN( MINWRK, MAXWRK )
             WORK( 1 ) = MAXWRK
             IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN
                INFO = -12
             END IF
          END IF
    *
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DGELSD', -INFO )
             RETURN
          ELSE IF( LQUERY ) THEN
             GO TO 10
          END IF
    *
    *     Quick return if possible.
    *
          IF( M.EQ.0 .OR. N.EQ.0 ) THEN
             RANK = 0
             RETURN
          END IF
    *
    *     Get machine parameters.
    *
          EPS = DLAMCH( 'P' )
          SFMIN = DLAMCH( 'S' )
          SMLNUM = SFMIN / EPS
          BIGNUM = ONE / SMLNUM
          CALL DLABAD( SMLNUM, BIGNUM )
    *
    *     Scale A if max entry outside range [SMLNUM,BIGNUM].
    *
          ANRM = DLANGE( 'M', M, N, A, LDA, WORK )
          IASCL = 0
          IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
    *
    *        Scale matrix norm up to SMLNUM.
    *
             CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
             IASCL = 1
          ELSE IF( ANRM.GT.BIGNUM ) THEN
    *
    *        Scale matrix norm down to BIGNUM.
    *
             CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
             IASCL = 2
          ELSE IF( ANRM.EQ.ZERO ) THEN
    *
    *        Matrix all zero. Return zero solution.
    *
             CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
             CALL DLASET( 'F', MINMN, 1, ZERO, ZERO, S, 1 )
             RANK = 0
             GO TO 10
          END IF
    *
    *     Scale B if max entry outside range [SMLNUM,BIGNUM].
    *
          BNRM = DLANGE( 'M', M, NRHS, B, LDB, WORK )
          IBSCL = 0
          IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
    *
    *        Scale matrix norm up to SMLNUM.
    *
             CALL DLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
             IBSCL = 1
          ELSE IF( BNRM.GT.BIGNUM ) THEN
    *
    *        Scale matrix norm down to BIGNUM.
    *
             CALL DLASCL( 'G', 0, 0, BNRM, BIGNUM, M, NRHS, B, LDB, INFO )
             IBSCL = 2
          END IF
    *
    *     If M < N make sure certain entries of B are zero.
    *
          IF( M.LT.N )
         $   CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
    *
    *     Overdetermined case.
    *
          IF( M.GE.N ) THEN
    *
    *        Path 1 - overdetermined or exactly determined.
    *
             MM = M
             IF( M.GE.MNTHR ) THEN
    *
    *           Path 1a - overdetermined, with many more rows than columns.
    *
                MM = N
                ITAU = 1
                NWORK = ITAU + N
    *
    *           Compute A=Q*R.
    *           (Workspace: need 2*N, prefer N+N*NB)
    *
                CALL DGEQRF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
         $                   LWORK-NWORK+1, INFO )
    *
    *           Multiply B by transpose(Q).
    *           (Workspace: need N+NRHS, prefer N+NRHS*NB)
    *
                CALL DORMQR( 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAU ), B,
         $                   LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
    *
    *           Zero out below R.
    *
                IF( N.GT.1 ) THEN
                   CALL DLASET( 'L', N-1, N-1, ZERO, ZERO, A( 2, 1 ), LDA )
                END IF
             END IF
    *
             IE = 1
             ITAUQ = IE + N
             ITAUP = ITAUQ + N
             NWORK = ITAUP + N
    *
    *        Bidiagonalize R in A.
    *        (Workspace: need 3*N+MM, prefer 3*N+(MM+N)*NB)
    *
             CALL DGEBRD( MM, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
         $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
         $                INFO )
    *
    *        Multiply B by transpose of left bidiagonalizing vectors of R.
    *        (Workspace: need 3*N+NRHS, prefer 3*N+NRHS*NB)
    *
             CALL DORMBR( 'Q', 'L', 'T', MM, NRHS, N, A, LDA, WORK( ITAUQ ),
         $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
    *
    *        Solve the bidiagonal least squares problem.
    *
             CALL DLALSD( 'U', SMLSIZ, N, NRHS, S, WORK( IE ), B, LDB,
         $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
             IF( INFO.NE.0 ) THEN
                GO TO 10
             END IF
    *
    *        Multiply B by right bidiagonalizing vectors of R.
    *
             CALL DORMBR( 'P', 'L', 'N', N, NRHS, N, A, LDA, WORK( ITAUP ),
         $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
    *
          ELSE IF( N.GE.MNTHR .AND. LWORK.GE.4*M+M*M+
         $         MAX( M, 2*M-4, NRHS, N-3*M, WLALSD ) ) THEN
    *
    *        Path 2a - underdetermined, with many more columns than rows
    *        and sufficient workspace for an efficient algorithm.
    *
             LDWORK = M
             IF( LWORK.GE.MAX( 4*M+M*LDA+MAX( M, 2*M-4, NRHS, N-3*M ),
         $       M*LDA+M+M*NRHS, 4*M+M*LDA+WLALSD ) )LDWORK = LDA
             ITAU = 1
             NWORK = M + 1
    *
    *        Compute A=L*Q.
    *        (Workspace: need 2*M, prefer M+M*NB)
    *
             CALL DGELQF( M, N, A, LDA, WORK( ITAU ), WORK( NWORK ),
         $                LWORK-NWORK+1, INFO )
             IL = NWORK
    *
    *        Copy L to WORK(IL), zeroing out above its diagonal.
    *
             CALL DLACPY( 'L', M, M, A, LDA, WORK( IL ), LDWORK )
             CALL DLASET( 'U', M-1, M-1, ZERO, ZERO, WORK( IL+LDWORK ),
         $                LDWORK )
             IE = IL + LDWORK*M
             ITAUQ = IE + M
             ITAUP = ITAUQ + M
             NWORK = ITAUP + M
    *
    *        Bidiagonalize L in WORK(IL).
    *        (Workspace: need M*M+5*M, prefer M*M+4*M+2*M*NB)
    *
             CALL DGEBRD( M, M, WORK( IL ), LDWORK, S, WORK( IE ),
         $                WORK( ITAUQ ), WORK( ITAUP ), WORK( NWORK ),
         $                LWORK-NWORK+1, INFO )
    *
    *        Multiply B by transpose of left bidiagonalizing vectors of L.
    *        (Workspace: need M*M+4*M+NRHS, prefer M*M+4*M+NRHS*NB)
    *
             CALL DORMBR( 'Q', 'L', 'T', M, NRHS, M, WORK( IL ), LDWORK,
         $                WORK( ITAUQ ), B, LDB, WORK( NWORK ),
         $                LWORK-NWORK+1, INFO )
    *
    *        Solve the bidiagonal least squares problem.
    *
             CALL DLALSD( 'U', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
         $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
             IF( INFO.NE.0 ) THEN
                GO TO 10
             END IF
    *
    *        Multiply B by right bidiagonalizing vectors of L.
    *
             CALL DORMBR( 'P', 'L', 'N', M, NRHS, M, WORK( IL ), LDWORK,
         $                WORK( ITAUP ), B, LDB, WORK( NWORK ),
         $                LWORK-NWORK+1, INFO )
    *
    *        Zero out below first M rows of B.
    *
             CALL DLASET( 'F', N-M, NRHS, ZERO, ZERO, B( M+1, 1 ), LDB )
             NWORK = ITAU + M
    *
    *        Multiply transpose(Q) by B.
    *        (Workspace: need M+NRHS, prefer M+NRHS*NB)
    *
             CALL DORMLQ( 'L', 'T', N, NRHS, M, A, LDA, WORK( ITAU ), B,
         $                LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
    *
          ELSE
    *
    *        Path 2 - remaining underdetermined cases.
    *
             IE = 1
             ITAUQ = IE + M
             ITAUP = ITAUQ + M
             NWORK = ITAUP + M
    *
    *        Bidiagonalize A.
    *        (Workspace: need 3*M+N, prefer 3*M+(M+N)*NB)
    *
             CALL DGEBRD( M, N, A, LDA, S, WORK( IE ), WORK( ITAUQ ),
         $                WORK( ITAUP ), WORK( NWORK ), LWORK-NWORK+1,
         $                INFO )
    *
    *        Multiply B by transpose of left bidiagonalizing vectors.
    *        (Workspace: need 3*M+NRHS, prefer 3*M+NRHS*NB)
    *
             CALL DORMBR( 'Q', 'L', 'T', M, NRHS, N, A, LDA, WORK( ITAUQ ),
         $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
    *
    *        Solve the bidiagonal least squares problem.
    *
             CALL DLALSD( 'L', SMLSIZ, M, NRHS, S, WORK( IE ), B, LDB,
         $                RCOND, RANK, WORK( NWORK ), IWORK, INFO )
             IF( INFO.NE.0 ) THEN
                GO TO 10
             END IF
    *
    *        Multiply B by right bidiagonalizing vectors of A.
    *
             CALL DORMBR( 'P', 'L', 'N', N, NRHS, M, A, LDA, WORK( ITAUP ),
         $                B, LDB, WORK( NWORK ), LWORK-NWORK+1, INFO )
    *
          END IF
    *
    *     Undo scaling.
    *
          IF( IASCL.EQ.1 ) THEN
             CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
             CALL DLASCL( 'G', 0, 0, SMLNUM, ANRM, MINMN, 1, S, MINMN,
         $                INFO )
          ELSE IF( IASCL.EQ.2 ) THEN
             CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
             CALL DLASCL( 'G', 0, 0, BIGNUM, ANRM, MINMN, 1, S, MINMN,
         $                INFO )
          END IF
          IF( IBSCL.EQ.1 ) THEN
             CALL DLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
          ELSE IF( IBSCL.EQ.2 ) THEN
             CALL DLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
          END IF
    *
       10 CONTINUE
          WORK( 1 ) = MAXWRK
          RETURN
    *
    *     End of DGELSD
    *
          END