Chimie Théorique » Opt'n Path

      SUBROUTINE DGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK,

     $                   WORK, LWORK, IWORK, INFO )

*

*  -- LAPACK driver 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            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 >= max(1, 3 * MINMN * NLVL + 11 * MINMN),

*          where MINMN = MIN( M,N ).

*          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

*

*  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, LIWORK, 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

      LIWORK = 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

         LIWORK = 3*MINMN*NLVL + 11*MINMN

         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 )

!     XXX: Ensure the Path 2a case below is triggered.  The workspace

!     calculation should use queries for all routines eventually.

               MAXWRK = MAX( MAXWRK,

     $              4*M+M*M+MAX( M, 2*M-4, NRHS, N-3*M ) )

            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

         IWORK( 1 ) = LIWORK

         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

      IWORK( 1 ) = LIWORK

      RETURN

*

*     End of DGELSD

*

      END