Chimie Théorique » Opt'n Path

      SUBROUTINE DGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,

     $                   WORK, LWORK, INFO )

*

*  -- LAPACK driver routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     November 2006

*

*     .. Scalar Arguments ..

      INTEGER            INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

      DOUBLE PRECISION   RCOND

*     ..

*     .. Array Arguments ..

      INTEGER            JPVT( * )

      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), WORK( * )

*     ..

*

*  Purpose

*  =======

*

*  DGELSY computes the minimum-norm solution to a real linear least

*  squares problem:

*      minimize || A * X - B ||

*  using a complete orthogonal factorization 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 routine first computes a QR factorization with column pivoting:

*      A * P = Q * [ R11 R12 ]

*                  [  0  R22 ]

*  with R11 defined as the largest leading submatrix whose estimated

*  condition number is less than 1/RCOND.  The order of R11, RANK,

*  is the effective rank of A.

*

*  Then, R22 is considered to be negligible, and R12 is annihilated

*  by orthogonal transformations from the right, arriving at the

*  complete orthogonal factorization:

*     A * P = Q * [ T11 0 ] * Z

*                 [  0  0 ]

*  The minimum-norm solution is then

*     X = P * Z' [ inv(T11)*Q1'*B ]

*                [        0       ]

*  where Q1 consists of the first RANK columns of Q.

*

*  This routine is basically identical to the original xGELSX except

*  three differences:

*    o The call to the subroutine xGEQPF has been substituted by the

*      the call to the subroutine xGEQP3. This subroutine is a Blas-3

*      version of the QR factorization with column pivoting.

*    o Matrix B (the right hand side) is updated with Blas-3.

*    o The permutation of matrix B (the right hand side) is faster and

*      more simple.

*

*  Arguments

*  =========

*

*  M       (input) INTEGER

*          The number of rows of the matrix A.  M >= 0.

*

*  N       (input) INTEGER

*          The number of columns of the matrix A.  N >= 0.

*

*  NRHS    (input) INTEGER

*          The number of right hand sides, i.e., the number of

*          columns of matrices B and X. NRHS >= 0.

*

*  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)

*          On entry, the M-by-N matrix A.

*          On exit, A has been overwritten by details of its

*          complete orthogonal factorization.

*

*  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, the N-by-NRHS solution matrix X.

*

*  LDB     (input) INTEGER

*          The leading dimension of the array B. LDB >= max(1,M,N).

*

*  JPVT    (input/output) INTEGER array, dimension (N)

*          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted

*          to the front of AP, otherwise column i is a free column.

*          On exit, if JPVT(i) = k, then the i-th column of AP

*          was the k-th column of A.

*

*  RCOND   (input) DOUBLE PRECISION

*          RCOND is used to determine the effective rank of A, which

*          is defined as the order of the largest leading triangular

*          submatrix R11 in the QR factorization with pivoting of A,

*          whose estimated condition number < 1/RCOND.

*

*  RANK    (output) INTEGER

*          The effective rank of A, i.e., the order of the submatrix

*          R11.  This is the same as the order of the submatrix T11

*          in the complete orthogonal factorization of A.

*

*  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.

*          The unblocked strategy requires that:

*             LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),

*          where MN = min( M, N ).

*          The block algorithm requires that:

*             LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),

*          where NB is an upper bound on the blocksize returned

*          by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,

*          and DORMRZ.

*

*          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.

*

*  INFO    (output) INTEGER

*          = 0: successful exit

*          < 0: If INFO = -i, the i-th argument had an illegal value.

*

*  Further Details

*  ===============

*

*  Based on contributions by

*    A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

*    E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

*    G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain

*

*  =====================================================================

*

*     .. Parameters ..

      INTEGER            IMAX, IMIN

      PARAMETER          ( IMAX = 1, IMIN = 2 )

      DOUBLE PRECISION   ZERO, ONE

      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )

*     ..

*     .. Local Scalars ..

      LOGICAL            LQUERY

      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,

     $                   LWKOPT, MN, NB, NB1, NB2, NB3, NB4

      DOUBLE PRECISION   ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,

     $                   SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE

*     ..

*     .. External Functions ..

      INTEGER            ILAENV

      DOUBLE PRECISION   DLAMCH, DLANGE

      EXTERNAL           ILAENV, DLAMCH, DLANGE

*     ..

*     .. External Subroutines ..

      EXTERNAL           DCOPY, DGEQP3, DLABAD, DLAIC1, DLASCL, DLASET,

     $                   DORMQR, DORMRZ, DTRSM, DTZRZF, XERBLA

*     ..

*     .. Intrinsic Functions ..

      INTRINSIC          ABS, MAX, MIN

*     ..

*     .. Executable Statements ..

*

      MN = MIN( M, N )

      ISMIN = MN + 1

      ISMAX = 2*MN + 1

*

*     Test the input arguments.

*

      INFO = 0

      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, M, N ) ) THEN

         INFO = -7

      END IF

*

*     Figure out optimal block size

*

      IF( INFO.EQ.0 ) THEN

         IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN

            LWKMIN = 1

            LWKOPT = 1

         ELSE

            NB1 = ILAENV( 1, 'DGEQRF', ' ', M, N, -1, -1 )

            NB2 = ILAENV( 1, 'DGERQF', ' ', M, N, -1, -1 )

            NB3 = ILAENV( 1, 'DORMQR', ' ', M, N, NRHS, -1 )

            NB4 = ILAENV( 1, 'DORMRQ', ' ', M, N, NRHS, -1 )

            NB = MAX( NB1, NB2, NB3, NB4 )

            LWKMIN = MN + MAX( 2*MN, N + 1, MN + NRHS )

            LWKOPT = MAX( LWKMIN,

     $                    MN + 2*N + NB*( N + 1 ), 2*MN + NB*NRHS )

         END IF

         WORK( 1 ) = LWKOPT

*

         IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN

            INFO = -12

         END IF

      END IF

*

      IF( INFO.NE.0 ) THEN

         CALL XERBLA( 'DGELSY', -INFO )

         RETURN

      ELSE IF( LQUERY ) THEN

         RETURN

      END IF

*

*     Quick return if possible

*

      IF( MN.EQ.0 .OR. NRHS.EQ.0 ) THEN

         RANK = 0

         RETURN

      END IF

*

*     Get machine parameters

*

      SMLNUM = DLAMCH( 'S' ) / DLAMCH( 'P' )

      BIGNUM = ONE / SMLNUM

      CALL DLABAD( SMLNUM, BIGNUM )

*

*     Scale A, B if max entries 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 )

         RANK = 0

         GO TO 70

      END IF

*

      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

*

*     Compute QR factorization with column pivoting of A:

*        A * P = Q * R

*

      CALL DGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),

     $             LWORK-MN, INFO )

      WSIZE = MN + WORK( MN+1 )

*

*     workspace: MN+2*N+NB*(N+1).

*     Details of Householder rotations stored in WORK(1:MN).

*

*     Determine RANK using incremental condition estimation

*

      WORK( ISMIN ) = ONE

      WORK( ISMAX ) = ONE

      SMAX = ABS( A( 1, 1 ) )

      SMIN = SMAX

      IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN

         RANK = 0

         CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )

         GO TO 70

      ELSE

         RANK = 1

      END IF

*

   10 CONTINUE

      IF( RANK.LT.MN ) THEN

         I = RANK + 1

         CALL DLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),

     $                A( I, I ), SMINPR, S1, C1 )

         CALL DLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),

     $                A( I, I ), SMAXPR, S2, C2 )

*

         IF( SMAXPR*RCOND.LE.SMINPR ) THEN

            DO 20 I = 1, RANK

               WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )

               WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )

   20       CONTINUE

            WORK( ISMIN+RANK ) = C1

            WORK( ISMAX+RANK ) = C2

            SMIN = SMINPR

            SMAX = SMAXPR

            RANK = RANK + 1

            GO TO 10

         END IF

      END IF

*

*     workspace: 3*MN.

*

*     Logically partition R = [ R11 R12 ]

*                             [  0  R22 ]

*     where R11 = R(1:RANK,1:RANK)

*

*     [R11,R12] = [ T11, 0 ] * Y

*

      IF( RANK.LT.N )

     $   CALL DTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),

     $                LWORK-2*MN, INFO )

*

*     workspace: 2*MN.

*     Details of Householder rotations stored in WORK(MN+1:2*MN)

*

*     B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)

*

      CALL DORMQR( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),

     $             B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )

      WSIZE = MAX( WSIZE, 2*MN+WORK( 2*MN+1 ) )

*

*     workspace: 2*MN+NB*NRHS.

*

*     B(1:RANK,1:NRHS) := inv(T11) * B(1:RANK,1:NRHS)

*

      CALL DTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', RANK,

     $            NRHS, ONE, A, LDA, B, LDB )

*

      DO 40 J = 1, NRHS

         DO 30 I = RANK + 1, N

            B( I, J ) = ZERO

   30    CONTINUE

   40 CONTINUE

*

*     B(1:N,1:NRHS) := Y' * B(1:N,1:NRHS)

*

      IF( RANK.LT.N ) THEN

         CALL DORMRZ( 'Left', 'Transpose', N, NRHS, RANK, N-RANK, A,

     $                LDA, WORK( MN+1 ), B, LDB, WORK( 2*MN+1 ),

     $                LWORK-2*MN, INFO )

      END IF

*

*     workspace: 2*MN+NRHS.

*

*     B(1:N,1:NRHS) := P * B(1:N,1:NRHS)

*

      DO 60 J = 1, NRHS

         DO 50 I = 1, N

            WORK( JPVT( I ) ) = B( I, J )

   50    CONTINUE

         CALL DCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )

   60 CONTINUE

*

*     workspace: N.

*

*     Undo scaling

*

      IF( IASCL.EQ.1 ) THEN

         CALL DLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )

         CALL DLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,

     $                INFO )

      ELSE IF( IASCL.EQ.2 ) THEN

         CALL DLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )

         CALL DLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,

     $                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

*

   70 CONTINUE

      WORK( 1 ) = LWKOPT

*

      RETURN

*

*     End of DGELSY

*

      END