Move LAPACK trunk into position.

1 files changed, 391 insertions, 0 deletions

diff --git a/SRC/sgelsy.f b/SRC/sgelsy.f
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+++ b/SRC/sgelsy.f
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+      SUBROUTINE SGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
+     $                   WORK, LWORK, 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
+      REAL               RCOND
+*     ..
+*     .. Array Arguments ..
+      INTEGER            JPVT( * )
+      REAL               A( LDA, * ), B( LDB, * ), WORK( * )
+*     ..
+*
+*  Purpose
+*  =======
+*
+*  SGELSY 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) REAL 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) REAL 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) REAL
+*          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) REAL 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 SGEQP3, STZRZF, STZRQF, SORMQR,
+*          and SORMRZ.
+*
+*          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 )
+      REAL               ZERO, ONE
+      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
+*     ..
+*     .. Local Scalars ..
+      LOGICAL            LQUERY
+      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKMIN,
+     $                   LWKOPT, MN, NB, NB1, NB2, NB3, NB4
+      REAL               ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
+     $                   SMAXPR, SMIN, SMINPR, SMLNUM, WSIZE
+*     ..
+*     .. External Functions ..
+      INTEGER            ILAENV
+      REAL               SLAMCH, SLANGE
+      EXTERNAL           ILAENV, SLAMCH, SLANGE
+*     ..
+*     .. External Subroutines ..
+      EXTERNAL           SCOPY, SGEQP3, SLABAD, SLAIC1, SLASCL, SLASET,
+     $                   SORMQR, SORMRZ, STRSM, STZRZF, 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, 'SGEQRF', ' ', M, N, -1, -1 )
+            NB2 = ILAENV( 1, 'SGERQF', ' ', M, N, -1, -1 )
+            NB3 = ILAENV( 1, 'SORMQR', ' ', M, N, NRHS, -1 )
+            NB4 = ILAENV( 1, 'SORMRQ', ' ', 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( 'SGELSY', -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 = SLAMCH( 'S' ) / SLAMCH( 'P' )
+      BIGNUM = ONE / SMLNUM
+      CALL SLABAD( SMLNUM, BIGNUM )
+*
+*     Scale A, B if max entries outside range [SMLNUM,BIGNUM]
+*
+      ANRM = SLANGE( 'M', M, N, A, LDA, WORK )
+      IASCL = 0
+      IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
+*
+*        Scale matrix norm up to SMLNUM
+*
+         CALL SLASCL( '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 SLASCL( '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 SLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
+         RANK = 0
+         GO TO 70
+      END IF
+*
+      BNRM = SLANGE( 'M', M, NRHS, B, LDB, WORK )
+      IBSCL = 0
+      IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
+*
+*        Scale matrix norm up to SMLNUM
+*
+         CALL SLASCL( '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 SLASCL( '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 SGEQP3( 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 SLASET( '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 SLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
+     $                A( I, I ), SMINPR, S1, C1 )
+         CALL SLAIC1( 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 STZRZF( 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 SORMQR( '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 STRSM( '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 SORMRZ( '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 SCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
+   60 CONTINUE
+*
+*     workspace: N.
+*
+*     Undo scaling
+*
+      IF( IASCL.EQ.1 ) THEN
+         CALL SLASCL( 'G', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
+         CALL SLASCL( 'U', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
+     $                INFO )
+      ELSE IF( IASCL.EQ.2 ) THEN
+         CALL SLASCL( 'G', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
+         CALL SLASCL( 'U', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
+     $                INFO )
+      END IF
+      IF( IBSCL.EQ.1 ) THEN
+         CALL SLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
+      ELSE IF( IBSCL.EQ.2 ) THEN
+         CALL SLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
+      END IF
+*
+   70 CONTINUE
+      WORK( 1 ) = LWKOPT
+*
+      RETURN
+*
+*     End of SGELSY
+*
+      END