OpenBLAS/lapack-netlib/SRC/sgelsx.f

*> \brief <b> SGELSX solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
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*> \endhtmlonly 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
*                          WORK, INFO )
* 
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
*       REAL               RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       REAL               A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine SGELSY.
*>
*> SGELSX 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**T [ inv(T11)*Q1**T*B ]
*>                 [        0         ]
*> where Q1 consists of the first RANK columns of Q.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of right hand sides, i.e., the number of
*>          columns of matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is 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.
*>          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.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,M,N).
*> \endverbatim
*>
*> \param[in,out] JPVT
*> \verbatim
*>          JPVT is INTEGER array, dimension (N)
*>          On entry, if JPVT(i) .ne. 0, the i-th column of A is an
*>          initial column, otherwise it is a free column.  Before
*>          the QR factorization of A, all initial columns are
*>          permuted to the leading positions; only the remaining
*>          free columns are moved as a result of column pivoting
*>          during the factorization.
*>          On exit, if JPVT(i) = k, then the i-th column of A*P
*>          was the k-th column of A.
*> \endverbatim
*>
*> \param[in] RCOND
*> \verbatim
*>          RCOND is 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.
*> \endverbatim
*>
*> \param[out] RANK
*> \verbatim
*>          RANK is 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension
*>                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup realGEsolve
*
*  =====================================================================
      SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
     $                   WORK, INFO )
*
*  -- LAPACK driver routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      REAL               A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IMAX, IMIN
      PARAMETER          ( IMAX = 1, IMIN = 2 )
      REAL               ZERO, ONE, DONE, NTDONE
      PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0, DONE = ZERO,
     $                   NTDONE = ONE )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
      REAL               ANRM, BIGNUM, BNRM, C1, C2, S1, S2, SMAX,
     $                   SMAXPR, SMIN, SMINPR, SMLNUM, T1, T2
*     ..
*     .. External Functions ..
      REAL               SLAMCH, SLANGE
      EXTERNAL           SLAMCH, SLANGE
*     ..
*     .. External Subroutines ..
      EXTERNAL           SGEQPF, SLABAD, SLAIC1, SLASCL, SLASET, SLATZM,
     $                   SORM2R, STRSM, STZRQF, 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
      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
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SGELSX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( MIN( M, N, 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 elements 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 100
      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 SGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), INFO )
*
*     workspace 3*N. 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 100
      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
*
*     Logically partition R = [ R11 R12 ]
*                             [  0  R22 ]
*     where R11 = R(1:RANK,1:RANK)
*
*     [R11,R12] = [ T11, 0 ] * Y
*
      IF( RANK.LT.N )
     $   CALL STZRQF( RANK, N, A, LDA, WORK( MN+1 ), INFO )
*
*     Details of Householder rotations stored in WORK(MN+1:2*MN)
*
*     B(1:M,1:NRHS) := Q**T * B(1:M,1:NRHS)
*
      CALL SORM2R( 'Left', 'Transpose', M, NRHS, MN, A, LDA, WORK( 1 ),
     $             B, LDB, WORK( 2*MN+1 ), INFO )
*
*     workspace 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 I = RANK + 1, N
         DO 30 J = 1, NRHS
            B( I, J ) = ZERO
   30    CONTINUE
   40 CONTINUE
*
*     B(1:N,1:NRHS) := Y**T * B(1:N,1:NRHS)
*
      IF( RANK.LT.N ) THEN
         DO 50 I = 1, RANK
            CALL SLATZM( 'Left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
     $                   WORK( MN+I ), B( I, 1 ), B( RANK+1, 1 ), LDB,
     $                   WORK( 2*MN+1 ) )
   50    CONTINUE
      END IF
*
*     workspace NRHS
*
*     B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
*
      DO 90 J = 1, NRHS
         DO 60 I = 1, N
            WORK( 2*MN+I ) = NTDONE
   60    CONTINUE
         DO 80 I = 1, N
            IF( WORK( 2*MN+I ).EQ.NTDONE ) THEN
               IF( JPVT( I ).NE.I ) THEN
                  K = I
                  T1 = B( K, J )
                  T2 = B( JPVT( K ), J )
   70             CONTINUE
                  B( JPVT( K ), J ) = T1
                  WORK( 2*MN+K ) = DONE
                  T1 = T2
                  K = JPVT( K )
                  T2 = B( JPVT( K ), J )
                  IF( JPVT( K ).NE.I )
     $               GO TO 70
                  B( I, J ) = T1
                  WORK( 2*MN+K ) = DONE
               END IF
            END IF
   80    CONTINUE
   90 CONTINUE
*
*     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
*
  100 CONTINUE
*
      RETURN
*
*     End of SGELSX
*
      END