cgelsx.f

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*> \brief <b> CGELSX solves overdetermined or underdetermined systems for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE CGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
*                          WORK, RWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
*       REAL               RCOND
*       ..
*       .. Array Arguments ..
*       INTEGER            JPVT( * )
*       REAL               RWORK( * )
*       COMPLEX            A( LDA, * ), B( LDB, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine CGELSY.
*>
*> CGELSX computes the minimum-norm solution to a complex 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 unitary 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**H [ inv(T11)*Q1**H*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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension
*>                      (min(M,N) + max( N, 2*min(M,N)+NRHS )),
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (2*N)
*> \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.
*
*> \ingroup complexGEsolve
*
*  =====================================================================
      SUBROUTINE cgelsx( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
     $                   WORK, RWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
      REAL               RCOND
*     ..
*     .. Array Arguments ..
      INTEGER            JPVT( * )
      REAL               RWORK( * )
      COMPLEX            A( LDA, * ), B( LDB, * ), WORK( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IMAX, IMIN
      parameter( imax = 1, imin = 2 )
      REAL               ZERO, ONE, DONE, NTDONE
      parameter( zero = 0.0e+0, one = 1.0e+0, done = zero,
     $                   ntdone = one )
      COMPLEX            CZERO, CONE
      parameter( czero = ( 0.0e+0, 0.0e+0 ),
     $                   cone = ( 1.0e+0, 0.0e+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
      REAL               ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
     $                   smlnum
      COMPLEX            C1, C2, S1, S2, T1, T2
*     ..
*     .. External Subroutines ..
      EXTERNAL           cgeqpf, claic1, clascl, claset, clatzm, ctrsm,
     $                   ctzrqf, cunm2r, slabad, xerbla
*     ..
*     .. External Functions ..
      REAL               CLANGE, SLAMCH
      EXTERNAL           clange, slamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, conjg, 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( 'CGELSX', -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 = clange( 'm', M, N, A, LDA, RWORK )
      IASCL = 0
.GT..AND..LT.      IF( ANRMZERO  ANRMSMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL CLASCL( 'g', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
         IASCL = 1
.GT.      ELSE IF( ANRMBIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL CLASCL( 'g', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
         IASCL = 2
.EQ.      ELSE IF( ANRMZERO ) THEN
*
*        Matrix all zero. Return zero solution.
*
         CALL CLASET( 'f', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
         RANK = 0
         GO TO 100
      END IF
*
      BNRM = CLANGE( 'm', M, NRHS, B, LDB, RWORK )
      IBSCL = 0
.GT..AND..LT.      IF( BNRMZERO  BNRMSMLNUM ) THEN
*
*        Scale matrix norm up to SMLNUM
*
         CALL CLASCL( 'g', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
         IBSCL = 1
.GT.      ELSE IF( BNRMBIGNUM ) THEN
*
*        Scale matrix norm down to BIGNUM
*
         CALL CLASCL( '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 CGEQPF( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ), RWORK,
     $             INFO )
*
*     complex workspace MN+N. Real workspace 2*N. Details of Householder
*     rotations stored in WORK(1:MN).
*
*     Determine RANK using incremental condition estimation
*
      WORK( ISMIN ) = CONE
      WORK( ISMAX ) = CONE
      SMAX = ABS( A( 1, 1 ) )
      SMIN = SMAX
.EQ.      IF( ABS( A( 1, 1 ) )ZERO ) THEN
         RANK = 0
         CALL CLASET( 'f', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
         GO TO 100
      ELSE
         RANK = 1
      END IF
*
   10 CONTINUE
.LT.      IF( RANKMN ) THEN
         I = RANK + 1
         CALL CLAIC1( IMIN, RANK, WORK( ISMIN ), SMIN, A( 1, I ),
     $                A( I, I ), SMINPR, S1, C1 )
         CALL CLAIC1( IMAX, RANK, WORK( ISMAX ), SMAX, A( 1, I ),
     $                A( I, I ), SMAXPR, S2, C2 )
*
.LE.         IF( SMAXPR*RCONDSMINPR ) 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
*
.LT.      IF( RANKN )
     $   CALL CTZRQF( 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**H * B(1:M,1:NRHS)
*
      CALL CUNM2R( 'left', 'conjugate 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 CTRSM( 'left', 'upper', 'No transpose', 'Non-unit', rank,
     $            nrhs, cone, a, lda, b, ldb )
*
      DO 40 i = rank + 1, n
         DO 30 j = 1, nrhs
            b( i, j ) = czero
   30    CONTINUE
   40 CONTINUE
*
*     B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
*
      IF( rank.LT.n ) THEN
         DO 50 i = 1, rank
            CALL clatzm( 'left', N-RANK+1, NRHS, A( I, RANK+1 ), LDA,
     $                   CONJG( 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
.EQ.            IF( WORK( 2*MN+I )NTDONE ) THEN
.NE.               IF( JPVT( I )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 )
.NE.                  IF( JPVT( K )I )
     $               GO TO 70
                  B( I, J ) = T1
                  WORK( 2*MN+K ) = DONE
               END IF
            END IF
   80    CONTINUE
   90 CONTINUE
*
*     Undo scaling
*
.EQ.      IF( IASCL1 ) THEN
         CALL CLASCL( 'g', 0, 0, ANRM, SMLNUM, N, NRHS, B, LDB, INFO )
         CALL CLASCL( 'u', 0, 0, SMLNUM, ANRM, RANK, RANK, A, LDA,
     $                INFO )
.EQ.      ELSE IF( IASCL2 ) THEN
         CALL CLASCL( 'g', 0, 0, ANRM, BIGNUM, N, NRHS, B, LDB, INFO )
         CALL CLASCL( 'u', 0, 0, BIGNUM, ANRM, RANK, RANK, A, LDA,
     $                INFO )
      END IF
.EQ.      IF( IBSCL1 ) THEN
         CALL CLASCL( 'g', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
.EQ.      ELSE IF( IBSCL2 ) THEN
         CALL CLASCL( 'g', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
      END IF
*
  100 CONTINUE
*
      RETURN
*
*     End of CGELSX
*
      END