cgelsy.f

*> \brief <b> CGELSY solves overdetermined or underdetermined systems for GE matrices</b>
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE CGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
* WORK, LWORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
* REAL RCOND
* ..
* .. Array Arguments ..
* INTEGER JPVT( * )
* REAL RWORK( * )
* COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGELSY 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.
*>
*> This routine is basically identical to the original xGELSX except
*> three differences:
*> o The permutation of matrix B (the right hand side) is faster and
*> more simple.
*> 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.
*> \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.
*> \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 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 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 (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK.
*> The unblocked strategy requires that:
*> LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
*> where MN = min(M,N).
*> The block algorithm requires that:
*> LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
*> where NB is an upper bound on the blocksize returned
*> by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR,
*> and CUNMRZ.
*>
*> 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.
*> \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
*
*> \par Contributors:
* ==================
*>
*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \n
*> E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
*> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \n
*>
* =====================================================================
SUBROUTINECGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
$WORK, LWORK, 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, LWORK, 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
PARAMETER ( ZERO =0.0E+0, ONE =1.0E+0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E+0, 0.0E+0 ),
 $ CONE = ( 1.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER I, IASCL, IBSCL, ISMAX, ISMIN, J, LWKOPT, MN,
 $ NB, NB1, NB2, NB3, NB4
REAL ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
 $ SMLNUM, WSIZE
COMPLEX C1, C2, S1, S2
* ..
* .. External Subroutines ..
EXTERNAL CCOPY, CGEQP3, CLAIC1, CLASCL, CLASET, CTRSM,
 $ CTZRZF, CUNMQR, CUNMRZ, SLABAD, XERBLA
* ..
* .. External Functions ..
INTEGER ILAENV
REAL CLANGE, SLAMCH
EXTERNAL CLANGE, ILAENV, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, REAL, CMPLX
* ..
* .. Executable Statements ..
*
 MN =MIN( M, N )
 ISMIN = MN +1
 ISMAX =2*MN +1
*
* Test the input arguments.
*
 INFO =0
 NB1 = ILAENV( 1, 'CGEQRF', '', M, N, -1, -1 )
 NB2 = ILAENV( 1, 'CGERQF', '', M, N, -1, -1 )
 NB3 = ILAENV( 1, 'CUNMQR', '', M, N, NRHS, -1 )
 NB4 = ILAENV( 1, 'CUNMRQ', '', M, N, NRHS, -1 )
 NB =MAX( NB1, NB2, NB3, NB4 )
 LWKOPT =MAX( 1, MN+2*N+NB*(N+1), 2*MN+NB*NRHS )
 WORK( 1 ) =CMPLX( LWKOPT )
 LQUERY = ( LWORK.EQ.-1 )
IF( M.LT.0 ) THEN
 INFO =-1
ELSEIF( N.LT.0 ) THEN
 INFO =-2
ELSEIF( NRHS.LT.0 ) THEN
 INFO =-3
ELSEIF( LDA.LT.MAX( 1, M ) ) THEN
 INFO =-5
ELSEIF( LDB.LT.MAX( 1, M, N ) ) THEN
 INFO =-7
ELSEIF( LWORK.LT.( MN+MAX( 2*MN, N+1, MN+NRHS ) ) .AND.
 $ .NOT.LQUERY ) THEN
 INFO =-12
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGELSY', -INFO )
RETURN
ELSEIF( LQUERY ) THEN
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 entries outside range [SMLNUM,BIGNUM]
*
 ANRM = CLANGE( 'M', M, N, A, LDA, RWORK )
 IASCL =0
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL CLASCL( 'G', 0, 0, ANRM, SMLNUM, M, N, A, LDA, INFO )
 IASCL =1
ELSEIF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL CLASCL( 'G', 0, 0, ANRM, BIGNUM, M, N, A, LDA, INFO )
 IASCL =2
ELSEIF( ANRM.EQ.ZERO ) THEN
*
* Matrix all zero. Return zero solution.
*
CALL CLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
 RANK =0
GO TO70
END IF
*
 BNRM = CLANGE( 'M', M, NRHS, B, LDB, RWORK )
 IBSCL =0
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
*
* Scale matrix norm up to SMLNUM
*
CALL CLASCL( 'G', 0, 0, BNRM, SMLNUM, M, NRHS, B, LDB, INFO )
 IBSCL =1
ELSEIF( BNRM.GT.BIGNUM ) 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 CGEQP3( M, N, A, LDA, JPVT, WORK( 1 ), WORK( MN+1 ),
 $ LWORK-MN, RWORK, INFO )
 WSIZE = MN +REAL( WORK( MN+1 ) )
*
* complex workspace: MN+NB*(N+1). 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
IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
 RANK =0
CALL CLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
GO TO70
ELSE
 RANK =1
END IF
*
10CONTINUE
IF( RANK.LT.MN ) 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 )
*
IF( SMAXPR*RCOND.LE.SMINPR ) THEN
DO20 I =1, RANK
 WORK( ISMIN+I-1 ) = S1*WORK( ISMIN+I-1 )
 WORK( ISMAX+I-1 ) = S2*WORK( ISMAX+I-1 )
20CONTINUE
 WORK( ISMIN+RANK ) = C1
 WORK( ISMAX+RANK ) = C2
 SMIN = SMINPR
 SMAX = SMAXPR
 RANK = RANK +1
GO TO10
END IF
END IF
*
* complex 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 CTZRZF( RANK, N, A, LDA, WORK( MN+1 ), WORK( 2*MN+1 ),
 $ LWORK-2*MN, INFO )
*
* complex workspace: 2*MN.
* Details of Householder rotations stored in WORK(MN+1:2*MN)
*
* B(1:M,1:NRHS) := Q**H * B(1:M,1:NRHS)
*
CALL CUNMQR( 'Left', 'Conjugate transpose', M, NRHS, MN, A, LDA,
 $ WORK( 1 ), B, LDB, WORK( 2*MN+1 ), LWORK-2*MN, INFO )
 WSIZE =MAX( WSIZE, 2*MN+REAL( WORK( 2*MN+1 ) ) )
*
* complex workspace: 2*MN+NB*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 )
*
DO40 J =1, NRHS
DO30 I = RANK +1, N
 B( I, J ) = CZERO
30CONTINUE
40CONTINUE
*
* B(1:N,1:NRHS) := Y**H * B(1:N,1:NRHS)
*
IF( RANK.LT.N ) THEN
CALL CUNMRZ( 'Left', 'Conjugate transpose', N, NRHS, RANK,
 $ N-RANK, A, LDA, WORK( MN+1 ), B, LDB,
 $ WORK( 2*MN+1 ), LWORK-2*MN, INFO )
END IF
*
* complex workspace: 2*MN+NRHS.
*
* B(1:N,1:NRHS) := P * B(1:N,1:NRHS)
*
DO60 J =1, NRHS
DO50 I =1, N
 WORK( JPVT( I ) ) = B( I, J )
50CONTINUE
CALL CCOPY( N, WORK( 1 ), 1, B( 1, J ), 1 )
60CONTINUE
*
* complex workspace: N.
*
* Undo scaling
*
IF( IASCL.EQ.1 ) 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 )
ELSEIF( IASCL.EQ.2 ) 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
IF( IBSCL.EQ.1 ) THEN
CALL CLASCL( 'G', 0, 0, SMLNUM, BNRM, N, NRHS, B, LDB, INFO )
ELSEIF( IBSCL.EQ.2 ) THEN
CALL CLASCL( 'G', 0, 0, BIGNUM, BNRM, N, NRHS, B, LDB, INFO )
END IF
*
70CONTINUE
 WORK( 1 ) =CMPLX( LWKOPT )
*
RETURN
*
* End of CGELSY
*
END