SUBROUTINE CGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
$ INFO )
*
* -- LAPACK driver routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* June 30, 1999
*
* .. Scalar Arguments ..
CHARACTER TRANS
INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), B( LDB, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* CGELS solves overdetermined or underdetermined complex linear systems
* involving an M-by-N matrix A, or its conjugate-transpose, using a QR
* or LQ factorization of A. It is assumed that A has full rank.
*
* The following options are provided:
*
* 1. If TRANS = 'N' and m >= n: find the least squares solution of
* an overdetermined system, i.e., solve the least squares problem
* minimize || B - A*X ||.
*
* 2. If TRANS = 'N' and m < n: find the minimum norm solution of
* an underdetermined system A * X = B.
*
* 3. If TRANS = 'C' and m >= n: find the minimum norm solution of
* an undetermined system A**H * X = B.
*
* 4. If TRANS = 'C' and m < n: find the least squares solution of
* an overdetermined system, i.e., solve the least squares problem
* minimize || B - A**H * X ||.
*
* 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.
*
* Arguments
* =========
*
* TRANS (input) CHARACTER
* = 'N': the linear system involves A;
* = 'C': the linear system involves A**H.
*
* 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 the matrices B and X. NRHS >= 0.
*
* A (input/output) COMPLEX array, dimension (LDA,N)
* On entry, the M-by-N matrix A.
* if M >= N, A is overwritten by details of its QR
* factorization as returned by CGEQRF;
* if M < N, A is overwritten by details of its LQ
* factorization as returned by CGELQF.
*
* LDA (input) INTEGER
* The leading dimension of the array A. LDA >= max(1,M).
*
* B (input/output) COMPLEX array, dimension (LDB,NRHS)
* On entry, the matrix B of right hand side vectors, stored
* columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS
* if TRANS = 'C'.
* On exit, B is overwritten by the solution vectors, stored
* columnwise:
* if TRANS = 'N' and m >= n, rows 1 to n of B contain the least
* squares solution vectors; the residual sum of squares for the
* solution in each column is given by the sum of squares of
* elements N+1 to M in that column;
* if TRANS = 'N' and m < n, rows 1 to N of B contain the
* minimum norm solution vectors;
* if TRANS = 'C' and m >= n, rows 1 to M of B contain the
* minimum norm solution vectors;
* if TRANS = 'C' and m < n, rows 1 to M of B contain the
* least squares solution vectors; the residual sum of squares
* for the solution in each column is given by the sum of
* squares of elements M+1 to N in that column.
*
* LDB (input) INTEGER
* The leading dimension of the array B. LDB >= MAX(1,M,N).
*
* WORK (workspace/output) COMPLEX array, dimension (LWORK)
* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
* LWORK (input) INTEGER
* The dimension of the array WORK.
* LWORK >= max( 1, MN + max( MN, NRHS ) ).
* For optimal performance,
* LWORK >= max( 1, MN + max( MN, NRHS )*NB ).
* where MN = min(M,N) and NB is the optimum block size.
*
* 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
*
* =====================================================================
*
* .. Parameters ..
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, TPSD
INTEGER BROW, I, IASCL, IBSCL, J, MN, NB, SCLLEN, WSIZE
REAL ANRM, BIGNUM, BNRM, SMLNUM
* ..
* .. Local Arrays ..
REAL RWORK( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL CLANGE, SLAMCH
EXTERNAL LSAME, ILAENV, CLANGE, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL CGELQF, CGEQRF, CLASCL, CLASET, CTRSM, CUNMLQ,
$ CUNMQR, SLABAD, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* Test the input arguments.
*
INFO = 0
MN = MIN( M, N )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.( LSAME( TRANS, 'N' ) .OR. LSAME( TRANS, 'C' ) ) ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
INFO = -6
ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
INFO = -8
ELSE IF( LWORK.LT.MAX( 1, MN+MAX( MN, NRHS ) ) .AND.
$ .NOT.LQUERY ) THEN
INFO = -10
END IF
*
* Figure out optimal block size
*
IF( INFO.EQ.0 .OR. INFO.EQ.-10 ) THEN
*
TPSD = .TRUE.
IF( LSAME( TRANS, 'N' ) )
$ TPSD = .FALSE.
*
IF( M.GE.N ) THEN
NB = ILAENV( 1, 'CGEQRF', ' ', M, N, -1, -1 )
IF( TPSD ) THEN
NB = MAX( NB, ILAENV( 1, 'CUNMQR', 'LN', M, NRHS, N,
$ -1 ) )
ELSE
NB = MAX( NB, ILAENV( 1, 'CUNMQR', 'LC', M, NRHS, N,
$ -1 ) )
END IF
ELSE
NB = ILAENV( 1, 'CGELQF', ' ', M, N, -1, -1 )
IF( TPSD ) THEN
NB = MAX( NB, ILAENV( 1, 'CUNMLQ', 'LC', N, NRHS, M,
$ -1 ) )
ELSE
NB = MAX( NB, ILAENV( 1, 'CUNMLQ', 'LN', N, NRHS, M,
$ -1 ) )
END IF
END IF
*
WSIZE = MAX( 1, MN + MAX( MN, NRHS )*NB )
WORK( 1 ) = REAL( WSIZE )
*
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGELS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( MIN( M, N, NRHS ).EQ.0 ) THEN
CALL CLASET( 'Full', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
RETURN
END IF
*
* Get machine parameters
*
SMLNUM = SLAMCH( 'S' ) / SLAMCH( 'P' )
BIGNUM = ONE / SMLNUM
CALL SLABAD( SMLNUM, BIGNUM )
*
* Scale A, B if max element 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
ELSE IF( ANRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL CLASCL( '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 CLASET( 'F', MAX( M, N ), NRHS, CZERO, CZERO, B, LDB )
GO TO 50
END IF
*
BROW = M
IF( TPSD )
$ BROW = N
BNRM = CLANGE( 'M', BROW, 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, BROW, NRHS, B, LDB,
$ INFO )
IBSCL = 1
ELSE IF( BNRM.GT.BIGNUM ) THEN
*
* Scale matrix norm down to BIGNUM
*
CALL CLASCL( 'G', 0, 0, BNRM, BIGNUM, BROW, NRHS, B, LDB,
$ INFO )
IBSCL = 2
END IF
*
IF( M.GE.N ) THEN
*
* compute QR factorization of A
*
CALL CGEQRF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least N, optimally N*NB
*
IF( .NOT.TPSD ) THEN
*
* Least-Squares Problem min || A * X - B ||
*
* B(1:M,1:NRHS) := Q' * B(1:M,1:NRHS)
*
CALL CUNMQR( 'Left', 'Conjugate transpose', M, NRHS, N, A,
$ LDA, WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
* B(1:N,1:NRHS) := inv(R) * B(1:N,1:NRHS)
*
CALL CTRSM( 'Left', 'Upper', 'No transpose', 'Non-unit', N,
$ NRHS, CONE, A, LDA, B, LDB )
*
SCLLEN = N
*
ELSE
*
* Overdetermined system of equations A' * X = B
*
* B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS)
*
CALL CTRSM( 'Left', 'Upper', 'Conjugate transpose',
$ 'Non-unit', N, NRHS, CONE, A, LDA, B, LDB )
*
* B(N+1:M,1:NRHS) = ZERO
*
DO 20 J = 1, NRHS
DO 10 I = N + 1, M
B( I, J ) = CZERO
10 CONTINUE
20 CONTINUE
*
* B(1:M,1:NRHS) := Q(1:N,:) * B(1:N,1:NRHS)
*
CALL CUNMQR( 'Left', 'No transpose', M, NRHS, N, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
SCLLEN = M
*
END IF
*
ELSE
*
* Compute LQ factorization of A
*
CALL CGELQF( M, N, A, LDA, WORK( 1 ), WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least M, optimally M*NB.
*
IF( .NOT.TPSD ) THEN
*
* underdetermined system of equations A * X = B
*
* B(1:M,1:NRHS) := inv(L) * B(1:M,1:NRHS)
*
CALL CTRSM( 'Left', 'Lower', 'No transpose', 'Non-unit', M,
$ NRHS, CONE, A, LDA, B, LDB )
*
* B(M+1:N,1:NRHS) = 0
*
DO 40 J = 1, NRHS
DO 30 I = M + 1, N
B( I, J ) = CZERO
30 CONTINUE
40 CONTINUE
*
* B(1:N,1:NRHS) := Q(1:N,:)' * B(1:M,1:NRHS)
*
CALL CUNMLQ( 'Left', 'Conjugate transpose', N, NRHS, M, A,
$ LDA, WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
SCLLEN = N
*
ELSE
*
* overdetermined system min || A' * X - B ||
*
* B(1:N,1:NRHS) := Q * B(1:N,1:NRHS)
*
CALL CUNMLQ( 'Left', 'No transpose', N, NRHS, M, A, LDA,
$ WORK( 1 ), B, LDB, WORK( MN+1 ), LWORK-MN,
$ INFO )
*
* workspace at least NRHS, optimally NRHS*NB
*
* B(1:M,1:NRHS) := inv(L') * B(1:M,1:NRHS)
*
CALL CTRSM( 'Left', 'Lower', 'Conjugate transpose',
$ 'Non-unit', M, NRHS, CONE, A, LDA, B, LDB )
*
SCLLEN = M
*
END IF
*
END IF
*
* Undo scaling
*
IF( IASCL.EQ.1 ) THEN
CALL CLASCL( 'G', 0, 0, ANRM, SMLNUM, SCLLEN, NRHS, B, LDB,
$ INFO )
ELSE IF( IASCL.EQ.2 ) THEN
CALL CLASCL( 'G', 0, 0, ANRM, BIGNUM, SCLLEN, NRHS, B, LDB,
$ INFO )
END IF
IF( IBSCL.EQ.1 ) THEN
CALL CLASCL( 'G', 0, 0, SMLNUM, BNRM, SCLLEN, NRHS, B, LDB,
$ INFO )
ELSE IF( IBSCL.EQ.2 ) THEN
CALL CLASCL( 'G', 0, 0, BIGNUM, BNRM, SCLLEN, NRHS, B, LDB,
$ INFO )
END IF
*
50 CONTINUE
WORK( 1 ) = REAL( WSIZE )
*
RETURN
*
* End of CGELS
*
END
