cgels.f

          SUBROUTINE CGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK,
         $                  INFO )
    *
    *  -- LAPACK driver routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. 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*1
    *          = '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, if INFO = 0, 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 the
    *          modulus 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 the modulus 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 (MAX(1,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
    *          > 0:  if INFO =  i, the i-th diagonal element of the
    *                triangular factor of A is zero, so that A does not have
    *                full rank; the least squares solution could not be
    *                computed.
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          REAL               ZERO, ONE
          PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
          COMPLEX            CZERO
          PARAMETER          ( CZERO = ( 0.0E+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, CTRTRS, 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 CTRTRS( 'Upper', 'No transpose', 'Non-unit', N, NRHS,
         $                   A, LDA, B, LDB, INFO )
    *
                IF( INFO.GT.0 ) THEN
                   RETURN
                END IF
    *
                SCLLEN = N
    *
             ELSE
    *
    *           Overdetermined system of equations A' * X = B
    *
    *           B(1:N,1:NRHS) := inv(R') * B(1:N,1:NRHS)
    *
                CALL CTRTRS( 'Upper', 'Conjugate transpose','Non-unit',
         $                   N, NRHS, A, LDA, B, LDB, INFO )
    *
                IF( INFO.GT.0 ) THEN
                   RETURN
                END IF
    *
    *           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 CTRTRS( 'Lower', 'No transpose', 'Non-unit', M, NRHS,
         $                   A, LDA, B, LDB, INFO )
    *
                IF( INFO.GT.0 ) THEN
                   RETURN
                END IF
    *
    *           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 CTRTRS( 'Lower', 'Conjugate transpose', 'Non-unit',
         $                   M, NRHS, A, LDA, B, LDB, INFO )
    *
                IF( INFO.GT.0 ) THEN
                   RETURN
                END IF
    *
                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