ZGELS (3lapack)

solve 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

SYNOPSIS

  SUBROUTINE ZGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO )

      CHARACTER     TRANS

      INTEGER       INFO, LDA, LDB, LWORK, M, N, NRHS

      COMPLEX*16    A( LDA, * ), B( LDB, * ), WORK( LWORK )

PURPOSE

  ZGELS 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*16 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 ZGEQRF; if M <  N, A
          is overwritten by details of its LQ factorization as returned by
          ZGELQF.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  B       (input/output) COMPLEX*16 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*16 array, dimension (LWORK)
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK   (input) INTEGER
          The dimension of the array WORK.  LWORK >= min(M,N) +
          MAX(1,M,N,NRHS).  For optimal performance, LWORK >= min(M,N) +
          MAX(1,M,N,NRHS) * NB where NB is the optimum block size.

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value