dgels(3S)

NAME
     DGELS - solve overdetermined or underdetermined real linear systems
     involving an M-by-N matrix A, or its transpose, using a QR or LQ
     factorization of A

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

         CHARACTER     TRANS

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

         DOUBLE        PRECISION A( LDA, * ), B( LDB, * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DGELS solves overdetermined or underdetermined real linear systems
     involving an M-by-N matrix A, or its 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 = 'T' and m >= n:  find the minimum norm solution of
        an undetermined system A**T * X = B.

     4. If TRANS = 'T' and m < n:  find the least squares solution of
        an overdetermined system, i.e., solve the least squares problem
                     minimize || B - A**T * 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;
             = 'T': the linear system involves A**T.

     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) DOUBLE PRECISION array, dimension (LDA,N)
             On entry, the M-by-N matrix A.  On exit, if M >= N, A is
             overwritten by details of its QR factorization as returned by
             DGEQRF; if M <  N, A is overwritten by details of its LQ
             factorization as returned by DGELQF.

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

     B       (input/output) DOUBLE PRECISION 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
             = 'T'.  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 =
             'T' and m >= n, rows 1 to M of B contain the minimum norm
             solution vectors; if TRANS = 'T' 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) DOUBLE PRECISION 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

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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