sgelsy(3S)

NAME
     SGELSY - compute the minimum-norm solution to a real linear least squares
     problem

SYNOPSIS
     SUBROUTINE SGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
                        LWORK, INFO )

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

         REAL           RCOND

         INTEGER        JPVT( * )

         REAL           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
     SGELSY computes the minimum-norm solution to a real 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
     orthogonal 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' [ inv(T11)*Q1'*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 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.
       o The permutation of matrix B (the right hand side) is faster and
         more simple.


ARGUMENTS
     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
             matrices B and X. NRHS >= 0.

     A       (input/output) REAL 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.

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

     B       (input/output) REAL array, dimension (LDB,NRHS)
             On entry, the M-by-NRHS right hand side matrix B.  On exit, the
             N-by-NRHS solution matrix X.

     LDB     (input) INTEGER
             The leading dimension of the array B. LDB >= max(1,M,N).

     JPVT    (input/output) 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 AP was the k-th column of
             A.

     RCOND   (input) 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.

     RANK    (output) 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.

     WORK    (workspace/output) REAL array, dimension (LWORK)
             On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
             The dimension of the array WORK.  The unblocked strategy requires
             that:  LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ), where MN = min( M, N
             ).  The block algorithm requires that:  LWORK >= MAX(
             MN+2*N+NB*(N+1), 2*MN+NB*NRHS ), where NB is an upper bound on
             the blocksize returned by ILAENV for the routines SGEQP3, STZRZF,
             STZRQF, SORMQR, and SORMRZ.

             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.

FURTHER DETAILS
     Based on contributions by
       A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
       E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
       G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain


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

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