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NAME

        ZGELSY  -  compute  the minimum-norm solution to a complex linear least
        squares problem
 

SYNOPSIS

        SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
                           LWORK, RWORK, INFO )
 
            INTEGER        INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
 
            DOUBLE         PRECISION RCOND
 
            INTEGER        JPVT( * )
 
            DOUBLE         PRECISION RWORK( * )
 
            COMPLEX*16     A( LDA, * ), B( LDB, * ), WORK( * )
 

PURPOSE

        ZGELSY  computes  the  minimum-norm  solution to a complex 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 con‐
        dition 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
        unitary 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 permutation of matrix B (the right hand side) is faster and
            more simple.
          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.
 

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) COMPLEX*16 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) COMPLEX*16 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 A*P was the k-th
                column of A.
 
        RCOND   (input) DOUBLE PRECISION
                RCOND is used to determine the effective rank of  A,  which  is
                defined  as  the order of the largest leading triangular subma‐
                trix 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  com‐
                plete orthogonal factorization of A.
 
        WORK    (workspace/output) COMPLEX*16 array, dimension (LWORK)
WORK(1) returns the optimal LWORK.
 
        LWORK   (input) INTEGER
                The  dimension  of  the  array  WORK.   The  unblocked strategy
                requires that: LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) where MN
                =  min(M,N).   The block algorithm requires that: LWORK >= MN +
                MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS  )  where  NB  is  an
                upper  bound  on  the blocksize returned by ILAENV for the rou‐
                tines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ.
 
                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.
 
        RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N)
 
        INFO    (output) INTEGER
                = 0: successful exit
                < 0: if INFO = -i, the i-th argument had an illegal value
        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
 

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