SGEQP3  -  compute a QR factorization with column pivoting
       of a matrix A


SYNOPSIS

       SUBROUTINE SGEQP3( M, N, A, LDA, JPVT, TAU,  WORK,  LWORK,
                          INFO )

           INTEGER        INFO, LDA, LWORK, M, N

           INTEGER        JPVT( * )

           REAL           A( LDA, * ), TAU( * ), WORK( * )


PURPOSE

       SGEQP3 computes a QR factorization with column pivoting of
       a matrix A: A*P = Q*R using Level 3 BLAS.


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.

       A       (input/output) REAL array, dimension (LDA,N)
               On entry, the M-by-N matrix A.  On exit, the upper
               triangle  of  the array contains the min(M,N)-by-N
               upper trapezoidal matrix R; the elements below the
               diagonal,  together  with the array TAU, represent
               the orthogonal matrix Q as a product  of  min(M,N)
               elementary reflectors.

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

       JPVT    (input/output) INTEGER array, dimension (N)
               On entry, if JPVT(J).ne.0, the J-th column of A is
               permuted  to  the front of A*P (a leading column);
               if JPVT(J)=0, the J-th column of A is a free  col­
               umn.   On exit, if JPVT(J)=K, then the J-th column
               of A*P was the the K-th column of A.

       TAU     (output) REAL array, dimension (min(M,N))
               The scalar factors of the elementary reflectors.

       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. LWORK >= 3*N+1.
               For optimal performance LWORK >= 2*N+(  N+1  )*NB,


               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 ille­
               gal value.


FURTHER DETAILS

       The matrix Q is represented as  a  product  of  elementary
       reflectors

          Q = H(1) H(2) . . . H(k), where k = min(m,n).

       Each H(i) has the form

          H(i) = I - tau * v * v'

       where  tau  is a real/complex scalar, and v is a real/com­
       plex vector with v(1:i-1) = 0 and v(i) =  1;  v(i+1:m)  is
       stored on exit in A(i+1:m,i), and tau in TAU(i).

       Based on contributions by
         G.  Quintana-Orti,  Depto.  de  Informatica, Universidad
       Jaime I, Spain
         X. Sun, Computer Science Dept., Duke University, USA


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