SRC/zggqrf.f(3)            Library Functions Manual            SRC/zggqrf.f(3)

NAME
       SRC/zggqrf.f

SYNOPSIS
   Functions/Subroutines
       subroutine zggqrf (n, m, p, a, lda, taua, b, ldb, taub, work, lwork,
           info)
           ZGGQRF

Function/Subroutine Documentation
   subroutine zggqrf (integer n, integer m, integer p, complex*16, dimension(
       lda, * ) a, integer lda, complex*16, dimension( * ) taua, complex*16,
       dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) taub,
       complex*16, dimension( * ) work, integer lwork, integer info)
       ZGGQRF

       Purpose:


            ZGGQRF computes a generalized QR factorization of an N-by-M matrix A
            and an N-by-P matrix B:

                        A = Q*R,        B = Q*T*Z,

            where Q is an N-by-N unitary matrix, Z is a P-by-P unitary matrix,
            and R and T assume one of the forms:

            if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
                            (  0  ) N-M                         N   M-N
                               M

            where R11 is upper triangular, and

            if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
                             P-N  N                           ( T21 ) P
                                                                 P

            where T12 or T21 is upper triangular.

            In particular, if B is square and nonsingular, the GQR factorization
            of A and B implicitly gives the QR factorization of inv(B)*A:

                         inv(B)*A = Z**H * (inv(T)*R)

            where inv(B) denotes the inverse of the matrix B, and Z**H denotes the
            conjugate transpose of matrix Z.

       Parameters
           N

                     N is INTEGER
                     The number of rows of the matrices A and B. N >= 0.

           M

                     M is INTEGER
                     The number of columns of the matrix A.  M >= 0.

           P

                     P is INTEGER
                     The number of columns of the matrix B.  P >= 0.

           A

                     A is COMPLEX*16 array, dimension (LDA,M)
                     On entry, the N-by-M matrix A.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(N,M)-by-M upper trapezoidal matrix R (R is
                     upper triangular if N >= M); the elements below the diagonal,
                     with the array TAUA, represent the unitary matrix Q as a
                     product of min(N,M) elementary reflectors (see Further
                     Details).

           LDA

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

           TAUA

                     TAUA is COMPLEX*16 array, dimension (min(N,M))
                     The scalar factors of the elementary reflectors which
                     represent the unitary matrix Q (see Further Details).

           B

                     B is COMPLEX*16 array, dimension (LDB,P)
                     On entry, the N-by-P matrix B.
                     On exit, if N <= P, the upper triangle of the subarray
                     B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
                     if N > P, the elements on and above the (N-P)-th subdiagonal
                     contain the N-by-P upper trapezoidal matrix T; the remaining
                     elements, with the array TAUB, represent the unitary
                     matrix Z as a product of elementary reflectors (see Further
                     Details).

           LDB

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

           TAUB

                     TAUB is COMPLEX*16 array, dimension (min(N,P))
                     The scalar factors of the elementary reflectors which
                     represent the unitary matrix Z (see Further Details).

           WORK

                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK

                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= max(1,N,M,P).
                     For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
                     where NB1 is the optimal blocksize for the QR factorization
                     of an N-by-M matrix, NB2 is the optimal blocksize for the
                     RQ factorization of an N-by-P matrix, and NB3 is the optimal
                     blocksize for a call of ZUNMQR.

                     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

                     INFO is INTEGER
                      = 0:  successful exit
                      < 0:  if INFO = -i, the i-th argument had an illegal value.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Further Details:


             The matrix Q is represented as a product of elementary reflectors

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

             Each H(i) has the form

                H(i) = I - taua * v * v**H

             where taua is a complex scalar, and v is a complex vector with
             v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
             and taua in TAUA(i).
             To form Q explicitly, use LAPACK subroutine ZUNGQR.
             To use Q to update another matrix, use LAPACK subroutine ZUNMQR.

             The matrix Z is represented as a product of elementary reflectors

                Z = H(1) H(2) . . . H(k), where k = min(n,p).

             Each H(i) has the form

                H(i) = I - taub * v * v**H

             where taub is a complex scalar, and v is a complex vector with
             v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
             B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
             To form Z explicitly, use LAPACK subroutine ZUNGRQ.
             To use Z to update another matrix, use LAPACK subroutine ZUNMRQ.

       Definition at line 213 of file zggqrf.f.

Author
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LAPACK                          Version 3.12.0                 SRC/zggqrf.f(3)