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

NAME
       SRC/zggrqf.f

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

Function/Subroutine Documentation
   subroutine zggrqf (integer m, integer p, integer n, 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)
       ZGGRQF

       Purpose:


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

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

            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 M <= N,  R = ( 0  R12 ) M,   or if M > N,  R = ( R11 ) M-N,
                             N-M  M                           ( R21 ) N
                                                                 N

            where R12 or R21 is upper triangular, and

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

            where T11 is upper triangular.

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

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

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

       Parameters
           M

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

           P

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

           N

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

           A

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

           LDA

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

           TAUA

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

           B

                     B is COMPLEX*16 array, dimension (LDB,N)
                     On entry, the P-by-N matrix B.
                     On exit, the elements on and above the diagonal of the array
                     contain the min(P,N)-by-N upper trapezoidal matrix T (T is
                     upper triangular if P >= N); the elements below the diagonal,
                     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,P).

           TAUB

                     TAUB is COMPLEX*16 array, dimension (min(P,N))
                     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 RQ factorization
                     of an M-by-N matrix, NB2 is the optimal blocksize for the
                     QR factorization of a P-by-N matrix, and NB3 is the optimal
                     blocksize for a call of ZUNMRQ.

                     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(m,n).

             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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
             A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
             To form Q explicitly, use LAPACK subroutine ZUNGRQ.
             To use Q to update another matrix, use LAPACK subroutine ZUNMRQ.

             The matrix Z is represented as a product of elementary reflectors

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

             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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
             and taub in TAUB(i).
             To form Z explicitly, use LAPACK subroutine ZUNGQR.
             To use Z to update another matrix, use LAPACK subroutine ZUNMQR.

       Definition at line 212 of file zggrqf.f.

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