zggrqf(3F)

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

SYNOPSIS
     SUBROUTINE ZGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO
                        )

         INTEGER        INFO, LDA, LDB, LWORK, M, N, P

         COMPLEX*16     A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK(
                        * )

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'

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


ARGUMENTS
     M       (input) INTEGER
             The number of rows of the matrix A.  M >= 0.

     P       (input) INTEGER
             The number of rows of the matrix B.  P >= 0.

     N       (input) INTEGER
             The number of columns of the matrices A and B. N >= 0.

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

     TAUA    (output) COMPLEX*16 array, dimension (min(M,N))
             The scalar factors of the elementary reflectors which represent
             the unitary matrix Q (see Further Details).  B
             (input/output) 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
             (input) INTEGER The leading dimension of the array B. LDB >=
             max(1,P).

     TAUB    (output) COMPLEX*16 array, dimension (min(P,N))
             The scalar factors of the elementary reflectors which represent
             the unitary matrix Z (see Further Details).  WORK
             (workspace/output) COMPLEX*16 array, dimension (LWORK) On exit,
             if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) 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.

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO=-i, the i-th argument had an illegal 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 - taua * v * v'

     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'

     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.