zggqrf(3F)

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

SYNOPSIS
     SUBROUTINE ZGGQRF( N, M, P, 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
     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'*(inv(T)*R)

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


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

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

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

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

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

     TAUB    (output) COMPLEX*16 array, dimension (min(N,P))
             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 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.

     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(n,m).
     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(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'

     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.