Provided by: liblapack-doc_3.3.1-1_all bug

NAME

       LAPACK-3  -  computes  a  generalized  RQ factorization of an M-by-N matrix A and a P-by-N
       matrix B

SYNOPSIS

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

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

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

PURPOSE

       CGGRQF 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.

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 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 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 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 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 array, dimension (MAX(1,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 CUNMRQ.
                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 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**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 CUNGRQ.
        To use Q to update another matrix, use LAPACK subroutine CUNMRQ.
        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 CUNGQR.
        To use Z to update another matrix, use LAPACK subroutine CUNMQR.

 LAPACK routine (version 3.3.1)             April 2011                            CGGRQF(3lapack)