cggrqf(3S)

NAME
     CGGRQF - compute 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(
                        * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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'

     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 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 (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'

     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'

     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.


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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