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.

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 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 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 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 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 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 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

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO=-i, the i-th argument had an illegal value.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  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.