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SUBROUTINE <a name="ZGGRQF.1"></a><a href="zggrqf.f.html#ZGGRQF.1">ZGGRQF</a>( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
$ LWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> INTEGER INFO, LDA, LDB, LWORK, M, N, P
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> COMPLEX*16 A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ),
$ WORK( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="ZGGRQF.19"></a><a href="zggrqf.f.html#ZGGRQF.1">ZGGRQF</a> computes a generalized RQ factorization of an M-by-N matrix A
</span><span class="comment">*</span><span class="comment"> and a P-by-N matrix B:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A = R*Q, B = Z*T*Q,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where Q is an N-by-N unitary matrix, Z is a P-by-P unitary
</span><span class="comment">*</span><span class="comment"> matrix, and R and T assume one of the forms:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> if M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
</span><span class="comment">*</span><span class="comment"> N-M M ( R21 ) N
</span><span class="comment">*</span><span class="comment"> N
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where R12 or R21 is upper triangular, and
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
</span><span class="comment">*</span><span class="comment"> ( 0 ) P-N P N-P
</span><span class="comment">*</span><span class="comment"> N
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where T11 is upper triangular.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> In particular, if B is square and nonsingular, the GRQ factorization
</span><span class="comment">*</span><span class="comment"> of A and B implicitly gives the RQ factorization of A*inv(B):
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A*inv(B) = (R*inv(T))*Z'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where inv(B) denotes the inverse of the matrix B, and Z' denotes the
</span><span class="comment">*</span><span class="comment"> conjugate transpose of the matrix Z.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> M (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of the matrix A. M >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> P (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of the matrix B. P >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of columns of the matrices A and B. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) COMPLEX*16 array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment"> On entry, the M-by-N matrix A.
</span><span class="comment">*</span><span class="comment"> On exit, if M <= N, the upper triangle of the subarray
</span><span class="comment">*</span><span class="comment"> A(1:M,N-M+1:N) contains the M-by-M upper triangular matrix R;
</span><span class="comment">*</span><span class="comment"> if M > N, the elements on and above the (M-N)-th subdiagonal
</span><span class="comment">*</span><span class="comment"> contain the M-by-N upper trapezoidal matrix R; the remaining
</span><span class="comment">*</span><span class="comment"> elements, with the array TAUA, represent the unitary
</span><span class="comment">*</span><span class="comment"> matrix Q as a product of elementary reflectors (see Further
</span><span class="comment">*</span><span class="comment"> Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAUA (output) COMPLEX*16 array, dimension (min(M,N))
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment"> represent the unitary matrix Q (see Further Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input/output) COMPLEX*16 array, dimension (LDB,N)
</span><span class="comment">*</span><span class="comment"> On entry, the P-by-N matrix B.
</span><span class="comment">*</span><span class="comment"> On exit, the elements on and above the diagonal of the array
</span><span class="comment">*</span><span class="comment"> contain the min(P,N)-by-N upper trapezoidal matrix T (T is
</span><span class="comment">*</span><span class="comment"> upper triangular if P >= N); the elements below the diagonal,
</span><span class="comment">*</span><span class="comment"> with the array TAUB, represent the unitary matrix Z as a
</span><span class="comment">*</span><span class="comment"> product of elementary reflectors (see Further Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array B. LDB >= max(1,P).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAUB (output) COMPLEX*16 array, dimension (min(P,N))
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors which
</span><span class="comment">*</span><span class="comment"> represent the unitary matrix Z (see Further Details).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment"> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LWORK (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The dimension of the array WORK. LWORK >= max(1,N,M,P).
</span><span class="comment">*</span><span class="comment"> For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
</span><span class="comment">*</span><span class="comment"> where NB1 is the optimal blocksize for the RQ factorization
</span><span class="comment">*</span><span class="comment"> of an M-by-N matrix, NB2 is the optimal blocksize for the
</span><span class="comment">*</span><span class="comment"> QR factorization of a P-by-N matrix, and NB3 is the optimal
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