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NAME SGGRQF - compute a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix B SYNOPSIS SUBROUTINE SGGRQF( M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, INFO ) INTEGER INFO, LDA, LDB, LWORK, M, N, P REAL A( LDA, * ), B( LDB, * ), TAUA( * ), TAUB( * ), WORK( * ) PURPOSE SGGRQF 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 M-by-M orthogonal matrix, and Z is an N-by-N orthogonal matrix. R and T assume one of the forms: if M <= N, or if M > N R = ( 0 R12 ) M, R = ( R11 ) M-N N-M M ( R21 ) N N where R12 or R21 is upper triangular, and if P >= N, or if P < N T = ( T11 ) N T = ( T11 T12 ) P ( 0 ) P-N P N-P N where T11 is an upper triangular matrix. In particular, if B is square and nonsingular, the GRQ fac- torization of A and B implicitly gives the RQ factorization of matrix A*inv(B): A*inv(B) = (R*inv(T))*Z' where inv(B) denotes the inverse of the matrix B, and Z' denotes the 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) REAL 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 subdiag- onal contain the M-by-N upper trapezoidal matrix R; the remaining elements, with the array TAUA, represent the orthogonal 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) REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details). B (input/output) REAL array, dimension (LDB,N) On entry, the P-by-N matrix B. On exit, the ele- ments 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 orthog- onal 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) REAL array, dimension (MIN(P,N)) The scalar factors of the elementary reflectors (see Further Details). WORK (workspace) REAL 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(1,N,M,P)*MAX(NB1,NB2,NB3), where NB1 is the optimal blocksize for the RQ factorization of the M-by-N matrix A. NB2 is the optimal blocksize for the QR factorization of the P-by-N matrix B. NB3 is the optimal blocksize for SORMRQ. INFO (output) INTEGER = 0: successful exit < 0: if INF0= -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 - taub * v * v' where taub is a real scalar, and v is a real 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 taub in TAUA(i). To form Q explicitly, use LAPACK subroutine SORGRQ. To use Q to update another matrix, use LAPACK subroutine SORMRQ. 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 - taua * v * v' where taua is a real scalar, and v is a real vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:M) is stored on exit in B(i+1:P,i), and taua in TAUB(i). To form Z explicitly, use LAPACK subroutine SORGQR. To use Z to update another matrix, use LAPACK subroutine SORMQR.