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SGGRQF(l)) SGGRQF(l)NAMESGGRQF - compute a generalized RQ factorization of an M-by-N matrix A and a P-by-N matrix BSYNOPSISSUBROUTINE 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( * )PURPOSESGGRQF 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 orthogonal matrix, Z is a P-by-P orthogonal 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 transpose of the matrix Z.ARGUMENTSM (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 sub- array 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 trape- zoidal 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 which represent the orthogonal matrix Q (see Further Details). B (input/output) REAL 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 orthogonal matrix Z as a product of elementary reflectors (see Fur- ther 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 which represent the orthogonal matrix Z (see Further Details). WORK (workspace/output) 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(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 fac- torization of a P-by-N matrix, and NB3 is the optimal blocksize for a call of SORMRQ. If LWORK =, 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 INF0=-1, the i-th argument had an illegal value.-iFURTHER DETAILSThe 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 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 taua 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 - taub * v * v' where taub is a real scalar, and v is a real 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 SORGQR. To use Z to update another matrix, use LAPACK subroutine SORMQR.LAPACK version 3.015 June 2000 SGGRQF(l)

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