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NAME SGGQRF - compute a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B SYNOPSIS SUBROUTINE SGGQRF( N, M, P, 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 SGGQRF computes a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B: A = Q*R, B = Q*T*Z, where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal matrix, and R and T assumes one of the forms: if N >= M, R = ( R11 ) M , or if N < M, R = ( R11 R12 ) N ( 0 ) N-M N M-N M where R11 is an upper triangular matrix, and if N <= P, T = ( 0 T12 ) N, or if N > P, T = ( T11 ) N-P P-N N ( T21 ) P P where T12 or T21 is a P-by-P upper triangular matrix. In particular, if B is square and nonsingular, the GQR fac- torization of A and B implicitly gives the QR factorization of inv(B)*A: inv(B)*A = Z'*(inv(T)*R) where inv(B) denotes the inverse of the matrix B, Z' denotes the transpose of matrix Z. ARGUMENTS N (input) INTEGER The number of rows of the matrices A and B. N >= 0. M (input) INTEGER The number of columns of the matrix A. M >= 0. P (input) INTEGER The number of columns of the matrix B. P >= 0. A (input/output) REAL array, dimension (LDA,M) On entry, the N-by-M matrix A. On exit, the ele- ments on and above the diagonal of the array contain the min(N,M)-by-M upper trapezoidal matrix R (R is upper triangular if N >= M); the elements below the diagonal, with the array TAUA, represent the orthog- onal matrix Q as a product of min(N,M) elementary reflectors (see Further Details). LDA (input) INTEGER The leading dimension of the array A. LDA >= MAX(1,N). TAUA (output) REAL array, dimension (MIN(N,M)) The scalar factors of the elementary reflectors (see Further Details). B (input/output) REAL array, dimension (LDB,P) On entry, the N-by-P matrix B. On exit, if N <= P, the upper triangle of the subarray B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T; if N > P, the elements on and above the (N-P)-th subdiag- onal contain the N-by-P upper trapezoidal matrix T; the remaining elements, with the array TAUB, represent the orthogonal matrix Z as a product of elementary reflectors (see Further Details). LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). TAUB (output) REAL array, dimension (MIN(N,P)) 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 QR factorization of an N- by-M matrix A. NB2 is the optimal blocksize for the RQ factorization of an N-by-P matrix B. NB3 is the optimal blocksize for calling SORMQR. 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(N,M). 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:N) is stored on exit in A(i+1:N,i), and taua in TAUA(i). To form Q explicitly, use LAPACK subroutine SORGQR. To use Q to update another matrix, use LAPACK subroutine SORMQR. The matrix Z is represented as a product of elementary reflectors Z = H(1) H(2) . . . H(k), where k = min(N,P). 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(P-k+i+1:P) = 0 and v(P-k+i) = 1; v(1:P-k+i-1) is stored on exit in B(N-k+i,1:P-k+i-1), and taub in TAUB(i). To form Z explicitly, use LAPACK subroutine SORGRQ. To use Z to update another matrix, use LAPACK subroutine SORMRQ.