SUBROUTINE SGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, $ LWORK, INFO ) * * -- LAPACK routine (version 3.1) -- * Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. * November 2006 * * .. Scalar Arguments .. INTEGER INFO, LDA, LDB, LWORK, M, N, P * .. * .. Array Arguments .. 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 assume 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 upper triangular, 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 upper triangular. * * In particular, if B is square and nonsingular, the GQR factorization * 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, and Z' denotes the * transpose of the 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 elements 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 orthogonal 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 which * represent the orthogonal matrix Q (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 subdiagonal * 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 which * represent the orthogonal matrix Z (see Further Details). * * WORK (workspace/output) REAL array, dimension (MAX(1,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 QR factorization * of an N-by-M matrix, NB2 is the optimal blocksize for the * RQ factorization of an N-by-P matrix, and NB3 is the optimal * blocksize for a call of SORMQR. * * 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 (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. * * ===================================================================== * * .. Local Scalars .. LOGICAL LQUERY INTEGER LOPT, LWKOPT, NB, NB1, NB2, NB3 * .. * .. External Subroutines .. EXTERNAL SGEQRF, SGERQF, SORMQR, XERBLA * .. * .. External Functions .. INTEGER ILAENV EXTERNAL ILAENV * .. * .. Intrinsic Functions .. INTRINSIC INT, MAX, MIN * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 NB1 = ILAENV( 1, 'SGEQRF', ' ', N, M, -1, -1 ) NB2 = ILAENV( 1, 'SGERQF', ' ', N, P, -1, -1 ) NB3 = ILAENV( 1, 'SORMQR', ' ', N, M, P, -1 ) NB = MAX( NB1, NB2, NB3 ) LWKOPT = MAX( N, M, P )*NB WORK( 1 ) = LWKOPT LQUERY = ( LWORK.EQ.-1 ) IF( N.LT.0 ) THEN INFO = -1 ELSE IF( M.LT.0 ) THEN INFO = -2 ELSE IF( P.LT.0 ) THEN INFO = -3 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -5 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( LWORK.LT.MAX( 1, N, M, P ) .AND. .NOT.LQUERY ) THEN INFO = -11 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'SGGQRF', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * QR factorization of N-by-M matrix A: A = Q*R * CALL SGEQRF( N, M, A, LDA, TAUA, WORK, LWORK, INFO ) LOPT = WORK( 1 ) * * Update B := Q'*B. * CALL SORMQR( 'Left', 'Transpose', N, P, MIN( N, M ), A, LDA, TAUA, $ B, LDB, WORK, LWORK, INFO ) LOPT = MAX( LOPT, INT( WORK( 1 ) ) ) * * RQ factorization of N-by-P matrix B: B = T*Z. * CALL SGERQF( N, P, B, LDB, TAUB, WORK, LWORK, INFO ) WORK( 1 ) = MAX( LOPT, INT( WORK( 1 ) ) ) * RETURN * * End of SGGQRF * END