.TH "SRC/dggqrf.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/dggqrf.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBdggqrf\fP (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)" .br .RI "\fBDGGQRF\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine dggqrf (integer n, integer m, integer p, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( * ) taua, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) taub, double precision, dimension( * ) work, integer lwork, integer info)" .PP \fBDGGQRF\fP .PP \fBPurpose:\fP .RS 4 .PP .nf !> !> DGGQRF 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**T*(inv(T)*R) !> !> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the !> transpose of the matrix Z\&. !> .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIN\fP .PP .nf !> N is INTEGER !> The number of rows of the matrices A and B\&. N >= 0\&. !> .fi .PP .br \fIM\fP .PP .nf !> M is INTEGER !> The number of columns of the matrix A\&. M >= 0\&. !> .fi .PP .br \fIP\fP .PP .nf !> P is INTEGER !> The number of columns of the matrix B\&. P >= 0\&. !> .fi .PP .br \fIA\fP .PP .nf !> A is DOUBLE PRECISION 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)\&. !> .fi .PP .br \fILDA\fP .PP .nf !> LDA is INTEGER !> The leading dimension of the array A\&. LDA >= max(1,N)\&. !> .fi .PP .br \fITAUA\fP .PP .nf !> TAUA is DOUBLE PRECISION array, dimension (min(N,M)) !> The scalar factors of the elementary reflectors which !> represent the orthogonal matrix Q (see Further Details)\&. !> .fi .PP .br \fIB\fP .PP .nf !> B is DOUBLE PRECISION 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)\&. !> .fi .PP .br \fILDB\fP .PP .nf !> LDB is INTEGER !> The leading dimension of the array B\&. LDB >= max(1,N)\&. !> .fi .PP .br \fITAUB\fP .PP .nf !> TAUB is DOUBLE PRECISION array, dimension (min(N,P)) !> The scalar factors of the elementary reflectors which !> represent the orthogonal matrix Z (see Further Details)\&. !> .fi .PP .br \fIWORK\fP .PP .nf !> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. !> .fi .PP .br \fILWORK\fP .PP .nf !> LWORK is 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 DORMQR\&. !> !> 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\&. !> .fi .PP .br \fIINFO\fP .PP .nf !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value\&. !> .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf !> !> 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**T !> !> 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 DORGQR\&. !> To use Q to update another matrix, use LAPACK subroutine DORMQR\&. !> !> 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**T !> !> 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 DORGRQ\&. !> To use Z to update another matrix, use LAPACK subroutine DORMRQ\&. !> .fi .PP .RE .PP .PP Definition at line \fB213\fP of file \fBdggqrf\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.