dggqrf - compute a generalized QR factorization of an N-by-M matrix A and an N-by-P matrix B.
SUBROUTINE DGGQRF( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK, * INFO) INTEGER N, M, P, LDA, LDB, LWORK, INFO DOUBLE PRECISION A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
SUBROUTINE DGGQRF_64( N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, * LWORK, INFO) INTEGER*8 N, M, P, LDA, LDB, LWORK, INFO DOUBLE PRECISION A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
SUBROUTINE GGQRF( [N], [M], [P], A, [LDA], TAUA, B, [LDB], TAUB, * [WORK], [LWORK], [INFO]) INTEGER :: N, M, P, LDA, LDB, LWORK, INFO REAL(8), DIMENSION(:) :: TAUA, TAUB, WORK REAL(8), DIMENSION(:,:) :: A, B
SUBROUTINE GGQRF_64( [N], [M], [P], A, [LDA], TAUA, B, [LDB], TAUB, * [WORK], [LWORK], [INFO]) INTEGER(8) :: N, M, P, LDA, LDB, LWORK, INFO REAL(8), DIMENSION(:) :: TAUA, TAUB, WORK REAL(8), DIMENSION(:,:) :: A, B
#include <sunperf.h>
void dggqrf(int n, int m, int p, double *a, int lda, double *taua, double *b, int ldb, double *taub, int *info);
void dggqrf_64(long n, long m, long p, double *a, long lda, double *taua, double *b, long ldb, double *taub, long *info);
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'*(inv(T)*R)
where inv(B)
denotes the inverse of the matrix B, and Z' denotes the
transpose of the matrix Z.
min(N,M)
elementary reflectors (see Further
Details).
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).
WORK(1)
returns the optimal LWORK.
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.
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
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.