Contents
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(*)
F95 INTERFACE
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
C INTERFACE
#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 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, and Z'
denotes the transpose of the matrix Z.
N (input) The number of rows of the matrices A and B. N >=
0.
M (input) The number of columns of the matrix A. M >= 0.
P (input) The number of columns of the matrix B. P >= 0.
A (input/output)
On entry, the N-by-M matrix A. On exit, the ele-
ments on and above the diagonal of the array con-
tain 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)
The leading dimension of the array A. LDA >=
max(1,N).
TAUA (output)
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Q (see
Further Details).
B (input/output)
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 tra-
pezoidal 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)
The leading dimension of the array B. LDB >=
max(1,N).
TAUB (output)
The scalar factors of the elementary reflectors
which represent the orthogonal matrix Z (see
Further Details).
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal
LWORK.
LWORK (input)
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)
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille-
gal 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.