dggrqf(3P) Sun Performance Library dggrqf(3P)NAMEdggrqf - compute a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B
SYNOPSIS
SUBROUTINE DGGRQF(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO)
INTEGER M, P, N, LDA, LDB, LWORK, INFO
DOUBLE PRECISION A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
SUBROUTINE DGGRQF_64(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
LWORK, INFO)
INTEGER*8 M, P, N, LDA, LDB, LWORK, INFO
DOUBLE PRECISION A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
F95 INTERFACE
SUBROUTINE GGRQF([M], [P], [N], A, [LDA], TAUA, B, [LDB], TAUB, [WORK],
[LWORK], [INFO])
INTEGER :: M, P, N, LDA, LDB, LWORK, INFO
REAL(8), DIMENSION(:) :: TAUA, TAUB, WORK
REAL(8), DIMENSION(:,:) :: A, B
SUBROUTINE GGRQF_64([M], [P], [N], A, [LDA], TAUA, B, [LDB], TAUB,
[WORK], [LWORK], [INFO])
INTEGER(8) :: M, P, N, LDA, LDB, LWORK, INFO
REAL(8), DIMENSION(:) :: TAUA, TAUB, WORK
REAL(8), DIMENSION(:,:) :: A, B
C INTERFACE
#include <sunperf.h>
void dggrqf(int m, int p, int n, double *a, int lda, double *taua, dou‐
ble *b, int ldb, double *taub, int *info);
void dggrqf_64(long m, long p, long n, double *a, long lda, double
*taua, double *b, long ldb, double *taub, long *info);
PURPOSEdggrqf computes a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B:
A = R*Q, B = Z*T*Q,
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 M <= N, R = ( 0 R12 ) M, or if M > N, R = ( R11 ) M-N,
N-M M ( R21 ) N
N
where R12 or R21 is upper triangular, and
if P >= N, T = ( T11 ) N , or if P < N, T = ( T11 T12 ) P,
( 0 ) P-N P N-P
N
where T11 is upper triangular.
In particular, if B is square and nonsingular, the GRQ factorization of
A and B implicitly gives the RQ factorization of A*inv(B):
A*inv(B) = (R*inv(T))*Z'
where inv(B) denotes the inverse of the matrix B, and Z' denotes the
transpose of the matrix Z.
ARGUMENTS
M (input) The number of rows of the matrix A. M >= 0.
P (input) The number of rows of the matrix B. P >= 0.
N (input) The number of columns of the matrices A and B. N >= 0.
A (input/output)
On entry, the M-by-N matrix A. On exit, if M <= N, the upper
triangle of the subarray A(1:M,N-M+1:N) contains the M-by-M
upper triangular matrix R; if M > N, the elements on and
above the (M-N)-th subdiagonal contain the M-by-N upper
trapezoidal matrix R; the remaining elements, with the array
TAUA, represent the orthogonal matrix Q as a product of ele‐
mentary reflectors (see Further Details).
LDA (input)
The leading dimension of the array A. LDA >= max(1,M).
TAUA (output)
The scalar factors of the elementary reflectors which repre‐
sent the orthogonal matrix Q (see Further Details).
B (input/output)
On entry, the P-by-N matrix B. On exit, the elements on and
above the diagonal of the array contain the min(P,N)-by-N
upper trapezoidal matrix T (T is upper triangular if P >= N);
the elements below the diagonal, with the array TAUB, repre‐
sent 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,P).
TAUB (output)
The scalar factors of the elementary reflectors which repre‐
sent 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 RQ factorization
of an M-by-N matrix, NB2 is the optimal blocksize for the QR
factorization of a P-by-N matrix, and NB3 is the optimal
blocksize for a call of DORMRQ.
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 INF0= -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(m,n).
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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
A(m-k+i,1:n-k+i-1), and taua in TAUA(i).
To form Q explicitly, use LAPACK subroutine DORGRQ.
To use Q to update another matrix, use LAPACK subroutine DORMRQ.
The matrix Z is represented as a product of elementary reflectors
Z = H(1)H(2) . . . H(k), where k = min(p,n).
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(1:i-1) = 0 and v(i) = 1; v(i+1:p) is stored on exit in B(i+1:p,i),
and taub in TAUB(i).
To form Z explicitly, use LAPACK subroutine DORGQR.
To use Z to update another matrix, use LAPACK subroutine DORMQR.
6 Mar 2009 dggrqf(3P)
