Contents
sggrqf - compute a generalized RQ factorization of an M-by-N
matrix A and a P-by-N matrix B
SUBROUTINE SGGRQF(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO)
INTEGER M, P, N, LDA, LDB, LWORK, INFO
REAL A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
SUBROUTINE SGGRQF_64(M, P, N, A, LDA, TAUA, B, LDB, TAUB, WORK,
LWORK, INFO)
INTEGER*8 M, P, N, LDA, LDB, LWORK, INFO
REAL 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, DIMENSION(:) :: TAUA, TAUB, WORK
REAL, 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, DIMENSION(:) :: TAUA, TAUB, WORK
REAL, DIMENSION(:,:) :: A, B
C INTERFACE
#include <sunperf.h>
void sggrqf(int m, int p, int n, float *a, int lda, float
*taua, float *b, int ldb, float *taub, int *info);
void sggrqf_64(long m, long p, long n, float *a, long lda,
float *taua, float *b, long ldb, float *taub, long
*info);
sggrqf 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 fac-
torization 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.
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 tra-
pezoidal matrix R; the remaining elements, with
the array TAUA, represent the orthogonal matrix Q
as a product of elementary 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 represent the orthogonal matrix Q (see
Further Details).
B (input/output)
On entry, the P-by-N matrix B. On exit, the ele-
ments on and above the diagonal of the array con-
tain 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, 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,P).
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 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 SORMRQ.
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 ille-
gal value.
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 SORGRQ.
To use Q to update another matrix, use LAPACK subroutine
SORMRQ.
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 SORGQR.
To use Z to update another matrix, use LAPACK subroutine
SORMQR.