sggqrf(3P) Sun Performance Library sggqrf(3P)NAMEsggqrf - compute a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B.
SYNOPSIS
SUBROUTINE SGGQRF(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO)
INTEGER N, M, P, LDA, LDB, LWORK, INFO
REAL A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
SUBROUTINE SGGQRF_64(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
LWORK, INFO)
INTEGER*8 N, M, P, LDA, LDB, LWORK, INFO
REAL 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, DIMENSION(:) :: TAUA, TAUB, WORK
REAL, 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, DIMENSION(:) :: TAUA, TAUB, WORK
REAL, DIMENSION(:,:) :: A, B
C INTERFACE
#include <sunperf.h>
void sggqrf(int n, int m, int p, float *a, int lda, float *taua, float
*b, int ldb, float *taub, int *info);
void sggqrf_64(long n, long m, long p, float *a, long lda, float *taua,
float *b, long ldb, float *taub, long *info);
PURPOSEsggqrf 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.
ARGUMENTS
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 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, repre‐
sent the orthogonal matrix Q as a product of min(N,M) elemen‐
tary 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 repre‐
sent 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
trapezoidal matrix T; the remaining elements, with the array
TAUB, represent the orthogonal matrix Z as a product of ele‐
mentary 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 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 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 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(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.
6 Mar 2009 sggqrf(3P)
