sggrqf - 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);
Oracle Solaris Studio Performance Library sggrqf(3P)
NAME
sggrqf - compute a generalized RQ factorization of an M-by-N matrix A
and a P-by-N matrix B
SYNOPSIS
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);
PURPOSE
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 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 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 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 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.
7 Nov 2015 sggrqf(3P)