cggqrf(3P) Sun Performance Library cggqrf(3P)NAMEcggqrf - compute a generalized QR factorization of an N-by-M matrix A
and an N-by-P matrix B.
SYNOPSIS
SUBROUTINE CGGQRF(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO)
COMPLEX A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
INTEGER N, M, P, LDA, LDB, LWORK, INFO
SUBROUTINE CGGQRF_64(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
LWORK, INFO)
COMPLEX A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
INTEGER*8 N, M, P, LDA, LDB, LWORK, INFO
F95 INTERFACE
SUBROUTINE GGQRF([N], [M], [P], A, [LDA], TAUA, B, [LDB], TAUB, [WORK],
[LWORK], [INFO])
COMPLEX, DIMENSION(:) :: TAUA, TAUB, WORK
COMPLEX, DIMENSION(:,:) :: A, B
INTEGER :: N, M, P, LDA, LDB, LWORK, INFO
SUBROUTINE GGQRF_64([N], [M], [P], A, [LDA], TAUA, B, [LDB], TAUB,
[WORK], [LWORK], [INFO])
COMPLEX, DIMENSION(:) :: TAUA, TAUB, WORK
COMPLEX, DIMENSION(:,:) :: A, B
INTEGER(8) :: N, M, P, LDA, LDB, LWORK, INFO
C INTERFACE
#include <sunperf.h>
void cggqrf(int n, int m, int p, complex *a, int lda, complex *taua,
complex *b, int ldb, complex *taub, int *info);
void cggqrf_64(long n, long m, long p, complex *a, long lda, complex
*taua, complex *b, long ldb, complex *taub, long *info);
PURPOSEcggqrf 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 unitary matrix, Z is a P-by-P unitary 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
conjugate transpose of 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 unitary 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 repre‐
sent the unitary 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 unitary matrix Z as a product of elemen‐
tary 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 unitary 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 CUNMQR.
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 complex scalar, and v is a complex 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 CUNGQR.
To use Q to update another matrix, use LAPACK subroutine CUNMQR.
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 complex scalar, and v is a complex 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 CUNGRQ.
To use Z to update another matrix, use LAPACK subroutine CUNMRQ.
6 Mar 2009 cggqrf(3P)
