Contents
zggqrf - compute a generalized QR factorization of an N-by-M
matrix A and an N-by-P matrix B.
SUBROUTINE ZGGQRF(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK, LWORK,
INFO)
DOUBLE COMPLEX A(LDA,*), TAUA(*), B(LDB,*), TAUB(*), WORK(*)
INTEGER N, M, P, LDA, LDB, LWORK, INFO
SUBROUTINE ZGGQRF_64(N, M, P, A, LDA, TAUA, B, LDB, TAUB, WORK,
LWORK, INFO)
DOUBLE 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(8), DIMENSION(:) :: TAUA, TAUB, WORK
COMPLEX(8), 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(8), DIMENSION(:) :: TAUA, TAUB, WORK
COMPLEX(8), DIMENSION(:,:) :: A, B
INTEGER(8) :: N, M, P, LDA, LDB, LWORK, INFO
C INTERFACE
#include <sunperf.h>
void zggqrf(int n, int m, int p, doublecomplex *a, int lda,
doublecomplex *taua, doublecomplex *b, int ldb,
doublecomplex *taub, int *info);
void zggqrf_64(long n, long m, long p, doublecomplex *a,
long lda, doublecomplex *taua, doublecomplex *b,
long ldb, doublecomplex *taub, long *info);
zggqrf 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 fac-
torization 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.
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 ele-
ments on and above the diagonal of the array con-
tain 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, represent
the unitary matrix Q as a product of min(N,M) ele-
mentary 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 represent 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 tra-
pezoidal matrix T; the remaining elements, with
the array TAUB, represent the unitary matrix Z as
a product of elementary 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 represent 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 ille-
gal value.
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.