ZGEQRF
Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
March 2008
March 2008
Purpose
ZGEQRF computes a QR factorization of a real M-by-N matrix A:
A = Q * R.
This is the left-looking Level 3 BLAS version of the algorithm.
A = Q * R.
This is the left-looking Level 3 BLAS version of the algorithm.
Arguments
| M |
(input) INTEGER
The number of rows of the matrix A. M >= 0.
|
| N |
(input) INTEGER
The number of columns of the matrix A. N >= 0.
|
| A |
(input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the M-by-N matrix A.
On exit, the elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details). |
| LDA |
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,M).
|
| TAU |
(output) COMPLEX*16 array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details). |
| WORK |
(workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
| LWORK |
(input) INTEGER
The dimension of the array WORK. The dimension can be divided into three parts. 1) The part for the triangular factor T. If the very last T is not bigger than any of the rest, then this part is NB x ceiling(K/NB), otherwise, NB x (K-NT), where K = min(M,N) and NT is the dimension of the very last T 2) The part for the very last T when T is bigger than any of the rest T. The size of this part is NT x NT, where NT = K - ceiling ((K-NX)/NB) x NB, where K = min(M,N), NX is calculated by NX = MAX( 0, ILAENV( 3, 'ZGEQRF', ' ', M, N, -1, -1 ) ) 3) The part for dlarfb is of size max((N-M)*K, (N-M)*NB, K*NB, NB*NB) So LWORK = part1 + part2 + part3 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) INTEGER
= 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(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).
Q = H(1) H(2) . . . H(k), where k = min(m,n).
Each H(i) has the form
H(i) = I - tau * v * v'
where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
and tau in TAU(i).