subroutine sgeqrf (M, N, A, LDA, TAU, WORK, LWORK, INFO)
SGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm.
SGEQRF VARIANT: left-looking Level 3 BLAS version of the algorithm. Purpose:
SGEQRF 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.
Parameters:
M is INTEGER
The number of rows of the matrix A. M >= 0.
N
N is INTEGER
The number of columns of the matrix A. N >= 0.
A
A is REAL 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
LDA is INTEGER
The leading dimension of the array A. LDA >= max(1,M).
TAU
TAU is REAL array, dimension (min(M,N))
The scalar factors of the elementary reflectors (see Further
Details).
WORK
WORK is REAL array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK
LWORK is 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, 'SGEQRF', ' ', 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
INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
Author:
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
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).
Definition at line 151 of file VARIANTS/qr/LL/sgeqrf.f.
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