SYNOPSIS

SUBROUTINE DGEQRF(

M, N, A, LDA, TAU, WORK, LWORK, INFO )

    

INTEGER INFO, LDA, LWORK, M, N

    

DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

DGEQRF computes a QR factorization of a real M-by-N matrix A: A = Q * R.

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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (min(M,N))

The scalar factors of the elementary reflectors (see Further Details).

WORK (workspace/output) DOUBLE PRECISION 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. LWORK >= max(1,N). For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. 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).