DGEQRF 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.

M

          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 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

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

TAU

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


          The scalar factors of the elementary reflectors (see Further


          Details).

WORK

          WORK is DOUBLE PRECISION 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, 'DGEQRF', ' ', 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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

  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).