!>
!> CGEQRF computes a QR factorization of a complex 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 COMPLEX 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 COMPLEX array, dimension (min(M,N))
!>          The scalar factors of the elementary reflectors (see Further
!>          Details).
!> 

WORK

!>          WORK is COMPLEX 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. LWORK >= 1 if MIN(M,N) = 0,
!>          otherwise 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, 'CGEQRF', ' ', 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).
!>
!>