ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
 matrix A so that elements below the k-th subdiagonal are zero. The
 reduction is performed by an unitary similarity transformation
 Q**H * A * Q. The routine returns the matrices V and T which determine
 Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
 This is an auxiliary routine called by ZGEHRD.

N

          N is INTEGER
          The order of the matrix A.

K

          K is INTEGER
          The offset for the reduction. Elements below the k-th
          subdiagonal in the first NB columns are reduced to zero.
          K < N.

NB

          NB is INTEGER
          The number of columns to be reduced.

A

          A is COMPLEX*16 array, dimension (LDA,N-K+1)
          On entry, the n-by-(n-k+1) general matrix A.
          On exit, the elements on and above the k-th subdiagonal in
          the first NB columns are overwritten with the corresponding
          elements of the reduced matrix; the elements below the k-th
          subdiagonal, with the array TAU, represent the matrix Q as a
          product of elementary reflectors. The other columns of A are
          unchanged. See Further Details.

LDA

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

TAU

          TAU is COMPLEX*16 array, dimension (NB)
          The scalar factors of the elementary reflectors. See Further
          Details.

T

          T is COMPLEX*16 array, dimension (LDT,NB)
          The upper triangular matrix T.

LDT

          LDT is INTEGER
          The leading dimension of the array T.  LDT >= NB.

Y

          Y is COMPLEX*16 array, dimension (LDY,NB)
          The n-by-nb matrix Y.

LDY

          LDY is INTEGER
          The leading dimension of the array Y. LDY >= N.

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  The matrix Q is represented as a product of nb elementary reflectors
     Q = H(1) H(2) . . . H(nb).
  Each H(i) has the form
     H(i) = I - tau * v * v**H
  where tau is a complex scalar, and v is a complex vector with
  v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
  A(i+k+1:n,i), and tau in TAU(i).
  The elements of the vectors v together form the (n-k+1)-by-nb matrix
  V which is needed, with T and Y, to apply the transformation to the
  unreduced part of the matrix, using an update of the form:
  A := (I - V*T*V**H) * (A - Y*V**H).
  The contents of A on exit are illustrated by the following example
  with n = 7, k = 3 and nb = 2:
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( a   a   a   a   a )
     ( h   h   a   a   a )
     ( v1  h   a   a   a )
     ( v1  v2  a   a   a )
     ( v1  v2  a   a   a )
  where a denotes an element of the original matrix A, h denotes a
  modified element of the upper Hessenberg matrix H, and vi denotes an
  element of the vector defining H(i).
  This subroutine is a slight modification of LAPACK-3.0's ZLAHRD
  incorporating improvements proposed by Quintana-Orti and Van de
  Gejin. Note that the entries of A(1:K,2:NB) differ from those
  returned by the original LAPACK-3.0's ZLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)

Gregorio Quintana-Orti and Robert van de Geijn, 'Improving the
performance of reduction to Hessenberg form,' ACM Transactions on Mathematical Software, 32(2):180-194, June 2006.