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zlahr2 examples
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Man Page Details
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 Online html documentation available at 
            http://www.netlib.org/lapack/explore-html/ 

 Definition:
 ===========

       SUBROUTINE ZLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )
 
       .. Scalar Arguments ..
       INTEGER            K, LDA, LDT, LDY, N, NB
       ..
       .. Array Arguments ..
       COMPLEX*16        A( LDA, * ), T( LDT, NB ), TAU( NB ),
      $                   Y( LDY, NB )
       ..
  

 Purpose:
 =============

  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.

 Arguments:
 ==========

  [in] N

          N is INTEGER
          The order of the matrix A.

  [in] 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.

  [in] NB

          NB is INTEGER
          The number of columns to be reduced.

  [in,out] 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.

  [in] LDA

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

  [out] TAU

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

  [out] T

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

  [in] LDT

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

  [out] Y

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

  [in] LDY

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

 Authors:
 ========

  Univ. of Tennessee 
  Univ. of California Berkeley 
  Univ. of Colorado Denver 
  NAG Ltd. 

 November 2011

 Further Details:
 =====================

  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 DLAHRD
  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 DLAHRD routine. (This
  subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)

 References:
 ================

  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.

