DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1)


 matrix A so that elements below the k-th subdiagonal are zero. The


 reduction is performed by an orthogonal similarity transformation


 Q**T * A * Q. The routine returns the matrices V and T which determine


 Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.


 This is an auxiliary routine called by DGEHRD.

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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)


          The scalar factors of the elementary reflectors. See Further


          Details.

T

          T is DOUBLE PRECISION 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 DOUBLE PRECISION 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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September 2012

  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**T


  where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).


  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.)

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.