clahr2.f(3)

NAME
clahr2.f -

SYNOPSIS

  Functions/Subroutines

subroutine clahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Function/Subroutine Documentation

  subroutine clahr2 (integerN, integerK, integerNB, complex, dimension( lda, * )A, integerLDA, complex, dimension( nb )TAU, complex, dimension( ldt, nb )T, integerLDT, complex, dimension( ldy, nb )Y, integerLDY)

CLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

Purpose:

     CLAHR2 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 CGEHRD.

Parameters:
    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 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 array, dimension (NB)
              The scalar factors of the elementary reflectors. See Further
              Details.

    T

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

    LDY

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

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    September 2012

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.

Definition at line 182 of file clahr2.f.

Author

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