LAPACK  3.4.0
LAPACK: Linear Algebra PACKage
Functions/Subroutines
zlahrd.f File Reference

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Functions/Subroutines

subroutine ZLAHRD (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
 ZLAHRD

Function/Subroutine Documentation

subroutine ZLAHRD ( INTEGER  N,
INTEGER  K,
INTEGER  NB,
COMPLEX*16, dimension( lda, * )  A,
INTEGER  LDA,
COMPLEX*16, dimension( nb )  TAU,
COMPLEX*16, dimension( ldt, nb )  T,
INTEGER  LDT,
COMPLEX*16, dimension( ldy, nb )  Y,
INTEGER  LDY 
)

ZLAHRD

Download ZLAHRD + dependencies [TGZ] [ZIP] [TXT]
Purpose: 

 ZLAHRD 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 a 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 OBSOLETE auxiliary routine. 
 This routine will be 'deprecated' in a  future release.
 Please use the new routine ZLAHR2 instead.
Parameters:
[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.
[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 >= max(1,N).
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
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   h   a   a   a )
     ( a   h   a   a   a )
     ( a   h   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).

Definition at line 170 of file zlahrd.f.

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