zlahrd.f File Reference

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Functions/Subroutines
subroutine	zlahrd (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
	ZLAHRD reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

---

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

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:

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   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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