zlahrd.f man page

zlahrd.f —

Synopsis

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 (integerN, integerK, integerNB, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( nb )TAU, complex*16, dimension( ldt, nb )T, integerLDT, complex*16, dimension( ldy, nb )Y, integerLDY)

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.

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:

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.

NB

NB is INTEGER
The number of columns to be reduced.

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.

LDA

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

TAU

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

T

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

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.

Author

Generated automatically by Doxygen for LAPACK from the source code.

Referenced By

zlahrd(3) is an alias of zlahrd.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK