# zlahrd (l) - Linux Manuals

## 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

## NAME

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## SYNOPSIS

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

- INTEGER K, LDA, LDT, LDY, N, NB

- COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

## 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 Qaq * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*Vaq, and also the matrix Y = A * V * T. This is an OBSOLETE auxiliary routine.This routine will be aqdeprecatedaq in a future release.

Please use the new routine ZLAHR2 instead.

## ARGUMENTS

- N (input) INTEGER
- The order of the matrix A.
- K (input) INTEGER
- The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero.
- NB (input) INTEGER
- The number of columns to be reduced.
- A (input/output) 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 (input) INTEGER The leading dimension of the array A. LDA >= max(1,N).
- TAU (output) COMPLEX*16 array, dimension (NB)
- The scalar factors of the elementary reflectors. See Further Details.
- T (output) COMPLEX*16 array, dimension (LDT,NB)
- The upper triangular matrix T.
- LDT (input) INTEGER
- The leading dimension of the array T. LDT >= NB.
- Y (output) COMPLEX*16 array, dimension (LDY,NB)
- The n-by-nb matrix Y.
- LDY (input) INTEGER
- The leading dimension of the array Y. LDY >= max(1,N).

## FURTHER DETAILS

The matrix Q is represented as a product of nb elementary reflectorsQ

Each H(i) has the form

H(i)

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*Vaq) * (A - Y*Vaq).

The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2:

(

(

(

(

(

(

(

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).