CLAHRD(l) — LAPACK auxiliary routine (version 2.0)

NAME

CLAHRD - reduce 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 CLAHRD(
N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

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

COMPLEX A( LDA, ∗ ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

PURPOSE

CLAHRD 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’ ∗ A ∗ Q. The routine returns the matrices V and T which determine Q as a block reflector I - V∗T∗V’, and also the matrix Y = A ∗ V ∗ T.

This is an auxiliary routine called by CGEHRD.

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

T (output) COMPLEX array, dimension (NB,NB)
The upper triangular matrix T.

LDT (input) INTEGER
The leading dimension of the array T. LDT >= NB.

Y (output) COMPLEX 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 reflectors

   Q = H(1) H(2) . . . H(nb).

Each H(i) has the form

   H(i) = I - tau ∗ v ∗ v’

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’) ∗ (A - Y∗V’).

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).