DLAHRD - reduce the first NB columns of a real general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero

SUBROUTINE DLAHRD(

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

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

DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

DLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal 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 DGEHRD.

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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (NB)

The scalar factors of the elementary reflectors. See Further Details.

T (output) DOUBLE PRECISION array, dimension (LDT,NB)

The upper triangular matrix T.

LDT (input) INTEGER

The leading dimension of the array T. LDT >= NB.

Y (output) DOUBLE PRECISION array, dimension (LDY,NB)

The n-by-nb matrix Y.

LDY (input) INTEGER

The leading dimension of the array Y. LDY >= N.

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

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

Each H(i) has the form

H(i) = I - tau * v * v'

where tau is a real scalar, and v is a real 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).