DLAHRD - reduce the first NB columns of a real general n-
by-(n-k+1) matrix A so that elements below the k-th subdi
agonal are zero
SYNOPSIS
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 )
PURPOSE
DLAHRD reduces the first NB columns of a real general n-
by-(n-k+1) matrix A so that elements below the k-th subdi
agonal are zero. The reduction is performed by an orthogo
nal 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.
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) 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 subdiago
nal 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.
(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.
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 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).
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