Householder's method for symmetric matrices

If we write
then
J. H. ~u : Householder's method ~or symmetric matrices 3 5 5
u~= [al,1; al,~; ...; al,,-2; ai, l-1 0;...; O] (9)
I- 2w, w~ = I ~' ~ (to)
Hi
1.1.2. In the above formulation it is assumed that we take either the upper
signs or the lower signs throughout. In practice the choice must be made so
as to give numerical stability and this is achieved by taking the plus sign in
equation (6) if at, l_l>=O and the minus sign otherwise.
It is important to take advantage of the symmetry of both Ai+ ~ and A~
when evaluating Ai+ 1 from equation (1), and to this end we compute only the
elements of Ai+ 1 on and below the diagonal. A convenient way of organising
the computation is as follows. We write
~ A~ u~+ (11)
----- A ~ -- u i H~ H~ H~
This may be simplified by introducing vectors p~ and qi and a scalar K i defined by
then we have
p~ =A~ udH~ (t2)
Ki=u~pd(2H3 (t3)
qi=P~-- Ki u~ (t4)
Equation (t5) is used only to compute the elements of rows (/) to (l--t) of
A~+~ on and below the diagonal. The elements of rows (l+t) to (n) are the
same as in A i; in row l the (1, l) element is the same as in A i while the (l, l -- t)
element becomes T a~.
From the form of A~ and u i it is evident that Pi is of the form
so that q~ is given by
PT=- EP~,~; P~,~; ..." P~,~-~; P~,~; 0; ...; o~
q~-~ [Pi, x --Kiui,1; Pi,2 --Kiui,2; ... ; Pi, l-x- Kiui, l-1; P~,z; 0; ... ; 0~.
If a~ is zero then we may take P~=I and no transformation is needed. This
case is given special treatment. We could in fact take P~= I if
a~,x + at,~ + ... +a~,~_~=0
since the l-th row and column are then already in tridiagonal form, but this
has not been done. Note that in this case the transformation with matrix P~
merely changes the sign of the (l, l -- ~) element.