Orthogonal Factorization - HP -15C Advanced Functions Handbook

Section 4: Using Matrix Operations 113
with
0.03
0.03
However, the correct least-squares solution is
b =
0.5000005
0.4999995
despite the fact that the calculated solution and the exact solution
satisfy the computed normal equations equally well.
The normal equations should be used only when the elements of X
are all small integers (say between -3000 and 3000) or when you
know that no perturbations in the columns x
;
of X of as much as
||x/||/10
4
could make those columns linearly dependent.
Orthogonal Factorization
The following orthogonal factorization method solves the least-
squares problem and is less sensitive to rounding errors than the
normal equation method. You might use this method when the
normal equations aren't appropriate.
Any n X p matrix X can be factored as X — Q
T
U, where Q is an
n X n orthogonal matrix characterized by Q
r
= Q"
1
and U is an
n X p upper-triangular matrix. The essential property of
orthogonal matrices is that they preserve length in the sense that
||Qr||| = (Qr)
r
(Qr)
= r
T
Q
T
Qr
T
— rr
rll
— IU|2
\F
Therefore, if r = y — Xb, it has the same length as
Qr = Qy - QXb = Qy - Ub.