HP HP-15C - Page 116

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Section 4: Using Matrix Operations
The upper-triangular matrix U and the product Qy can be written
as
u=
(p rows)
(n —prows)
and Qy =
(p rows)
(n— p rows)
Then
UHlQrllJ
HiQy-Ub||£
> Ml
with equality when g — Ub = 0. In other words, the solution to the
ordinary least-squares problem is any solution to Ub — g and the
minimal sum of squares is \\f\\p. This is the basis of all numerically
sound least-squares programs.
You can solve the unconstrained least-squares problem in two
steps:
1. Perform the orthogonal factorization of the augmented
n X (p + 1) matrix
= QTV
where Q = Q l, and retain only the upper-triangular factor
V, which you can then partition as
U g
0 q
0 0
(p rows)
(1 row)
(n — p — 1 rows)
A T. (1 column)
1 (p columns)
Only the first p + 1 rows (and columns) of V need to be
retained. (Note that Q here is not the same as that mentioned
earlier, since this Q must also transform y.)