HP HP-15C - Page 117

Section 4: Using Matrix Operations
2. Solve the following system for b:
115
U g
0 q
0
-Q
(If q = 0, replace it by any small nonzero number, say 10 ".)
The -1 in the solution matrix automatically appears; it
requires no additional calculations.
In the absence of rounding errors, q = ±||y — Xb||^-; this may
be inaccurate if \q\s too small, say smaller than ||y||/106. If
you desire a more accurate estimate of ||y — Xb||^, you can
calculate it directly from X, y, and the computed solution b.
For the weighted least-squares problem, replace X and y by WX
and Wy, where W is the diagonal matrix containing the weights.
For the linearly constrained least-squares problem, you must
recognize that constraints may be inconsistent. In addition, they
can't always be satisfied exactly by a calculated solution because
of rounding errors. Therefore, you must specify a tolerance t such
that the constraints are said to be satisfied when ||Cb — d|| < t.
Certainly t > ||d||/1010 for 10-digit computation, and in some cases
a much larger tolerance must be used.
Having chosen t, select a weight factor w that satisfies w > \\y\\/t.
For convenience, choose w to be a power of 10 somewhat bigger
than ||y||/t. Then w\\Cb - d|| > ||y|| unless ||Cb - d|| < t.
However, the constraint may fail to be satisfied for one of two
reasons:
• Nob exists for which || Cb —d|| < t.
• The leading columns of C are nearly linearly dependent.
Check for the first situation by determining whether a solution
exists for the constraints alone. When [wC wd] has been factored
to Q[U g], solve this system for b
(k rows) U g
(p 4-1 — & rows) I 0 diag(q)
0
-Q
(p rows)
(1 row)
using any small nonzero number q. If the computed solution b
satisfies Cb ~ d, then the constraints are not inconsistent.