HP HP-15C Advanced Functions Handbook

116 Section 4: Using Matrix Operations
The second situation is rarely encountered and can be avoided. It
shows itself by causing at least one of the diagonal elements of U
to be much smaller than the largest element above it in the same
column, where U is from the orthogonal factorization wC = QU.
To avoid this situation, reorder the columns of wC and X and
similarly reorder the elements (rows) of b. The reordering can be
chosen easily if the troublesome diagonal element of U is also
much smaller than some subsequent element in its row. Just swap
the corresponding columns in the original data and refactor the
weighted constraint equations. Repeat this procedure if necessary.
For example, if the factorization of wC gives
1.0 2.0 0.5 -1.5 0.3
0 0.02 0.5 3.0 0.1
0 0 2.5 1.5 -1.2
then the second diagonal element is much smaller than the value
2.0 above it. This indicates that the first and second columns in the
original constraints are nearly dependent. The diagonal element is
also much smaller than the subsequent value 3.0 in its row. Then
the second and fourth columns should be swapped in the original
data and the factorization repeated.
It is always prudent to check for consistent constraints. The test for
small diagonal elements of U can be done at the same time.
Finally, using U and g as the first k rows, add rows corresponding
to X and y. (Refer to Least-Squares Using Successive Rows on
page 140 for additional information.) Then solve the unconstrained
least-squares problem with
and y •¥
wd
y
Provided the calculated solution b satisfies ||Cb — d|| < t, that
solution will also minimize ||y — Xb|| subject to the constraint
Cb~d.

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