HP 15c - Manual (Page 97)

Section 4: Using Matrix Operations 97
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.)
2. Solve the following system for b:
.
1
ˆ





















q q
0 b
0
g U
(If q = 0, replace it by any small nonzero number, say 10
−99 .) The −1 in the solution
matrix automatically appears; it requires no additional calculations.
In the absence of rounding errors, q = ±||y – Xb||
F
; this may be inaccurate if |q| is too
small, say smaller than ||y||/l0
6
. If you desire a more accurate estimate of ||y – Xb||
F
,
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||/10
10
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:

No b 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
) row 1 (
) rows (
1 ) diag( ) rows 1 (
) rows ( p
q q k p
k





















 
0 b
0
g U
using any small nonzero number q. If the computed solution b satisfies Cb ≈ d, then the
constraints are not inconsistent.
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