User Manual

Section 4: Using Matrix Operations 95
and
.
97.999,9
03.000,10







yX
T
However, when rounded to 10 digits,
,
1010
1010
1010
1010








XX
T
which is the same as what would be calculated if X were rounded to five significant digits to
the largest element:
.
00
00
00
000,100000,100













X
The HP-15C solves X
T
Xb = X
T
y (perturbing the singular matrix as described on page 99) and
gets







060000.0
060001.0
b
With
.
03.0
03.0






 XbXyX
TT
However, the correct least-squares solution is







4999995.0
5000005.0
b
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
j
of X of
as much as ||x
j
||/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.