Is Backward Error Analysis A Good Idea - HP -15C Advanced Functions Handbook

204 Appendix: Accuracy of Numerical Calculations
Of course, none of these horrible things could happen if X were not
so nearly singular. Because ||X|| > 10
10
, a change in X
amounting to less than one unit in the 10th significant digit of ||X||
could make X singular; such a change might replace one of the
diagonal elements 0.00002 of X by zero. Since X is so nearly
singular, the accuracy of 1 1/x|(X) in this case rather exceeds what
might be expected in general. What makes this example special is
bad scaling; X was obtained from an unexceptional matrix
X =
2. -5. 5.000003 -4.5X10'
12
0 5. -5.000003 4.5 X10"
12
0 0 2 . -5.000003
0 0 0 5 . 2
by multiplying each row and each column by a carefully chosen
power of 10. Compensatory division of the columns and rows of the
equally unexceptional matrix
X
0.5 0.5 p q
0 0.2 0.5000003 0.4807698077.
0 0 0.5 0.4807695192.
0 0 0 0.1923076923.
yielded X"
1
, withp = q = 0. The HP-15C calculates Q/x](X) = X"
1
except that q = 0 is replaced by q — 9.6 X 10~
n
, a negligible change.
This illustrates how drastically the perceived quality of computed
results can be altered by scaling. (Refer to section 4 for more
information about scaling.)
Is Backward Error Analysis a Good Idea?
The only good thing to be said for backward error analysis is that it
explains internal errors in a way that liberates a system's user
from having to know about internal details of the system. Given
two tolerances, one upon the input noise 8x and one upon the
output noise 8f, the user can analyze the consequences of internal
noise in
= (f+8f)(x
by studying the noise propagation properties of the ideal system /
without further reference to the possibly complex internal structure
of 1?.