Appendix: Accuracy Of Numerical Calculations; Misconceptions About Errors - HP -15C Advanced Functions Handbook

Appendix
Accuracy of
Numerical Calculations
Misconceptions About Errors
Error is not sin, nor is it always a mistake. Numerical error is
merely the difference between what you wish to calculate and what
you get. The difference matters only if it is too big. Usually it is
negligible; but sometimes error is distressingly big, hard to
explain, and harder to correct. This appendix focuses on errors,
especially those that might be large—however rare. Here are some
examples.
Example 1: A Broken Calculator. Since ( \ f x )
2
= x whenever
, we expect also
50 50
roots squares
should equal x too.
A program of 100 steps can evaluate the expression f ( x ) for any
positive x. When x = 10 the HP-15C calculates 1 instead. The error
10 — 1 = 9 appears enormous considering that only 100 arithmetic
operations were performed, each one presumably correct to 10
digits. What the program actually delivers instead of f ( x ) = x turns
out to be
1
1 forx^l
0
which seems very wrong. Should this calculator be repaired?
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