2-1
CHAPTER 2
NUMERICAL ERRORS
All numerical calculations are prone to numerical errors. This
chapter examines two important sources of error:
round-off
error
and
truncation error
. Round-off errors are related to the
discrete representation of numbers in computers. Truncation
errors are related to the numerical algorithms and formulae
used in the calculations.
I. Error definitions
A. Absolute vs. Relative
Numerical methods are all approximation methods. They
give approximated values rather than true values. The
absolute error is defined as:
Value
True
Value
ed
Approximat
Error
Absolute
−
=
However, it is more convenient to use the relative error:
Value
ed
Approximat
Value
True
Value
ed
Approximat
Value
True
Value
True
Value
ed
Approximat
Error
Relative
−
≈
−
=
The second expression is used when the true value is
unknown, which is typically the case.
EE 3108 Semster B 2007/2008
S C Chan
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Example:
What is the relative error in using 0.33 to
approximate
3
1
?
Solution:
The relative error is:
01
.
0
3
1
3
1
33
.
0
−
=
−
.
We can also express the relative error in percentage as:
%
%=
×
.
1
100
01
0
−
−
.
B. Accuracy vs. Precision
When experiments or calculations are performed repeatedly,
the concepts of accuracy and precision are important:
•
Accuracy
measures
the
proximity
between
the
calculated values
and the
true value
.
•
Precision
measures the consistency among
repeated
calculations
.
The concepts are illustrated in the following diagram.

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