Chapter 01.02
Measuring Errors
After reading this chapter, you should be able to:
1.
find the true and relative true error,
2.
find the approximate and relative approximate error,
3.
relate the absolute relative approximate error to the number of significant digits
at least correct in your answers, and
4.
know the concept of significant digits.
In any numerical analysis, errors will arise during the calculations.
To be able to deal
with the issue of errors, we need to
(A)
identify where the error is coming from, followed by
(B)
quantifying the error, and lastly
(C)
minimize the error as per our needs.
In this chapter, we will concentrate on item (B), that is, how to quantify errors.
Q
: What is true error?
A
: True error denoted by
t
E
is the difference between the true value (also called the exact
value) and the approximate value.
True Error
=
True value – Approximate value
Example 1
The derivative of a function
)
(
x
f
at a particular value of
x
can be approximately calculated
by
h
x
f
h
x
f
x
f
)
(
)
(
)
(

+
≈
′
of
)
2
(
f
′
For
x
e
x
f
5
.
0
7
)
(
=
and
3
.
0
=
h
, find
a) the approximate value of
)
2
(
f
′
b) the true value of
)
2
(
f
′
c) the true error for part (a)
Solution
a)
h
x
f
h
x
f
x
f
)
(
)
(
)
(

+
≈
′
01.02.1
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Chapter 01.02
For
2
=
x
and
3
.
0
=
h
,
3
.
0
)
2
(
)
3
.
0
2
(
)
2
(
f
f
f

+
≈
′
3
.
0
)
2
(
)
3
.
2
(
f
f

=
3
.
0
7
7
)
2
(
5
.
0
)
3
.
2
(
5
.
0
e
e

=
3
.
0
028
.
19
107
.
22

=
265
.
10
=
b) The exact value of
)
2
(
f
′
can be calculated by using our knowledge of differential calculus.
x
e
x
f
5
.
0
7
)
(
=
x
e
x
f
5
.
0
5
.
0
7
)
(
'
×
×
=
x
e
5
.
0
5
.
3
=
So the true value of
)
2
(
'
f
is
)
2
(
5
.
0
5
.
3
)
2
(
'
e
f
=
5140
.
9
=
c) True error is calculated as
t
E
= True value – Approximate value
265
.
10
5140
.
9

=
75061
.
0

=
The magnitude of true error does not show how bad the error is.
A true error of
722
.
0

=
t
E
may seem to be small, but if the function given in the Example 1 were
,
10
7
)
(
5
.
0
6
x
e
x
f

×
=
the true error in calculating
)
2
(
f
′
with
,
3
.
0
=
h
would be
.
10
75061
.
0
6

×

=
t
E
This value of
true error is smaller, even when the two problems are similar in that they use the same value
of the function argument,
2
=
x
and the step size,
3
.
0
=
h
.
This brings us to the definition of
relative true error.
Q
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 Spring '07
 Gallivan
 Scientific Notation, Numerical Analysis, $100, Maclaurin, Hillsborough County, 0.069527%

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