Post on 14-Jul-2016
description
The Islamic University of Gaza
Faculty of Engineering
Civil Engineering Department
Numerical Analysis
ECIV 3306
Chapter 4
1
Truncation Errors and Taylor Series
Introduction
Truncation errors
• Result when approximations are used to represent exact mathematical procedure
• For example:
2
3
Taylor Series - Definition
• Mathematical Formulation used widely in numerical methods to express functions in an approximate fashion……. Taylor Series.
• It is of great value in the study of numerical methods.
• It provides means to predict a functional value at one point in terms of:
- the function value
- its derivatives at another point
Taylor’s Theorem
Where:
ii xxh 1
1)1(
)!1(
)(
n
n
n hn
fR
4
n
n
i
n
iiiii R
n
hxfhxfhxfhxfxfxf
!
)(.......
!3
)(
!2
)('')()()(
)(3)3(2'
1
General Expression
Rn is the remainder term to account
for all terms from n+1 to infinity.
And is a value of x that lies somewhere between xi and xi+1
Taylor’s Theorem
)()( 1 ii xfxf
!2
)('')()()(
2'
1
hxfhxfxfxf i
iii
5
Zero-order approximation: only true if xi+1 and xi are very close to each other
First-order approximation: in form of a straight line
Second-order approximation:
Any smooth function can be approximated as
a polynomial
hxfxfxf iii )()()( '
1
Taylor’s Theorem - Remainder Term
Remainder Term: What is ξ?
h
Rf o)(' oii Rxfxf )()1(
6
If Zero-order approximation:
Taylor Series - ExampleUse zero-order to fourth-order Taylor series expansions to approximate the function.
f(x)= -0.1x4 – 0.15x3 – 0.5x2 – 0.25x +1.2
From xi = 0 with h =1. Predict the function’s value at xi+1 =1.
Solution f(xi)= f(0)= 1.2 , f(xi+1)= f(1) = 0.2 ………exact solution
• Zero- order approx. (n=0) f(xi+1)=1.2
Et = 0.2 – 1.2 = -1.0
• First- order approx. (n=1) f(xi+1)= 0.95
)()1( ii xfxf
hxfxfxf iii )()()1( '
7
3 2 '
'( ) 0.4 - 0.45 - - 0.25 , (0) 0.25
( 1) 1.2 - 0.25 0.95
0.2 0.95 - 0.75
i
t
f x x x x f
f x h
E
Taylor Series - Example• Second- order approximation (n=2) f(xi+1)= 0.45
Third-order approximation (n=3) f(xi+1)= 0.3
!2
)('')()()1(
2' hxf
hxfxfxf iiii
2 (3) 3
' ''( ) ( )( 1) ( ) ( )
2! 3!
i i
i i i
f x h f x hf x f x f x h
8
2 ''
2
''( ) -1.2 - 0.9 -1 , (0) -1
( 1) 1.2 - 0.25 - 0.5 0.45
0.2 0.45 - 0.25t
f x x x f
f xi h h
E
3 3
2 3
( ) - 2.4 - 0.9 , (0) 0.9
( 1) 1.2 - 0.25 - 0.5 - 0.15 0.3
0.2 - 0.3 - 0.1t
f x x f
f xi h h h
E
Taylor Series - Example
• Fourth-order approximation (n = 4) f(xi+1)= 0.2
f( xi+1)= 1.2 - 0.25h - 0.5 h2 – 0.15h3 – 0.1h 4= 0.2
Et = 0.2 – 0.2 = 0
The remainder term (R4) = 0
because the fifth derivative of the fourth-order polynomial is zero.
!4
)(
!3
)(
!2
)('')()()1(
4)4(3)3(2' hxfhxfhxf
hxfxfxf iiiiii
5)5(
4!5
)(h
fR
9
10
Approximation using Taylor Series Expansion
The nth-order Approximation
Taylor Series
In General, the n-th order Taylor Series will be exact for n-th order polynomial.
For other differentiable and continuous functions,such as exponentials and sinusoids, a finite number ofterms will not yield an exact estimate. Each additionalterm will contribute some improvement.
Truncation error is decreased by addition of terms tothe Taylor series.
If h is sufficiently small, only a few terms may berequired to obtain an approximation close enough tothe actual value for practical purposes.
11
12
13
Example 2
14
Example 2
15
Example 2
16
Example 3
Example 3
17
x h one term two terms three terms four terms five terms six terms3.0 0.0 170 170.0 170.00 170.00 170.00 170.003.2 0.2 170 234.8 244.52 245.22 245.24 245.243.4 0.4 170 299.6 338.48 344.05 344.43 344.44
3.6 0.6 170 364.4 451.88 470.67 472.62 472.693.8 0.8 170 429.2 584.72 629.26 635.41 635.744.0 1.0 170 494.0 737.00 824.00 839.00 840.004.2 1.2 170 558.8 908.72 1059.06 1090.16 1092.654.4 1.4 170 623.6 1099.88 1338.61 1396.23 1401.614.6 1.6 170 688.4 1310.48 1666.83 1765.14 1775.62
4.8 1.8 170 753.2 1540.52 2047.90 2205.37 2224.265.0 2.0 170 818.0 1790.00 2486.00 2726.00 2758.005.2 2.2 170 882.8 2058.92 2985.30 3336.68 3388.225.4 2.4 170 947.6 2347.28 3549.97 4047.63 4127.265.6 2.6 170 1012.4 2655.08 4184.19 4869.66 4988.47
5.8 2.8 170 1077.2 2982.32 4892.14 5814.13 5986.236.0 3.0 170 1142.0 3329.00 5678.00 6893.00 7136.006.2 3.2 170 1206.8 3695.12 6545.94 8118.80 8454.346.4 3.4 170 1271.6 4080.68 7500.13 9504.63 9958.996.6 3.6 170 1336.4 4485.68 8544.75 11064.18 11668.846.8 3.8 170 1401.2 4910.12 9683.98 12811.69 13604.047.0 4.0 170 1466.0 5354.00 10922.00 14762.00 15786.00
18
Example 3