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Transcript of The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical...
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
ni
nii
iiiR
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
hxfxfxf iii)()()( '
1
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
Taylor’s Theorem - Remainder Term
Remainder Term: What is ξ ?
h
Rf o)(' oii
Rxfxf
)()(1
6
If Zero- order approximation:
Taylor Series - Example Use 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 f(x)= -0.4x3 – 0.45x2 – x – 0.25, f ’(0)= -0.25 f( xi+1)= 1.2- 0.25h = 0.95 Et = 0.2 - 0.95 = -0.75
)()(1 ii
xfxf
hxfxfxf iii)()()( '
1
7
Taylor Series - Example• Second- order approximation (n=2) f(xi+1)= 0.45
f ’’(x) = -1.2 x2 – 0.9x -1 , f ’’(0)= -1
f( xi+1)= 1.2 - 0.25h - 0.5 h2 = 0.45
Et = 0.2 – 0.45 = -0.25
• Third-order approximation (n=3) f(xi+1)= 0.3
f( xi+1)= 1.2 - 0.25h - 0.5 h2 – 0.15h3 = 0.3
Et = 0.2 – 0.3 = -0.1
!2
)('')()()(
2'
1
hxfhxfxfxf i
iii
!3
)(
!2
)('')()()(
3(3)2'
1
hxfhxfhxfxfxf ii
iii
8
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
)('')()()(
4)4(3)3(2'
1
hxfhxfhxfhxfxfxf iii
iii
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 of terms will not yield an exact estimate. Each additional term will contribute some improvement.
(see example 4.2)
11
Taylor Series
• Truncation error is decreased by addition of terms to the Taylor series.
• If h is sufficiently small, only a few terms may be required to obtain an approximation close enough to the actual value for practical purposes.
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