Upload
oderamadhani
View
220
Download
0
Embed Size (px)
Citation preview
7/25/2019 Ch 4 NonLinearEquations
1/10
PPS-UB-PAT-TM-2010 Chapter 6 1
Simple Fixed-point Iteration
...2,1,k,given)()(0)(
1
okk xxgxxxgxf
Bracketing methods are convergent.
Fixed-point methods may sometimediverge, depending on the stating point(initial guess) and how the function
behaves.
Rearrange the function so that x is on theleft side of the equation:
7/25/2019 Ch 4 NonLinearEquations
2/10
PPS-UB-PAT-TM-2010 Chapter 6 2
xxg
or
xxgor
xxg
xxxxf
21)(
2)(
2)(
02)(
2
2
Example:
7/25/2019 Ch 4 NonLinearEquations
3/10
PPS-UB-PAT-TM-2010 Chapter 6 3
Convergence
x=g(x) can be expressed
as a pair of equations:
y1=x
y2=g(x) (component
equations)
Plot them separately.
Figure 6.2
7/25/2019 Ch 4 NonLinearEquations
4/10
PPS-UB-PAT-TM-2010 Chapter 6 4
Conclusion Fixed-point iteration converges if
x)f(x)linetheof(slope1)( xg
When the method converges, the error isroughly proportional to or less than the error
of the previous step, therefore it is calledlinearly convergent.
7/25/2019 Ch 4 NonLinearEquations
5/10
PPS-UB-PAT-TM-2010 Chapter 6 5
Newton-Raphson Method Most widely used method.
Based on Taylor series expansion:
)(
)(
)(0
g,Rearrangin
0)f(xwhenxofvaluetheisrootThe
!2)()()()(
1
1
1i1i
3
2
1
i
iii
iiii
iiii
xf
xfxx
xx)(xf)f(x
xOx
xfxxfxfxf
Newton-Raphson formula
Solve for
7/25/2019 Ch 4 NonLinearEquations
6/10
PPS-UB-PAT-TM-2010 Chapter 6 6
A convenient method for
functions whose
derivatives can be
evaluated analytically. It
may not be convenientfor functions whose
derivatives cannot be
evaluated analytically.
Fig. 6.5
7/25/2019 Ch 4 NonLinearEquations
7/10PPS-UB-PAT-TM-2010 Chapter 6 7
Fig. 6.6
7/25/2019 Ch 4 NonLinearEquations
8/10PPS-UB-PAT-TM-2010 Chapter 6 8
The Secant Method
A slight variation of Newtons method for functionswhose derivatives are difficult to evaluate. For thesecases the derivative can be approximated by a
backward finite divided difference.
,3,2,1)()(
)(
)()()(
1
1
1
1
1
1
ixfxf
xxxfxx
xfxf
xx
xf
ii
ii
iii
ii
ii
i
7/25/2019 Ch 4 NonLinearEquations
9/10PPS-UB-PAT-TM-2010 Chapter 6 9
Requires two initial
estimates of x , e.g, xo,
x1. However, because f(x)is not required to change
signs between estimates,
it is not classified as a
bracketing method.
The secant method has
the same properties as
Newtons method.
Convergence is not
guaranteed for all xo, f(x).
Fig. 6.7
7/25/2019 Ch 4 NonLinearEquations
10/10PPS-UB-PAT-TM-2010 Chapter 6 10
Fig. 6.8