Numerical Analysis - Advanced Topics in Root Finding - Hanyang University Jong-Il Park.
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Transcript of Numerical Analysis - Advanced Topics in Root Finding - Hanyang University Jong-Il Park.
Numerical AnalysisNumerical Analysis- Advanced Topics in - Advanced Topics in Root Root
Finding Finding --
Hanyang University
Jong-Il Park
Department of Computer Science and Engineering, Hanyang University
Summary: Root Finding Summary: Root Finding
Department of Computer Science and Engineering, Hanyang University
Error analysis of N-R methodError analysis of N-R method
Taylor series:
Newton-Raphson method:
At the true solution xr :
--- (1)
--- (2)(1) (2) :
Let
Quadraticconvergence!
Department of Computer Science and Engineering, Hanyang University
Error Analysis of Secant MethodError Analysis of Secant Method
Convergence [Jeeves, 1958]
More efficient than N-R method if the calculation of f’(x) is complex
Modified secant method
i
iiii x
xfxxfxf
)()(
)('
)('
)(1
i
iii xf
xfxx
)()(
)(1
iii
iiii xfxxf
xfxxx
Department of Computer Science and Engineering, Hanyang University
Multiple rootsMultiple roots
Bracketing methods cannot cope with multiple roots
Open methods can find multiple roots But the speed is slow in many cases f’(x)0 will cause a problem(divide by zero)
f(x) will always reach zero before f’(x) [Ralston and Rabinowitz, 1978]
if a zero check for f(x) is incorporated into the program, the computation can be terminated before f’(x) reaches zero
Alternative way using u(x)=f(x)/f’(x)
)('')()]('[
)(')(21
iii
iiii xfxfxf
xfxfxx
Department of Computer Science and Engineering, Hanyang University
Polynomial evaluationPolynomial evaluation
Bad method
Worst method
Best method
C code
Department of Computer Science and Engineering, Hanyang University
Polynomial differentiationPolynomial differentiation
or
Department of Computer Science and Engineering, Hanyang University
N-th derivativesN-th derivatives
Department of Computer Science and Engineering, Hanyang University
Polynomial deflationPolynomial deflation
Multiplication by (x-a)
Synthetic division by (x-a)
Department of Computer Science and Engineering, Hanyang University
Bairstow’s method Bairstow’s method
Deflation method
Find r and s such that b0=b1=0 Efficient routine using synthetic division exists Good initial guess of r, s is important Complex roots can be evaluated Suitable for root polishing
In Numerical Recipes in Cvoid qroot();
nnn xaxaaxf 10)(
012
322 )())(()( brxbxbxbbsrxxxf n
nn
Read Sect.7.5.
Department of Computer Science and Engineering, Hanyang University
Laguerre methodLaguerre method
Deflation method Algorithm derivation
In Numerical Recipes in C: zroots() calls laguer();
Department of Computer Science and Engineering, Hanyang University
Application of Root Finding:Application of Root Finding: Electric circuit design(1/2)Electric circuit design(1/2)
Problem: Find the proper R to dissipate energy to 1% at a specified rate(t=0.05s), given L=5H, C=10-4 F.
Solution:
0
2)2/(
2
1cos)(
q
qt
L
R
LCeRf LRt
Department of Computer Science and Engineering, Hanyang University
Application of Root Finding:Application of Root Finding: Electric circuit design(2/2)Electric circuit design(2/2)
01.005.001.02000cos)( 2005.0 ReRf R
• Reasonable initial range for R:0< R < 400
• To achieve r.e. of 10-4 %
Bisection method: 21 iterations Other methods: ? • To achieve r.e. of 10-6 %
Bisection method: ? iterations Other methods: ? [Homework]
Department of Computer Science and Engineering, Hanyang University
Homework #4Homework #4
01.005.001.02000cos)( 2005.0 ReRf R
Find the root of f(R)=0 and the number of iterations when the r.e.=10-4 and 10-6 respectively.
Solve the problems: 8-31, 32
Explain the concept of “pointer to function” and describe how you use it in your homework #3.
[Due: 10/17]