Numerical Analysis
- Advanced Topics in Root
Finding -
Hanyang University
Jong-Il Park
Summary: Root Finding
Department of Computer Science and Engineering, Hanyang University
Error analysis of N-R method
Taylor series:
Newton-Raphson method:
--- (1)
At the true solution xr :
--- (2)
(1)  (2) :
Let
Quadratic
convergence!
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Error Analysis of Secant Method
 Convergence [Jeeves, 1958]

More efficient than N-R method if the calculation of f’(x)
is complex
 Modified secant method
f ' ( xi ) 
x i 1  x i 
f ( xi )
f ' ( xi )
f ( xi  xi )  f ( xi )
 xi
x i 1  x i 
 xi f ( xi )
f ( xi   xi )  f ( xi )
Department of Computer Science and Engineering, Hanyang University
Multiple 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)

x i 1  x i 
f ( xi ) f ' ( xi )
[ f ' ( x i )]  f ( x i ) f ' ' ( x i )
2
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Polynomial evaluation
 Bad method
 Worst method
 Best method
C code
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Polynomial differentiation
or
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N-th derivatives
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Polynomial deflation
 Multiplication by (x-a)
 Synthetic division by (x-a)
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Bairstow’s method
 Deflation method
f n ( x )  a 0  a1 x    a n x
n
f n ( x )  ( x  rx  s )( b 2  b3 x    b n x
2





n2
)  b1 ( x  r )  b0
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 C
void qroot();
 Read Sect.7.5.
Department of Computer Science and Engineering, Hanyang University
Laguerre 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:
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:
f (R)  e
 Rt /( 2 L )

cos 


2 
R
q



 t 
LC  2 L   q 0

1
Department of Computer Science and Engineering, Hanyang University
Application of Root Finding:
Electric circuit design(2/2)
f (R)  e
 0 . 005 R
cos


2000  0 . 01 R 0 . 05  0 . 01
2
• 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 #4
[Due: 10/17]
 Find the root of f(R)=0 and the number of iterations
when the r.e.=10-4 and 10-6 respectively.
f (R)  e
 0 . 005 R
cos


2000  0 . 01 R 0 . 05  0 . 01
2
 Solve the problems: 8-31, 32
 Explain the concept of “pointer to function” and
describe how you use it in your homework #3.
Department of Computer Science and Engineering, Hanyang University