Numerical Analysis
- Root Finding I-
Hanyang University
Jong-Il Park
Root Finding
F(t)
t
F(t)=0
 Analytic하게 근을 구할 수 없을 때 수치해법 필요
eg. F(t) = e-t(3cos2t + 4sin2t)
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General procedure of root finding
Step 1. 해의 근방을 찾는 과정




graphical method
incremental search
experience
solution of a simplified model
Step 2. 정확한 해를 찾는 과정


Bracketing method
Open method
Graphical method
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Bracketing method
 구간 [a,b]의 양끝점 사이에 근이
존재한다고 가정하고, 구간의 폭을
체계적으로 좁혀감
 정확한 근으로의 수렴성이 좋음
 종류
 bisection method
 linear interpolation method
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Open method
 임의의 시작점에서 시작
 규칙적 반복계산으로 근을 구함
 초기값에 따라 발산가능성 있음
 bracketing method보다 수렴속도 빠름
 종류
 fixed-point iteration
 Newton-Raphson method
 Secant method
 Muller method
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대략적 구간 찾기
 Graphical method
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대략적 구간 찾기
Incremental search method
 초기값 x와 dx 사이의 함수값의 부호 조사
 변화가 생기면 그 구간 안에 해가 존재
 난점:
 dx의 크기
• 작으면 긴 계산시간, 크면 해를 놓침

중근일 경우
• 해를 놓칠 가능성 큼
 양끝점의 도함수를 계산
if 서로 다른 부호 => 극소점 존재
 정확한 해를 찾기 위해 반드시 프로그램의 앞부분에
있어야 함
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Bisection method(I)
=Half Interval method = Bolzano method
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Bisection method(II)
- 한번 반복할 때마다 구간이 1/2로 감소
- 간단하고, 오차범위의 확인이 용이
n 번 반복시,
최대오차 < (초기구간간격/2n)
- 수렴속도가 느린 편임
- 다중근의 경우에 부호변화가 없으므로
적절치 못함
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Linear interpolation method(I)
= False position method
 Principle
Algorithm
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Linear interpolation method(II)
Algorithm
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Convergence speed
 Bisection보다 대체로
빠른 수렴성
 수렴 속도가 곡률에
의존(경우에 따라 매우
느린 수렴)
 Modified linear
interpolation method
제안됨
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Modified linear interpolation
0.5f(x2 )
x2
Slow
convergence
x2
Faster
convergence
Modified
Linear
Interpolation
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Open methods
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Newton-Raphson method(I)
 원리
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Newton-Raphson method(II)
 Fast convergence – quadratic convergence
 도함수 계산이 복잡할 때는 비효율적
 Initial guess가 중요함
 Troubles
 Cycling
 Wandering
 Overshooting
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Secant method(I)
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Secant method(II)
 Linear interpolation과 비슷한 복잡성
 update rule만 다름
 Linear interpolation 보다 빠른 수렴성
 Newton-Raphson method 보다 우수한 안정성
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Comparison: Convergence speed
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Fixed point iteration(I)
 원리
f(x)=0  x=g(x)

Iteration rule
xk+1=g(xk)
 Convergent
if contraction mapping
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Fixed point iteration(II)
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Muller method(I)
 Generalization of the Secant method
 Fast convergence
Secant
Muller
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Muller method(II)
 Algorithm
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Error analysis
 Linear convergence
 Quadratic convergence
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Accelerating convergence
 Aitken’s D2 method
Rapid convergence!
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Fail-safe methods
 Combination of Newton and Bisection
 if(out of bound || small reduction of bracket)
update by Bisection method
 rtsafe.c in NR in C
 Ridder's method (good performance)
 zriddr.c in NR in C
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Homework #3 (I)
[Due: 9/26]
 Programming: Find the roots of the Bessel function Jo
in the interval [1.0, 10.0] using the following methods:
a) Bisection (rtbis.c)
b) Linear interpolation (rtflsp.c)
c) Secant (rtsec.c)
d) Newton-Raphson (rtnewt.c)
e) Newton with bracketing (rtsafe.c)
 Hint
 First, use bracketing routine (zbrak.c).
 Then, call proper routines in NR in C.
-6 x
 Set xacc=10
 Use “pointer to function” to write succinct source
codes
(Cont’)
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Homework #3 (II)
 Write source codes for Muller method (muller.c) and
do the same job as above.
 Discuss the convergence speed of the methods.
 Solve the following problems using the routine of
rtsafe.c in NR in C
 10 e-x sin(2p x)-2= 0, on [0.1,1]


x2 - 2xe-x + e-2x = 0, on [0,1]
cos(x+sqrt(2))+x(x/2+sqrt(2)) = 0, on [-2,-1]
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