MATLAB and Numerical Analysisbml.pusan.ac.kr/LectureFrame/Lecture/Undergraduates... · 2018-04-04 · MATLAB and Numerical Analysis. Radiological Imaging Sciences. Graph plot. Y

MATLAB and Numerical Analysis

Radiological Imaging SciencesSchool of Mechanical Engineering

Pusan National University

[email protected]


Radiological Imaging Sciences

Review

무명함수

>> fun = @(x,y) x^2 + y^2; % 𝑓𝑓 𝑥𝑥, 𝑦𝑦 = 𝑥𝑥2 + 𝑦𝑦2

>> fun(3,4)

>> ans

= 25

If문

>> if a == b

>> c=d;

>> end

만약 a=b 라면, c=d로계산하시오

for문

>> for i = 1:5

>> x(i+1)=x(i)+1;

>> end

순서도와동일

시작

x=x+1

i=5

출력



Review

nlinfit

>> fun = @(p,x) I(1)*exp(-p*x); 𝑓𝑓 𝑡𝑡 = 𝐼𝐼 0 × 𝑒𝑒−𝜇𝜇𝑡𝑡

>> t = [0:2:10];

>> I = [0.9726 0.8001 0.6582 0.5415 0.4455 0.3665];

>> b0 = 1;

>> mu = nlinfit(t,I,fun,b0);

>> figure,plot(t,I,’ob’); hold on;

>> plot(t,fun(mu,t),’r’);



Bisection

2ul

rxx

x+

=

추정된근의값



Bisection

다음의방법들을사용하여 𝑓𝑓 𝑥𝑥 = 𝑒𝑒−𝑥𝑥 − 𝑥𝑥의근을구하고, 참백분율상대오차를하나의그래프에함께그려비교하시오. 반복횟수에따른오차의특성을살피고, 각방법들간의차이점에대해논하시오.

참값으로 0.56714329를사용.

𝑓𝑓 𝑥𝑥 = 𝑒𝑒−𝑥𝑥 − 𝑥𝑥

𝑥𝑥𝑙𝑙 = 0, 𝑥𝑥𝑢𝑢 = 1

𝑦𝑦𝑙𝑙 = 𝑓𝑓(𝑥𝑥𝑙𝑙),𝑦𝑦𝑢𝑢 = 𝑓𝑓(𝑥𝑥𝑢𝑢)

함수및초기치설정 𝑥𝑥𝑥𝑥𝑖𝑖 =𝑥𝑥𝑥𝑥𝑖𝑖 + 𝑥𝑥𝑥𝑥𝑖𝑖

2

iteration

𝑦𝑦𝑥𝑥𝑖𝑖 = 𝑓𝑓(𝑥𝑥𝑥𝑥𝑖𝑖)

만약 𝑦𝑦𝑥𝑥𝑖𝑖 × 𝑦𝑦𝑥𝑥𝑖𝑖 < 0이면,

𝑥𝑥𝑥𝑥𝑖𝑖+1 = 𝑥𝑥𝑥𝑥𝑖𝑖만약 𝑦𝑦𝑥𝑥𝑖𝑖 × 𝑦𝑦𝑥𝑥𝑖𝑖 > 0이면,

𝑥𝑥𝑥𝑥𝑖𝑖+1 = 𝑥𝑥𝑥𝑥𝑖𝑖



𝑒𝑒𝑡𝑡𝑖𝑖 = 100 ×𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡 − 𝑥𝑥𝑥𝑥𝑖𝑖𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡

𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡 = 0.56714329최종결과

𝑥𝑥𝑥𝑥𝑡𝑡𝑒𝑒𝑒𝑒, 𝑒𝑒𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 획득



Bisection coding





False position


참값으로 0.56714329를사용.


𝑥𝑥𝑙𝑙 = 0, 𝑥𝑥𝑢𝑢 = 1

𝑦𝑦𝑙𝑙 = 𝑓𝑓(𝑥𝑥𝑙𝑙),𝑦𝑦𝑢𝑢 = 𝑓𝑓(𝑥𝑥𝑢𝑢)

함수및초기치설정 𝑥𝑥𝑥𝑥𝑖𝑖 = 𝑥𝑥𝑥𝑥𝑖𝑖 − 𝑦𝑦𝑥𝑥𝑖𝑖 ×𝑥𝑥𝑥𝑥𝑖𝑖 − 𝑥𝑥𝑥𝑥𝑖𝑖𝑦𝑦𝑥𝑥𝑖𝑖 − 𝑦𝑦𝑥𝑥𝑖𝑖

iteration

𝑦𝑦𝑥𝑥𝑖𝑖 = 𝑓𝑓(𝑥𝑥𝑥𝑥𝑖𝑖)

만약 𝑦𝑦𝑥𝑥𝑖𝑖 × 𝑦𝑦𝑥𝑥𝑖𝑖 < 0이면,

𝑥𝑥𝑥𝑥𝑖𝑖+1 = 𝑥𝑥𝑥𝑥𝑖𝑖만약 𝑦𝑦𝑥𝑥𝑖𝑖 × 𝑦𝑦𝑥𝑥𝑖𝑖 > 0이면,




𝑒𝑒𝑡𝑡𝑖𝑖 = 100 ×𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡 − 𝑥𝑥𝑥𝑥𝑖𝑖𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡


𝑥𝑥𝑥𝑥𝑡𝑡𝑒𝑒𝑒𝑒, 𝑒𝑒𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 획득



Newton-Raphson

)(')(

1i

iii xf

xfxx −=+

추정된근의값

1

0)()('+−−

=ii

ii xx

xfxf



Newton-Raphson


참값으로 0.56714329를사용.


𝑥𝑥 = 0

𝑦𝑦 = 𝑓𝑓(𝑥𝑥)

함수및초기치설정 𝑥𝑥𝑖𝑖+1 = 𝑥𝑥𝑖𝑖 −𝑦𝑦𝑖𝑖𝑦𝑦𝑦1

iteration

𝑦𝑦𝑖𝑖+1 = 𝑓𝑓(𝑥𝑥𝑖𝑖+1)

𝑒𝑒𝑡𝑡𝑖𝑖 = 100 ×𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡 − 𝑥𝑥𝑖𝑖+1

𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡


𝑥𝑥𝑡𝑡𝑒𝑒𝑒𝑒, 𝑒𝑒𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 획득𝑓𝑓′(𝑥𝑥) = −𝑒𝑒−𝑥𝑥 − 1

𝑦𝑦𝑦 = 𝑓𝑓𝑦(𝑥𝑥)

𝑒𝑒𝑡𝑡1 = 100 ×𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡 − 𝑥𝑥1𝑥𝑥𝑡𝑡𝑡𝑡𝑢𝑢𝑡𝑡

𝑦𝑦𝑦𝑖𝑖+1 = 𝑓𝑓𝑦(𝑥𝑥𝑖𝑖+1)



Newton-Raphson coding





Fixed-point iteration


참값으로 0.56714329를사용.


𝑥𝑥 = 0

𝑦𝑦 = 𝑔𝑔(𝑥𝑥)

함수및초기치설정 𝑥𝑥𝑖𝑖+1 = 𝑦𝑦𝑖𝑖

iteration

𝑦𝑦𝑖𝑖+1 = 𝑔𝑔(𝑥𝑥𝑖𝑖+1)




𝑥𝑥𝑡𝑡𝑒𝑒𝑒𝑒, 𝑒𝑒𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 획득𝑔𝑔(𝑥𝑥) = 𝑒𝑒−𝑥𝑥




Secant


참값으로 0.56714329를사용.


𝑥𝑥1 = 1𝑦𝑦𝑦 = 𝑓𝑓(𝑥𝑥1)

함수및초기치설정 𝑥𝑥𝑖𝑖+1 = 𝑥𝑥𝑖𝑖 −𝛿𝛿 × 𝑥𝑥𝑖𝑖 × 𝑦𝑦𝑦𝑖𝑖𝑦𝑦𝑦𝑖𝑖 − 𝑦𝑦𝑦𝑖𝑖

iteration

𝑦𝑦𝑦𝑖𝑖+1 = 𝑓𝑓(𝑥𝑥𝑖𝑖+1)




𝑥𝑥𝑡𝑡𝑒𝑒𝑒𝑒, 𝑒𝑒𝑡𝑡𝑡𝑡𝑒𝑒𝑒𝑒 획득


𝑦𝑦𝑦 = 𝑓𝑓 𝑥𝑥1 + 𝛿𝛿𝑥𝑥1

𝛿𝛿 = 0.000001𝑦𝑦𝑦𝑖𝑖+1 = 𝑓𝑓 𝑥𝑥𝑖𝑖+1 + 𝛿𝛿 × 𝑥𝑥𝑖𝑖+1



Structure & Save

Mat 파일로저장

Structure



Graph plot

1) Load

2) Plotting



Graph plot

3) Axis control



Graph plot

4) Labeling



Graph plot

5) Legend

6) save



Graph plot

Y축 log-scale로 plot 그래프선굵기한그래프에여러 line plot

Line의형태및색Legend (범례)

격자x & y 축범위설정 x 축범위설정및해당값설정

x 축설명y 축설명

타이틀

저장

Mat파일불러오기

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Documents

MATLAB and Numerical Analysisbml.pusan.ac.kr/LectureFrame/Lecture/Undergraduates... · 2018-04-04 · MATLAB and Numerical Analysis. Radiological Imaging Sciences. Graph plot. Y