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Lecture 6II While Loop

Lecture 6II While Loop. While loop Root Finding GUI tool Design. Inline : f(x) = sin(x). fs = input('f(x) = ','s'); f = inline(fs);. >> f(pi/2) ans = 1. Roots. Mathematics x is a root of f(x) if f(x) = 0 Numerical analysis x is a root of f(x) if abs(f(x)) < eps. >> sin(pi)

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Lecture 6II While Loop

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  1. Lecture 6II While Loop • While loop • RootFinding • GUI tool Design 軟體實作與計算實驗

  2. Inline : f(x) = sin(x) fs = input('f(x) = ','s'); f = inline(fs); >> f(pi/2) ans = 1 軟體實作與計算實驗

  3. Roots • Mathematics • x is a root of f(x) if f(x) = 0 • Numerical analysis • x is a root of f(x) if abs(f(x)) < eps >> sin(pi) ans = 1.2246e-016 >> abs(sin(pi)) < eps ans = 1 軟體實作與計算實驗

  4. Existence of roots • f(x) is well defined over interval [a,b] • If f(x) is continuous and f(a)f(b) < 0 there exists at least one root within [a,b] f(b) a b f(a) 軟體實作與計算實驗

  5. Bipartition • Partition interval [a,b] to two intervals such that [a,b]=[a,c] U [c,b], where c = (a+b) / 2 c = (a+b)/2 a b 軟體實作與計算實驗

  6. Proposition If f(a)f(b) < 0 it holds that f(a)f(c) < 0 or f(c)f(b) < 0 c = (a+b)/2 a b 軟體實作與計算實驗

  7. Proof f(a)f(b) < 0, f(c) ≠ 0 (I) f(a) > 0 and f(b) < 0 f(c)f(a) < 0 or f(c)f(b) < 0 (II) f(a) < 0 and f(b) > 0 f(c)f(a) < 0 or f(c)f(b) < 0 軟體實作與計算實驗

  8. Target interval If f(a)f(b) < 0 it holds that f(a)f(c) < 0 or f(c)f(b) < 0 If f(a)f(c) < 0, choose [a c] as target interval If f(c)f(b) < 0, choose [c b] as target interval c = (a+b)/2 a b 軟體實作與計算實驗

  9. Binary search • Target interval [t1 t2] • If halting condition holds, halt • a ← t1, b ← t2, c ← (t1+t2)/2 • If f(a)f(c) < 0, t1 ← a, t2 ← c choose [a c] • Iff(c)f(b) < 0, t1 ← c, t2 ← b choose [c b] 軟體實作與計算實驗

  10. Halting condition • c=(t1+t2)/2 • f(c) is close enough to zero • Implementation • abs(f(c)) < eps 軟體實作與計算實驗

  11. Zero finding Input t1,t2,f c=(t1+t2)/2 • Flow Chart: abs(f(c)) > eps T Return c a=t1;b=t2;c=(a+b)/2 if f(a)*f(c) < 0 t1=a;t2=c; else t1=c;t2=b; end c=(t1+t2)/2; 軟體實作與計算實驗

  12. my_fzero.m 軟體實作與計算實驗

  13. New 軟體實作與計算實驗

  14. Edit Text 軟體實作與計算實驗

  15. Addition 軟體實作與計算實驗

  16. 軟體實作與計算實驗

  17. 軟體實作與計算實驗

  18. function pushbutton1_Callback(hObject, eventdata, handles) % hObject handle to pushbutton1 (see GCBO) % eventdata reserved - to be defined in a future version of MATLAB % handles structure with handles and user data (see GUIDATA) s1=get(handles.edit1,'String'); s2=get(handles.edit2,'String'); x=str2double(s1)+str2double(s2); s3=num2str(x); set(handles.edit3,'String',s3); return 軟體實作與計算實驗

  19. 軟體實作與計算實驗

  20. GUI tool Add.fig Add.m 軟體實作與計算實驗

  21. ADD16 A toolbox for addition of hexadecimal numbers 軟體實作與計算實驗

  22. 軟體實作與計算實驗

  23. Symbolic integration • A toolbox for demonstrating symbolic integration 軟體實作與計算實驗

  24. 軟體實作與計算實驗

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