Analisa Numerik Sistem Persamaan Non-Linear 1

2 Pendahuluan Diberikan f(x) = 0, terdefinisi pada suatu interval [a, b]. Cari x* sedemikian shg. f(x*) 0.  Metoda Bisection  Metoda Regula Falsi (RF) Modifikasi RF Bentuk umum Iterasi-titik tetap (fixed point) Perbaikan Secant Newton

3 Metoda Bisection Algoritma 3.1. Metoda Bisection : f (x) kontinu pada [a 0, b 0 ] dan f(a 0 ) f(b 0 ) <= 0 for n = 0, 1, 2,..., until satisfied do m = (a n + b n ) / 2 if f(a n )f(m) ≤ 0, a n+1 = a n, b n+1 = m else a n+1 = m, b n+1 = b n end for f(x) punya x* pada [a n+1, b n+1 ]

4 Metoda Regula Falsi Bisection lambat dlm. mendekati x*, mk. titik m di bisection ditentukan dng. memakai bobot rata-rata w = [f(b n )a n – f(a n )b n ] / [f(b n ) – f(a n )] metoda ini disebut regula falsi Algoritma 3.2. Metoda Regula Falsi : f(x) kontinu pd. [a0, b0] dan f(a0)f(b0) < 0 for n = 0, 1, 2,... until satisfied do w = [f(b n )a n – f(a n )b n ] / [f(b n ) – f(a n )] if f(an)f(w) ≤ 0, an+1 = an, bn+1 = w else an+1 = w, bn+1 = bn end for

5 Metoda Regula Falsi

6 Modifikasi Regula Falsi Algoritma 3.3. Modifikasi Regula Falsi f(x) kontinu pada [a 0, b 0 ], f(a 0 )f(b 0 ) < 0 F = f(a 0 ), G = f(b 0 ), w 0 = a 0 for n = 0, 1, 2,..., until satisfied, do : w n+1 = (Ga n – Fb n ) / (G – F) if f(a n )f(w n+1 ) ≤ 0, a n+1 = a n, b n+1 = w n+1, G = f(w n+1 ) if f(an)f(wn+1) ≥ 0, F = F/2 else an+1 = wn+1, F = f(wn+1), bn+1 = bn if f(wn)f(wn+1) ≥ 0, G = G/2 end if end for f(x) punya x* pada [a n+1, b n+1 ]

7 Modifikasi Regula Falsi