Algoritma Metode Bagi dua / Bisection

Algoritma Metode Bagi dua / Bisection

1. Untuk n = 0, 1, 2 …… sampai selesai ß kriteria pemutusan

2. ambil m = (a_n + b_n )/2    iterasi (bil. kecil tertentu)

3. Kalau f (a_n) f (m) < 0, ambil a_n+1 = a_n ; b_n+1 = m

4. Jika f (a_n) f (m)>0  ambil a_n+1 = m ; b_n+1 = b_n

5. Jika  f(a_n) f(b_n) = 0 maka a_n dan b_n merupakan akarnya, hentikan perhitungan

f (x) punya akar dalam [a_n+1, b_n+1]

dengan membandingkan nilai  m lama dan  m baru pada suatu batas ketelitian tertentu, misal dibuat:
       

Apabila e_a sudah lebih kecil dari ketelitian yang diinginkan, maka perhitungan dihentikan.

Dari algoritma diatas dapat diturunkan program metode bisection, ada dua buah program:

1.      program utama disimpan dengan nama bisection.m

2.      program fungsi disimpan dengan nama fbi.m(program ini akan dipanggil oleh program utama)

Program utama dengan nama Bisection.m

%Nama File Bisection.m

clear;

clc;

galat = 0.001;

bawah = input('Batas Bawah : ');

atas = input('Batas Atas : ');

nilai = 1;

no = 0;

m0=bawah;

clc;

fprintf ('Taksiran batas bawah : %5.3f\n', bawah);

fprintf ('Taksiran batas atas : %5.3f\n', atas);

fprintf ('=========================\n');

fprintf ('Iterasi  (bawah+atas)/2 Galat      Interval\n');

fprintf ('=========================\n');

while nilai> galat

   no = no +1;

   fbawah=feval('fbi',bawah);

   m=(bawah+atas)/2;

   ftengah=feval('fbi',m);

   if fbawah*ftengah==0

      disp('m adalah akarnya');

   elseif fbawah*ftengah<0

      atas=m;

      else

         bawah=m;

         end

nilai=abs(m0-m);

fprintf ('  %3d     %8.5f    %8.5f     [%8.5f ; %8.5f]\n', no, m, nilai, bawah, atas);

m0=m;

   end

   fprintf ('==========================\n');

fprintf ('Pada iterasi ke-%1d, Selisih Interval < %5.3f\n', no, galat);

fprintf ('jadi, akar persamaannya adalah %7.5f\n', m);

fungsi yang diberi nama fbi.m

%Nama Fungsi fbi.m

function [y]=f(x)

y=x^3+x^2-8*x-10;

Advantages and Disadvantages of Bisection method.

Advantages:
- The method is quite simple
- Easy to make the program
- A small error rate
- Always be obtained root
weakness
- Convergence is rather slow
- Convergence is linear
- If enough roots in the interval, then the result is unpredictable

Bisection method of numerically

Often called the method of Bolzano. If there are signs of change in function on the interval, then the function value is calculated at the midpoint. Then the root location is determined at the midpoint of the interval where the sign changes.
The first stage of this process is to establish the value of a and b as a boundary segment of the value of unknown function. Limits a and b give you a price for the function f (x) for x = a and x = b. The next step is to check whether f (a). F (b) <0.
If these conditions are fulfilled, then there is the function of roots in this segment. If not, return must set the value of a and b so that in accordance with the multiplication of f (a) x f(b) <0.
With the formula m = (a + b) / 2, checked whether the absolute value of f (m) <10-6 (margin of error deviation). If true, the value of x = m is the solution sought. If not met, set new limits by changing the value of b = m if f (a) * f (m) <0, and replace if m = f (a) xf (m)> 0; the process of discovering new m done with the procedure has been described.
             Looking back to the drawing below:
given f (x) = 0; continuous nature and limits of the interval [a, b], f (a) f (b) <= 0

Figure 1. Determination of the midpoint m interval Bisection method

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