# Newton Method and Bisection Method – Matlab Scripts

Here are 2 scripts on Newtons method and the Bisection method.

Newtons methods is based on using the derivative at a point to help calculate a closer value and this continues to find a closer and closer value.

The bisection method, as the name suggests, halves the distance between 2 points continuously until the point in between is accurate enough.

The bisection method is quite slow compared to newtons method as it only halves the distance, yet newtons method uses the gradient to approximate quickly with fewer iterations.

## Newtons Method

function x = newton_method(f_str,df_str,x0,n)
% in the form f(x) = 0

f = inline(f_str);
df = inline(df_str);
disp(‘Number of Iterations =’)
disp(0)

x = x0;
xn = 0;

disp(‘Xn =’)
disp(x)
disp(‘f(x) =’)
disp(f(x))
disp(‘f”(x) =’)
disp(df(x))

for i = 1:n
xn = x – (f(x)/df(x));
if x == xn
disp(‘This is the Highest Accuracy Achievable’)
i = i-1;
break
end
x = xn;
disp(‘Number of Iterations =’)
disp(i)
disp(‘Xn =’)
disp(x)
disp(‘f(x) =’)
disp(f(x))
disp(‘f”(x) =’)
disp(df(x))
end
disp(‘Number of Iterations Done:’)
disp(i)
end

## Bisection Method

function x = bisection(f_str,x0,x1,n)
% in the form f(x) = 0

f = inline(f_str);
a = x0;
b = x1;
disp(‘Number of Iterations =’)
disp(0)
disp(‘a =’)
disp(a)
disp(‘b =’)
disp(b)

x = (a+b)/2;
disp(‘x =’)
disp(x)
disp(‘f(a)=’)
disp(f(a))
disp(‘f(b)=’)
disp(f(b))
disp(‘f(x)=’)
disp(f(x))

for i = 1:(n)
if (f(x) > 0 && f(b) > 0) || (f(x) < 0 && f(b) < 0)
b = x;
elseif (f(x) > 0 && f(a) > 0) || (f(x) < 0 && f(a) < 0)
a = x;
else disp(‘This is the Highest Accuracy Achievable’)
i = i-1;
break
end
disp(‘Number of Iterations =’)
disp(i)
disp(‘a =’)
disp(a)
disp(‘b =’)
disp(b)

x = (a+b)/2;
disp(‘x =’)
disp(x)
disp(‘f(a)=’)
disp(f(a))
disp(‘f(b)=’)
disp(f(b))
disp(‘f(x)=’)
disp(f(x))
end
disp(‘Number of Iterations Done:’)
disp(i)
end

### Other Iterative and Mathematical Method using Matlab and also other Mathematical Examples:

>>> Romberg Numerical Integration – Matlab Script

>>> Simpson’s Rule and Trapezoidal Rule of Numerical Integration – Matlab Scripts

>>> Secant Method of finding Roots – Matlab Script

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