NEWTON-RAPSHON METHOD

The Newton-Rapshon is an open method and is one of the most used because of its speed and effectiveness. This is a variant of the fixed point method. The difference is found in the iteration equation that is given by:

According to the geometric definition, the method works as explained below:

1. Start by drawing a tangent line to the graph of f(X) at the point (X0,f(X0)), where X0 is the initial approximation.

2. The previous tangent line intersects the x-axis at a point (X1,0), so with this value we draw a tangent line at the point: (X1,f(X1)) of the curve f (X).

3. Steps 1 and 2 are repeated and consequently the cut points of the tangent lines approach more and more the root of f (X), when the method converges.

PSEUDOCODE OF NEWTON-RAPSHON

write ('Newton-Rapshon method')

read fx, df, x0, tol, niter

fx = convert the input fx of the user to function

df = convert the input df of the user to function

counter = 0

error = tol + 1

while error> tol and fx (x0) ≠ 0 and df (x0) ≠ 0 and counter <niter do

counter = counter + 1

write table with this information (Ite, Xn, f (Xn), df (Xn), error)

x1 = x0-f (x0) / df (x0)

error = | x1-x0 |

x0 = x1

end while

if fx (x0) == 0

write ('This number is root: x0')

else if

write ('This number x1 is an approximation to a root with a tolerance of: tol')

else if

write ('It is a possible multiple root')

else

write ('Failure the number of iterations')

end

CODE OF NEWTON-RAPSHON

clc

clear all

disp ('Newton-Rapshon Method');

fx = input ('Enter the function:', 's');

df = input ('Enter the derivative of the function:', 's');

x0 = input ('Enter the initial approach:');

tol = input ('Enter a tolerance:');

niter = input ('Enter the maximum number of iterations:');

fx = inline (fx);

df = inline (df);

counter = 0;

error = tol + 1;

while error> tol && fx (x0) ~ = 0 && df (x0) ~ = 0 && counter <niter

counter = counter + 1;

fprintf ('Iter:% 2i \ t Xn =% f \ tf (Xn) =% f \ t df (Xn) =% f \ t Error =% f \ n', counter, x0, fx (x0), df (x0), error)

x1 = x0-fx (x0) / df (x0);

error = abs (x1-x0);

x0 = x1;

end

if fx (x0) == 0

disp ('This number is root:')

disp (x0)

elseif error <tol

disp ('This number ... is an approximation to a root with a tolerance of:')

disp (x1)

disp (tol)

elseif df (x0) == 0

disp ('It is a possible multiple root:')

disp (x1)

else

disp ('Failure the number of iterations')

end