Question

The Ostrowski method for finding a single root of ?(?)=0 is given by
Initial guess ?0 ??=??−?(??)?′(??), ??+1=??−?(??)?′(??) ?(??)?(??)−2?(??).
a) Write MATLAB or OCTAVE coding to implement the Ostrowski method.
(Hint: You may use the coding of Newton Method given in Moodle pages)
b) Use your coding to find a root of the equation
(?−2)2−ln(?)=0
With initial guess ?0=1.0 and ?0 = 3.0.
Write or print the results in your Homework sheet.

Answer 1

%%Matlab code for finding root using newton secant bisection and false
clear all
close all

%function for which root have to find
fun=@(x) (x-2).^2-log(x);
fprintf('For the function f(x)=')
disp(fun)
xx=linspace(0,4);
yy=fun(xx);

plot(xx,yy)
xlabel('x')
ylabel('f(x)')
title('x vs. f(x) plot')
box on; grid on;
[root1]=newton_method(fun,1,1000);
[root2]=newton_method(fun,3,1000);
hold on
plot(root1,fun(root1),'r*')
plot(root2,fun(root2),'r*')

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Matlab function for Newton Method
function [root]=newton_method(fun,x0,maxit)
syms x
g1(x) =diff(fun,x);   %1st Derivative of this function
xx=x0;            %initial guess]
fprintf('\nRoot using Newton method for initial guess %f\n',x0)
%Loop for all intial guesses
    n=5*10^-15; %error limit for close itteration
    for i=1:maxit
        x2=double(xx-(fun(xx)./g1(xx))); %Newton Raphson Formula
        cc=double(fun(x2));                 %Error
        err(i)=cc;
        xx=x2;
        if cc<=n
            break
        end
      
    end
    fprintf('\tAfter %d iteration root using Newton method is %f\n',i,xx)
    root=xx;
end
  
%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%