Question

find the root(s) of the following functions using both Newton's method and the secant method, using tol = eps.

3 Find the root s of the following functions using both Newton's ulethod and the anat inethod using tol epa. . You will vood to experiment with the parameters po, pl, ad maxits. . For each root, visualize the iteration history of both methods by plotting the albsolute errors, as a function . Label the two curves (Newton's method and secaut method) using a legend, desacribing the po and p (a) f:z2, which has two roots, z2 of itcrution mumbser, on a (single) semilogy plot values you used, eg, "Newton's (po 7)" and Secant (po-0, pl 1)" Rewrite the iteration formula to verify it is the saine as the Babylonian method (see worksheet ง Q1 (a)). Explsin the problem with the starting guess po " 0 in terims of the derivative of f. (b) f:z tanb(r), which has oue root, r 0 Characterixe (rouglhly) wbeu the methods converge or diverge in terms of the starting xuesses, (E g. try Newtou's metbod with po -1.08 aud 1.09.) (c) fi-2r +2, which has oue root, z 1.769- (d) f: z + +阪, wlich has cine root,エー0. (la Matlab, use nthroot (x,3) mausul of x"(1/3).) (e) (Challeng; unograded f: What happens with Newtou's method wheu po 0 or 17 duewhich has a root at r 0. Can you find the root? (Explain what you olsxerve:) Nobe that t

Answer 1

function [root] = newtonRaphMethod(f,fd,x0,tol,maxits)
figure
root_not_found = true;
iter = 1;
xn = x0;
while iter<maxits
  
% X = x - f/f'   
%Xn = xn - (f(xn)/fd(xn));
% simplified to Babylonian method
Xn = (xn + 2 / xn) / 2;
%termination criterion when |X-x|<tol
abs_error = abs(Xn-xn);

if abs_error <tol
root = Xn;
root_not_found = false;
break;
end
  
  
%plotting absolute error vs iterations
errors(iter) = abs_error;
iters = 1:iter;
semilogy(iters,errors,'-*')
xlabel('iteration')
ylabel('absolute error')
title(['Newton''s (p0 = ' num2str(x0) ')'])
grid on

xn = Xn;
iter = iter+1;
end
if root_not_found
fprintf(['\n Root not found. Method failed at ' num2str(iter) ' iterations \n'])
root =[];
end
end
------------------------------------------
function root = secantMethod(f,x0,x1,tol,maxits)
figure
x(1) = x0;
x(2) = x1;
n = 2;
iter = 1;
root_not_found = true;
while iter<maxits;
  
n = n+1;
x(n) = x(n-1) - ( ( f(x(n-1) )*( x(n-1) - x(n-2) ))/( f(x(n-1)) - f(x(n-2)) ) );


  
%termination criterion |x(n)-x(n-1)|<tol
abs_error = abs(x(n)-x(n-1));
if abs_error<tol
root = x(n-1);
root_not_found = false;
break;
end
  
  
  
%plotting absolute error vs iterations
errors(iter) = abs_error;
iters = 1:iter;
semilogy(iters,errors,'-*')
xlabel('iteration')
ylabel('absolute error')
title(['Secant (p0 = ' num2str(x0) ', p1 = ' num2str(x1) ')'])
grid on
  
  
iter = iter+1;
end
if root_not_found
fprintf(['\n Root not found. Method failed at ' num2str(iter) ' iterations \n'])
root = [];
end
end
--------------------------------------------------------------------------
%command
clear
clc
close all
% let f(x) = x^2-4
f = @(x) x.^2 - 2
% plotting the function for visual
x_min = -5;
x_max = 5;
spacing = 100; % 1/100
x = linspace(x_min,x_max,spacing);
plot(x,f(x))
title('f(x) = x^2 - 2')
xlabel('x')
ylabel('f(x)')
grid on
%setting termination criteria
maxits = 100;
tol = 1e-6;

%definig derivative fp and x_init initial guess for newton method
fp = @(x) 2*x;
x_init = -2.5;
%calling newton function to compute root
root_newton = newtonRaphMethod(f,fp,x_init,tol,maxits)
%{
% X = x - f/f'

root newton -1.4142 root ecant -1.4142 (x) tanh (x) root newton 1.4142 root secant

f(x) = x2-2 20 15 10 -5 -5 -3-2 0 13 45

Newton's (po =-2.5) 10 10 O 40-2 辷10. CD 10 10 10-5 1.5 2 2.5 iteration 3 3.5 4

Secant 10 10 O 40-2 .3 D 10 10-4 10-5 1 1.5 22.5 33.5 4 4.5 5 iteration

5 4 3 2 1864 202468 3

Newton's (p0= 1.09) 10 -1 10 5 10 -3 10 104 1 1.2 14 1.6 18 22.224 2.6 2.8 3 iteration

Secant (po =-1 , p1 = 1) 10 米 10.1 L 0 0.2 040.6 0.8 1 12 1416 18 2 iteration