Question

Please code in MatLab or Octave

Output should match Sample Output in Second Picture

Thank you

5. In this problem we will investigate using the Secant Method to approximate a root of a function f(r). The Secant Method is an iterative approach that begins with initial guesses , and r2. Then, for n > 3, the Secant Method generates approximations of a root of f(z) as In-1-In-2 n=En-1-f (x,-1) f(Fn-1)-f(-2) any iteration, the absolute error in the approximation can be approximated as For The algorithm iterates until the error approximation is less than some threshold e. In this question we will study the function f(x) = - -72++6. Write a program that defines a "math" (anonymous) function for f. The main program should then call a function get input which should prompt the user for an initial guess r . prompt the user for an initial guess I2 keep prompting the user and giving error messages until a positive value is entered for e return the initial guesses and e value The main program should then call the function secant method which should take in a function f, the initial guesses, and the e value starting from the initial guesses, run the Secant Method while the error approxima- tion is larger than e call the function plot results return the final root approximation, the n value at which it occurred, and the final corresponding error approximation The function plot results should take in a vector of root approximations produced by the Secant Method (including the initial guesses), a vector of error approximations produced by the Secant Method, and the final iteration value n obtained by the Secant Method create a plot of the root approximation vs. iteration number and save it as a png via print-dpng'. 'secant method.approx.png'); create a semilogy plot of the error approximation vs. iteration number and save it as a png via print-dpng'. (note that semilogy is an alternative to the plot function that uses the exact same syntax we've been using for plot) secant method error.png')

Answer 1

We have developed a MATLAB code for the secant method satisfying the requirements.

Command Window >> main Enter an initial guess xl: 2 Enter an initial guess x2: 5 enter a value for epsilon: 0 Invalid epsilon 0 enter a value for epsilon: -2 Invalid epsilon -2 enter a value for epsilon: le-6 An approximation of the root is x (12)-2.000000 An approximation of the absolute error is 9.894228e-09

Iteration Number vs Root Approximation 1 2 3 5 9 10 11 Iteration Number Root Approximation

Iteration Number vs Error Approximation 1D 100 10 102 103 104 10 10 10 10 10 3 4 5 6 9 10 11 12 Iteration Number

MATLAB code

f=@(x) x^4-x^3-7*x^2+x+6;
[x1,x2,epsilon]=get_input();
[x,curr_error,it]=secant_method(f,x1,x2,epsilon);
plot_result(x,curr_error,it)
  
  
function [x1,x2,epsilon]=get_input()

fprintf('\n\n');
prompt1='Enter an initial guess x1: '; x1= input(prompt1);
prompt2='Enter an initial guess x2: '; x2= input(prompt2);
prompt3='enter a value for epsilon: '; epsilon= input(prompt3);

while epsilon<=0
disp(['Invalid epsilon ',num2str(epsilon)]);
prompt3='enter a value for epsilon: ';
epsilon= input(prompt3);
end
end
  
function [x,curr_error,it]=secant_method(f,x1,x2,epsilon)
  
x(1)=x1;
x(2)=x2;
k=2;
curr_error(1)=3;

while curr_error(k-1)>epsilon
k=k+1;
xn1=x(k-2);
xn2=x(k-1);
x(k)=xn1-f(xn1)*(xn1-xn2)/(f(xn1)-f(xn2));
curr_error(k-1)=abs(x(k)-x(k-1));
end
fprintf('\nAn approximation of the root is x(%g)=%f',k,x(k));
fprintf('\nAn approximation of the absolute error is %e\n\n',curr_error(end));
it=1:k;
end
  
function plot_result(x,curr_error,it)

figure(1);
plot(it,x,'LineWidth',1.5);
xlabel('Iteration Number','fontsize',14)
ylabel('Root Approximation','fontsize',14)
title('Iteration Number vs Root Approximation','fontsize',14)
print('-dpng','secant_method_approx.png');


figure(2);
semilogy(it(2:end),curr_error,'LineWidth',1.5);
xlabel('Iteration Number','fontsize',14)
ylabel('Error Approximation','fontsize',14)
title('Iteration Number vs Error Approximation','fontsize',14)
print('-dpng','secant_method_error.png');
end