Several Numerical methods have been developed to deal with problems that may not

Question

Several Numerical methods have been developed to deal with problems that may not have analytical solutions, or when such solutions become difficult to obtain. Among these methods are Newton-Raphson and Secant methods. In this homework, you will have to find the solution (root) for the given function using both methods. The function is:
f(x)=10+e^(-x)-5x+sin?(x?^2)

Newton-Raphson method:
1. Start with an initial guess x0
2. Find x1 as follows: x_1= x_0-(f(x_0))/(f ´(x_0))
where (f(x)) ´ is the derivative of f(x) with respect to x
3. Keep finding new values for x using the previous value of x until the difference between any two successive values is negligible:
x_(n+1)= x_n-(f(x_n))/(f ´(x_n))
tolerance= |x_(n+1)-x_n |
4.Choose tolerance =10-7 and catch all the values the method generates for x


Secant method:
1. This method doesn’t require knowledge of the derivative, instead it requires two guess points x0 and x1
2. Find x2 as follows: x_2= x_1-f(x_1)(((x_1-x_0 ))/((f(x_1 )-f(x_0)) ))
3. Keep finding new values for x using the last two generated values of x until the difference between any two successive values is negligible:
x_(n+1)= x_n-f(x_n)(((x_n-x_(n-1) ))/((f(x_n )-f(x_(n-1))) ))
tolerance= |x_(n+1)-x_n |
4. Choose tolerance =10-7 and catch all the values the method generates for x




Finally, compare between the two methods by comparing the number of iterations each method requires before it converges. For example, start from a far guess point (say 100) and see how fast the method finds the solution.

Please turn in your homework printed in a report format (i.e: cover page, introduction, codes, figures of the function and the x values and comments). You can use the publish html files feature but you need to add introduction and comments.

Explanation / Answer

Here are the basic codes for the two methods. The way you wrote the function seems odd, my assumption is you meant to write sin(x^2) and randomly threw question marks in for some reason. If I am wrong it doesn't matter, just plug the correct equation into the codes below:

% Basic Newton-Raphson code
clear
clc
format long
%function:
f=inline('10+exp(-x)-5*x+sin(x^2)')
%derivative of function:
fp=inline('-exp(-x)-5+2*x*cos(x^2)');
i=1;
x(i)=100;
test=1;
while test
x(i+1)=x(i)-f(x(i))/fp(x(i));
test=abs(x(i+1)-x(i))>10^(-7);
i=i+1;
end
fprintf('The root is %2.7f after %.0f iterations ',x(i),i)
fprintf('Examine the vector x from the workspace ')
fprintf('or command window to track the iterations. ')



%Basic secant method code:
clear
clc
format long
f=inline('10+exp(-x)-5*x+sin(x^2)')
i=2;
x(i-1)=0;
x(i)=100;
test=1;
while test
x(i+1)=x(i)-(f(x(i))*(x(i)-x(i-1)))/(f(x(i))-f(x(i-1)));
test=abs(x(i+1)-x(i))>10^(-7 );
i=i+1;
end
fprintf('The root is %2.7f after %.0f iterations ',x(i),i)
fprintf('Examine the vector x from the workspace ')
fprintf('or command window to track the iterations. ')

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