MATLAB The following question needs to be programmed in MATL

MATLAB) The following question needs to be programmed in MATLAB.

Please make sure the it has codes related to the ratio and the convergence that is stated in the question as well!

10. Test the secant method on an example in which r, f\'(r), and f \"(r) are known in advance. Monitor the ratios en 1-1/(enen to see whether they converge to en-1) f\' r). The function f (x) arctan x is suit able for this experiment.

Solution

The Matlab function Seant_method.m

function Secant_method(f,x0,x1,tol,n) % secant method for finding root of f(x)
iter=1;% intial iteration value
u=f(x0);% function value at x0
v=f(x1);% function value at x1
err(iter)=abs(x1-x0);% error
% displaying the result in tabular form
disp(\' iter      f(x)            x     e(n+1)/(e(n)e(n-1))\')
disp(\'___________________________________________________\')
fprintf(\'%2.0f %12.6f %12.6f\ \',iter-1,u,x0); %result to the screen
while (err(iter)>tol)&&(iter<=n)&&((v-u)~=0) % loop to until converg or reach N iter
      iter=iter+1;% increase number of iteration
      x=x1-v*(x1-x0)/(v-u);% new approximation of root
      x0=x1; % replace the old value in x0 with x1
      u=v; % replace the old functional value f(x0) with f(x1)
      x1=x;% replace the old value in x1 with new value
      v=f(x1);% function value at new x1
      err(iter)=abs(x1-x0);% error
      if iter < 3
          fprintf(\'%2.0f %12.6f %12.6f\ \',iter-1,u,x0); %result to the screen
      else
          fprintf(\'%2.0f %12.6f %12.6f %12.6f\ \',iter-1,u,x0,err(iter)/(err(iter-1)*err(iter-2))); %result to the screen
      end
end
if ((v-u)==0)% Checking devision by zero occured or not
      error(\' Division by zero\')% if yes display error message
elseif (iter>n)% checking method diverge or not
      error(\' Method failed to converge\')% if yes display message
end
end

Calling the function

>> f =@(x) atan(x);
>> df =@(x) 1./(x.^2+1);
>> ddf = @(x) (-2*x)./(x.^2+1).^2;
>> x0 = -10;x1 = 2;tol = 10^-15;n=100;
>> Secant_method(f,x0,x1,tol,n)
iter      f(x)            x     e(n+1)/(e(n)e(n-1))
___________________________________________________
0     -1.471128    -10.000000
1      1.107149      2.000000
2     -1.263671     -3.152971     0.044418
3     -0.385999     -0.406386     0.085349
4      0.675691      0.801560     0.231717
5      0.032776      0.032787     0.042204
6     -0.006404     -0.006404     0.212623
7      0.000002      0.000002     0.007353
8     -0.000000     -0.000000     0.002134
9      0.000000      0.000000     0.000001
>> fprintf(\'The ratio -ddf/(2 df) at root x = 0 is %f\ \',-ddf(0)/(2*df(0)));
The ratio -ddf/(2 df) at root x = 0 is 0.000000

>>