Question

1. Develop a well-structured MATLAB function to compute the Maclaurin series expansion

   for the cosine function and name the function cosMac. The function should have the following features:

   1. Iterate until the relative error (variable name “relErr” ) falls below a stopping criterion OR exceeds a maximum number of iterations (variable name“maxIter”) for a given value of x.

   2. Include a default value of relErr = 1e-6 if the user does not enter the value (use nargin function).

   3. Include a default value for maxIter = 100 if the user does not enter the value (use nargin function).

   4. Return the estimate of cos(x), the approximate relative error, the number of iterations, and the true relative error (that you can calculate based on the built- in cosine function).

Answer 1

%Matlab code for Maclaurin series
clear all
close all

%example of cosMac function
x=pi;
[val,tr_er,rel_err,iter]=cosMac(x);
fprintf('For x=%f, cos(x)=%f with relative error=%e and iteration=%d\n',x,val,rel_err,iter)

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

%function for Maclaurin series for cos(x)
function [val,tr_er,rel_err,iter]=cosMac(x,maxIter,relErr)

    if nargin < 3
        relErr = 10^-6;
    end
  
    if nargin < 2
        maxIter = 100;
    end
    err=1;k=1;vv(1)=1; val=1;
  
    while err>=relErr
        k=k+1;
        if mod(k,2)==1
            vv(k)=((x^(2*(k-1)))/(factorial(2*(k-1))));
        else
            vv(k)=-((x^(2*(k-1)))/(factorial(2*(k-1))));
        end
        val=val+vv(k);
        err=abs(abs(vv(k-1))-abs(vv(k)));
        if k>=maxIter
            break
        end
      
    end
  
  
    tr_er=(cos(x)-val);
  
    rel_err=err;
    iter=k;
end

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