Question

Numerically apply the composite Simpson’s rule to compute   with evenly spaced nodes with n = 10, 20, 40, 80. Note that different coefficients should be placed in front of the function values at nodes with odd and even subscripts. Compute absolute errors. Please attach your MATLAB code and complete the table below:

1 0 n Result by the composite Simpson's rule Absolute error Absolute errorxn4 10 20 40 80

Answer 1

Matlab code for finding integration clear all close all %functions for which integration have to do func-e (x) 1./ (1+x) tupper and lower limit of integrations a 0;b-1; All n values n [10 20 40 801 fprintf(n nFor the function f(x)) disp (func) %Exact integration for function I_ext integral (func,a,b) tprint f ("\nExact integration value for a=%f to b=%f is %f. n',a,b,I_ext) %loop for finding integrations fprintfntn t Simpson\t error t error* n 4\n' for i= 1:length(n) all integration values val-simpson ( func,a,b,N); er1-abs (val-Iext); fprint f ("\n\t%d \t %f \t \t %e.\n",N,va 1 , er 1 , er2 ) %e end 各各各各各各各各各各各各음음各各음 各各음음各各음음各各음음各各음음各各 음各各음음各各음음各各음음各各음음各 %%Matlab function for Simpson integration function val-simpson (func,a, b, N) % func is the function for integration a is the lower limit of integration % b is the upper limit of integration N number of rectangles to be used splits interval a to b into N+1 subintervals xx-linspace (a, b, N+1); dx-xx ( 2 )-xx ( 1 ) ; %x interval val-(dx/3) ( double (func(xx (1)))+double (func (xx (end)))) %loop for Riemann integration for i-2:length (xx)-1 if mod(i,2)0 val-val+ (dx/3) *4 double(func (xx1)); else val-val+ (dx/3) *2 double(func (xx1)); end end 888888888888888888 End of Code 융8888888888888888옹

For the function f(x)- (x)1./(1+x) Exact integration value for a=0.000000 to b= 1.000000 is 0.693147. n Simpson error error*n 4 10 0.693150 3.050129e-06 3.050129e-02. 3·1056 83e-02. 20 0.693147 1.941052e-07 40 0.693147 1.218801e-08 3.120131e-02. 80 0.693147 7.626413e-10 3. 123779e-02. Published with MATLAB® R2018a

%%Matlab code for finding integration
clear all
close all
%functions for which integration have to do
func=@(x) 1./(1+x) ;
%upper and lower limit of integrations
a=0;b=1;
%All n values
n=[10 20 40 80];

fprintf('\n\nFor the function f(x)= ')
disp(func)
%Exact integration for function
I_ext = integral(func,a,b);
fprintf('\nExact integration value for a=%f to b=%f is %f.\n',a,b,I_ext)

%loop for finding integrations
fprintf('\n\tn \t Simpson\t error\t error*n^4\n')
for i=1:length(n)
    N=n(i);
    %all integration values
    val=simpson(func,a,b,N);
    er1=abs(val-I_ext);
    er2=er1*N^4;
    fprintf('\n\t%d \t %f \t %e \t %e.\n',N,val,er1,er2)
end
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%    
%%Matlab function for Simpson integration
function val=simpson(func,a,b,N)
    % func is the function for integration
    % a is the lower limit of integration
    % b is the upper limit of integration
    % N number of rectangles to be used
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    val=(dx/3)*(double(func(xx(1)))+double(func(xx(end))));
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            if mod(i,2)==0
                val=val+(dx/3)*4*double(func(xx1));
            else
                val=val+(dx/3)*2*double(func(xx1));             
            end
        end    
end

%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%