Question

Calculate , and apply the composite trapezoidal rule to numerically compute it with evenly spaced nodes where n = 10, 20, 40, 80. Compute absolute numerically errors. Please post your code in MATLAB and present your results in the table below:

1 0 n Result by the composite trapezoidal rule Absolute error Absolute errorxn2 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 Trapizoidal\t errort error*n 2\n') for i-l:length (n) all integration values val-trapizoidal(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 Trap izoidal integration function val-trapizoidal ( 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 val-0; %splits interval a to b into N+1 subintervals xx-linspace (a,b, N+1) dx-xx ( 2 )-xx ( 1 ) ; %x interval %loop for Riemann integration for i-2:length (xx)-1 val-val+dx*double(func (xx1)) end val-val+dx (0.5*double (func (xx (1)))+0.5double (func (xx (end)))); end 용욤욤응%욤욤응%号号%%号号%%욤 End of Code %용융응용용융응용욤응응용욤응응용욤 For the function f(x)-

Exact integration value for a=0.000000 to b= 1 .000000 is 0.693147. n Trapizoidal error error*n 2 10 0.693771 6.242226e-04 6.242226e-02. 20 0.693303 1.562012e-04 6.248049e-02. 40 0.693186 3.905945e-05 6.249512e-02 80 0.693157 9.765434e-06 6.249878e-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 Trapizoidal\t error\t error*n^2\n')
for i=1:length(n)
    N=n(i);
    %all integration values
    val=trapizoidal(func,a,b,N);
    er1=abs(val-I_ext);
    er2=er1*N^2;
    fprintf('\n\t%d \t %f \t %e \t %e.\n',N,val,er1,er2)
end
%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%    
%%Matlab function for Trapizoidal integration
function val=trapizoidal(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
    val=0;
    %splits interval a to b into N+1 subintervals
    xx=linspace(a,b,N+1);
    dx=xx(2)-xx(1); %x interval
    %loop for Riemann integration
        for i=2:length(xx)-1
            xx1=xx(i);
            val=val+dx*double(func(xx1));
        end
       val=val+dx*(0.5*double(func(xx(1)))+0.5*double(func(xx(end))));
end

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