# The estimation error is strangely obtained from Simpson's 1/3 rule

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Hello, I am looking for the estimation error of Simpson's rule of thirds. When dx is 1.01, the integral value is 2908800 and the error is 0.01, but the estimation error is 5.5567e-09. Where did it go wrong?

clc; clear all; close all;

a = -1.93317553181561E-24;

b = 3.788630291530091e-21;

c = -2.3447910280083294e-18

d = -0.019531249999999518;

e = 18.74999999999999

fun = @(t) a.*t.^5 + b.*t.^4 + c.*t.^3 + d.*t.^2+e.*t

x = 0:0.1:960;

fx =fun(x);

n=960;

dx=1.01

int=0;

for i =1:n

plot(x, fx,'k','linewidth',2);

mid=((i-1)+i)/2;

fx_mid = fun(mid);

fx_left = fun(i-1);

fx_right = fun(i);

area_temp = dx/6*(fx_left +4*fx_mid+fx_right);

int = int + area_temp;

x_segment = linspace(i-1, i,100);

Px = fx_left * ((x_segment-mid).*(x_segment-i))/((i-1-mid)*(i-1-i))...

+ fx_mid*((x_segment-i+1)).*(x_segment-i)/((mid-i+1)*(mid-i))...

+ fx_right * ((x_segment-i+1).*(x_segment-mid))/((i-i+1)*(i-mid));

area(x_segment,Px); hold on;

end

C=480;

E_a = -((960.^5)/(2880.*1.01.^4)).*(a.*120.*C+24.*b);%Is there a problem here?

disp('E_a');

disp(E_a);

disp(int);

int_true = 2880000

rel_error=norm(int_true-int)/norm(int_true);

disp('rel_error');

disp(rel_error);

##### 1 Comment

Steven Lord
on 23 May 2024 at 13:51

### Accepted Answer

Ayush Anand
on 22 May 2024 at 8:11

Edited: Ayush Anand
on 22 May 2024 at 8:18

There is a mistake with your formulae for estimation of . The formula for the error estimate in Simpson's rule is generally given by:

where ξ is some number in the interval [a, b], n is the number of intervals, and is the fourth derivative of the function being integrated.

Given your function: ; the fourth derivative of would be:

So, given the interval and with , the formulae for looks like:

Replacing formulae for in your code with this should give you the correct results.

You can check this out for reading more on Simpson's rule: https://www.mathworks.com/matlabcentral/fileexchange/72526-simpson-s-1-3-rule-composite

##### 3 Comments

Ayush Anand
on 23 May 2024 at 4:49

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