Composite Boole's Rule

Create a function to numerically integrate the equation using the Newton-Cotes Composite Boole's Rule,

The input would be in the sequence, a, b, and n. Name the function "newton_cotes(a, b, n)". The "n" is the number of panels.

can someone teach me how to code this composite boole's rule..

Answer 1

[Prasanna]

Hi Muhammad,

To create a function to numerically integrate equation using the newton-cotes composite boole’s rule, you can perform the following steps:

• Create a function that takes in 4 inputs: the function that is the integrand, the lower limit ‘a’, the upper limit ‘b’ and the number of panels ‘n’.
• Calculate the width of each subinterval.
• Initialize the integral result ‘r’, calculate the first point calculation by setting the first evaluation point ‘x’ to the first subinterval endpoint.
• Iterate through each panel and accumulate the integral contributions. Once the main loop is done, evaluate the last few points and add the contribution from the right endpoint.
• The accumulated result is then scald by the factor h*2/45 to finalize the integral value according to Boole’s rule.

A sample MATLAB code for the same is as follows:

function r = compositeboole(f,a,b,n)
    h = (b - a) / (n * 4);
    r = 7 * f(a);
    x = a + h;
    for i = 1 : n-1
        r = r + 32 * f(x);
        x = x + h;
        r = r + 12 * f(x);
        x = x + h;
        r = r + 32 * f(x);
        x = x + h;
        r = r + 14 * f(x);
        x = x + h;
    end
    r = r + 32 * f(x);
    x = x + h;
    r = r + 12 * f(x);
    x = x + h;
    r = r + 32 * f(x);
    r = r + 7 * f(b);
    r = r * h*2/45;
end

The function can then be called with the corresponding values for a,b, and n with the function f integrand as the following:

f = @(x) 2 * x .* exp(-x.^2);

For more information regarding Boole’s rule, refer the following resource:

• Composite Boole's rule: https://www.mathworks.com/matlabcentral/answers/515177-composite-boole-s-rule

Hope this helps!