Performs numerical integration on a data set in the form of a txt file. Uses Simpson's Rule where applicable, and the Trapezoidal Rule in all other cases.
Simpson's 1/3 Rule:
Approximates a function using 2nd order polynomials.
For n equal segments (i.e., n+1 evenly spaced data points), the definite integral can be approximated as follows. Note that n must be even.
Here (b-a)/n is the spacing between data points.
Simpson's 3/8 Rule:
Approximates a function using 3rd order polynomials.
For n equal segments (i.e., n+1 evenly spaced data points), the definite integral can be approximated as follows. Note that n must be an integer multiple of 3.
Here (b-a)/n is the spacing between data points.
Trapezoidal Rule:
Approximates a function using trapezoids.
For n equal segments (i.e., n+1 evenly spaced data points), the definite integral can be approximated as follows.
Here (b-a)/n is the spacing between data points.
Test code:
for Npts = logspace(1, 4, 4) + 1
x = pi*linspace(0, 1, Npts);
y = sin(x);
[method, result] = Simpson(x, y);
fprintf('n = %-6d \tTrapz = %.6f \t%s = %.6f\n', Npts-1, trapz(x, y), method, result)
end
n = 10 Trapz = 1.983524 Simpson's 1/3 Rule = 2.000110
n = 100 Trapz = 1.999836 Simpson's 1/3 Rule = 2.000000
n = 1000 Trapz = 1.999998 Simpson's 1/3 Rule = 2.000000
n = 10000 Trapz = 2.000000 Simpson's 1/3 Rule = 2.000000
for Npts = logspace(1, 4, 4)
x = pi*linspace(0, 1, Npts);
y = sin(x);
[method, result] = Simpson(x, y);
fprintf('n = %-6d \tTrapz = %.6f \t%s = %.6f\n', Npts-1, trapz(x, y), method, result)
end
n = 9 Trapz = 1.979651 Simpson's 3/8 Rule = 2.000382
n = 99 Trapz = 1.999832 Simpson's 3/8 Rule = 2.000000
n = 999 Trapz = 1.999998 Simpson's 3/8 Rule = 2.000000
n = 9999 Trapz = 2.000000 Simpson's 3/8 Rule = 2.000000
Resources:
https://web.engr.oregonstate.edu/~webbky/MAE4020_5020_files/Section%208%20Integration.pdf