Algorithm_of_Calculation

Comparison of integration methods (rectangles, trapezoids, Simpson's rule) and selection of the most advantageous method.

Project Description

This project implements three numerical methods for calculating the definite integral of a function. The implemented methods are:

• Rectangle Method
• Trapezoid Method
• Simpson's Method

Each of these methods approximates the value of the integral by dividing the interval into smaller subintervals and summing the calculated values.

Files

'obliczenia.py': The main Python script containing the implementation of all three methods, as well as tests for different values of the number of subdivisions ( n ).

Test Results

Method	n = 5	n = 10	n = 100	n = 10000
Rectangles	-0.25469535390745	-0.44240000419352	-0.60896323443218	-0.6271490889782426
Trapezoids	-0.34453845351618	-0.48732155399789	-0.61345538941262	-0.6271940105280469
Simpson's	-1.24131003239092	-0.94198684815441	-0.6593670768964	-0.627653543229868

Conclusions

Method Accuracy:

• The rectangle method is the least accurate of the three methods, especially for smaller values of ( n ).
• The trapezoid method provides better approximations than the rectangle method but is less accurate than Simpson's method.
• Simpson's method, which uses quadratic interpolation, is the most accurate, especially for larger values of ( n ).

Convergence of Results:

• All three methods approach the true value of the integral as the number of subdivisions ( n ) increases.
• For ( n = 10000 ), the results of all methods are very close to the true value of the integral, confirming the correctness of the implementation.

Best Method:

• In general, Simpson's method provides the best results with the fewest iterations due to its higher accuracy from quadratic interpolation.
• In this specific case for the function ( f(x) ) in the interval ([-2, 1]), Simpson's method delivers the results closest to the true value of the integral even for smaller values of ( n ).