Add- simpson's rule

[Erennn7]

PR: Simpson’s Rule for Numerical Integration in R

This PR introduces a fully documented implementation of Simpson’s Rule in R, a numerical integration technique that approximates the definite integral of a function using quadratic polynomials.
Simpson’s Rule provides higher accuracy than the Trapezoidal Rule by approximating the area under the curve with parabolic arcs rather than straight line segments.

Overview

The simpson_rule function divides the integration interval ([a, b]) into an even number of subintervals and applies quadratic interpolation to estimate the integral.
Key features include:

• Handles arbitrary continuous functions and numeric intervals
• Provides both standard and vectorized implementations for improved performance
• Computes approximate integrals and can compare against exact values if known
• Supports convergence studies by varying the number of subintervals
• Demonstrates accuracy and applicability for common functions, including sin(x), x², e^x, and probability density functions

Simpson’s Rule is widely used in applications such as:

• Scientific computing and physics simulations
• Signal processing and data analysis
• Probability, statistics, and areas under curves
• Computer graphics and visualization

Complexity

• Time Complexity: O(n), where n = number of subintervals
• Space Complexity: O(n) for storing x-values

[Copilot]

Pull Request Overview

This PR adds an implementation of Simpson's Rule for numerical integration in R, providing both a standard and vectorized approach to approximate definite integrals using quadratic polynomial interpolation.

Key Changes:

• Implements simpson_rule() function with input validation and error handling
• Adds a vectorized version simpson_rule_vectorized() for improved performance
• Provides comprehensive examples demonstrating convergence, accuracy, and various use cases

[Copilot]

Pull Request Overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 1 comment.

[Copilot]

Pull Request Overview

Copilot reviewed 1 out of 1 changed files in this pull request and generated 1 comment.

[Erennn7]

@siriak please have a look

[siriak]

Looks good, thanks!