Lecture 1 Code2_1 - Simpson's Rule Implementation

In Lecture 1, we will introduce the %code2_1 implementation of Simpson's Rule. Simpson's Rule is a numerical integration method used for approximating definite integrals. The algorithm works by dividing the integration interval into multiple subintervals and approximating the function using quadratic polynomials over each subinterval. This method typically provides greater accuracy compared to traditional rectangular and trapezoidal rules, particularly when dealing with functions that exhibit complex curvature patterns. The implementation involves calculating the integral approximation using the formula: ∫f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + 2f(xₙ-₂) + 4f(xₙ-₁) + f(xₙ)], where h is the subinterval width and n must be even. Through studying %code2_1, you will gain a deeper understanding of Simpson's Rule implementation, including handling of even-numbered subintervals, weight coefficients (1,4,2 pattern), and error analysis. This knowledge will enhance your numerical computation skills and provide practical experience in implementing advanced integration algorithms for complex mathematical problems.