Numerical Methods

Mathematical Principles & Theorems

Analyzes numerical algorithms for root-finding and definite integration: (1) Bisection method: brackets root in \([a, b]\), linear convergence rate 1/2; (2) Secant method: superlinear convergence order \(\approx 1.618\); (3) Newton-Raphson: quadratic convergence \(|e_{k+1}| \le C |e_k|^2\); (4) Trapezoidal rule quadrature \(\int_a^b f(x)dx \approx \frac{h}{2}[f(a) + 2\sum f(x_i) + f(b)]\) with error \(\mathcal{O}(h^2)\); (5) Simpson's 1/3 Rule: parabolic interpolation with error \(\mathcal{O}(h^4)\).

Operating Instructions

• Select numerical method category: Root Finding (Bisection, Secant, Newton) or Numerical Quadrature.
• Enter mathematical function \(f(x)\), interval bounds \([a, b]\), and step/subdivision count \(N\).
• Click Execute to run numerical iterations and generate convergence tables.
• Inspect the interactive plot displaying tangent lines, secant brackets, or Simpson parabolic interpolants.