Numerical Methods
Root-finding, ODE/PDE solvers, interpolation, Monte Carlo.
Technical Articles
01
Nonlinear Equation Solvers: Bisection, Newton-Raphson & Secant Methods
Convergence order analysis of bisection, Newton-Raphson local quadratic proofs, and golden ratio secant convergence.
02
Numerical ODE Integration: Runge-Kutta, A-Stability & Stiff Systems
Explicit vs implicit Euler stability regions, RK4 Butcher tableaus, A-stability proofs, and stiff BDF solvers.
03
PDE Numerical Solvers: Von Neumann Stability & CFL Conditions
Finite difference discretization, Courant-Friedrichs-Lewy (CFL) stability proofs, and Crank-Nicolson schemes.
04
Function Interpolation: Polynomial Basis & Runge's Phenomenon
Lagrange and Newton divided differences, Runge oscillation proof, and C^2 cubic spline interpolation.
05
Monte Carlo Integration & Variance Reduction Techniques
Asymptotic O(N^{-1/2}) convergence, importance sampling Radon-Nikodym weights, control variates, and antithetic sampling.