Numerical Analysis

Numerical error arises in computer calculations. Types include rounding error, truncation error, and propagation error.

Newton-Raphson method is an iterative technique for finding roots of f(x) = 0. It uses tangent lines for fast convergence.

Numerical integration approximates definite integral values. Methods include trapezoidal rule and Simpson's rule.

Interpolation estimates values between given data points. Methods include polynomial interpolation and spline interpolation.

Numerical ODE methods approximate solutions to differential equations that are difficult to solve analytically.

Bisection method is the simplest root-finding method for continuous functions. It halves the interval, narrowing toward sign changes.

Runge-Kutta methods are high-precision techniques for numerical solutions of differential equations. Fourth-order (RK4) is most widely used.

Matrix factorization expresses a matrix as product of simpler matrices. Used for solving systems, eigenvalue computation, and data compression.

Finite difference method approximates derivatives with differences, converting differential equations to algebraic equations. Fundamental for numerical PDE solutions.