≈Numerical Analysis
Error analysis, interpolation, numerical integration, numerical DE
Subfields
Error Analysis
Round-off error, truncation error, condition number
Interpolation
Polynomial interpolation, splines, Lagrange interpolation
Numerical Integration
Trapezoidal rule, Simpson's rule, Gaussian quadrature
Numerical Differential Equations
Euler method, Runge-Kutta, finite difference, finite element
Numerical Linear Algebra
LU decomposition, QR decomposition, SVD, iterative methods
Concepts
Numerical Error
Numerical error arises in computer calculations. Types include rounding error, truncation error, and propagation error.
Newton-Raphson Method
Newton-Raphson method is an iterative technique for finding roots of f(x) = 0. It uses tangent lines for fast convergence.
Numerical Integration
Numerical integration approximates definite integral values. Methods include trapezoidal rule and Simpson's rule.
Interpolation
Interpolation estimates values between given data points. Methods include polynomial interpolation and spline interpolation.
Numerical ODE Methods
Numerical ODE methods approximate solutions to differential equations that are difficult to solve analytically.
Bisection Method
Bisection method is the simplest root-finding method for continuous functions. It halves the interval, narrowing toward sign changes.
Runge-Kutta Methods
Runge-Kutta methods are high-precision techniques for numerical solutions of differential equations. Fourth-order (RK4) is most widely used.
Matrix Factorization
Matrix factorization expresses a matrix as product of simpler matrices. Used for solving systems, eigenvalue computation, and data compression.
Finite Difference Method
Finite difference method approximates derivatives with differences, converting differential equations to algebraic equations. Fundamental for numerical PDE solutions.