Numerical Analysis
Although mathematics is profound, not all problems in the world can be solved with just paper and pen. Numerical Analysis employs machinery to solve problems approximately and is an indispensable tool in applied mathematics research.
Functions and Integration
Differentiation
Interpolation
- Divided Differences $f[x_0, \ldots, x_n]$
- Interpolation
- Polynomial Interpolation
- Hermite Interpolation
- Splines
Function Approximation
- Function Approximation
- Chebyshev Polynomials of the First Kind $T_n$
- Chebyshev Polynomials of the Second Kind $U_n$
- Relationship between Chebyshev Polynomials of the First and Second Kind
- Chebyshev Expansion
- Chebyshev Nodes
Numerical Integration
- Numerical Integration
- Trapezoidal Rule $I_n^1(f)$
- Simpson’s Rule $I_n^2(f)$
- Newton-Cotes Integration Formulas $I_n^p(f)$
- Variable Substitution Trick for Computing Improper Integrals Numerically
- Gaussian Quadrature
- Laguerre Polynomials $L_n$
- Hermite Polynomials $H_{e_n}$, $H_n$
- Gaussian Quadrature for Computing Improper Integrals Numerically
Equations
Numerical Solutions of Algebraic Equations
Numerical Solutions of Differential Equations
- Lipschitz Condition
- Euler Method
- Multistep Methods
- Parasitic Solutions
- Trapezoidal Method
- Richardson Error Estimation
- Adams Methods
- Root Condition of Multistep Methods
- A-Stability
- Explicit Runge-Kutta Methods (ERK)
- Implicit Runge-Kutta Methods (IRK)
Solution of Partial Differential Equations
- Heat Equation: FDM
- Heat Equation: The Method of Lines
- Wave Equation: FDM
- Wave Equation: $k$-space method
- Wave Equation: Absorbing Boundary Condition
References
- Atkinson. (1989). An Introduction to Numerical Analysis(2nd Edition)
All posts
- Numerical Solution of the Wave Equation: Finite Difference Method (FDM)
- Numerical Solution of the Wave Equation: K-Space Method
- 룽게-쿠타 메소드에서 계수 결정하는 방법
- Numerical Analysis in Differences
- First kind Chebyshev polynomials
- Second Kind Chebyshev Polynomials
- Relationship between the First and Second Kind Chebyshev Polynomials
- Rate of Convergence in Numerical Analysis
- Bisection Method
- Newton-Raphson Method
- Differential Stages in Numerical Analysis
- Secant Method
- Mueller Method
- Newton's Method for Solving Nonlinear Systems
- Interpolation in Numerical Analysis
- Polynomial Interpolation
- Lagrange's Formula Derivation
- Derivation of Newton's Forward Difference Formula
- Hermite-Genocchi Formula
- Hermite Interpolation
- Numerical Analysis in Splines
- B-Splines in Numerical Analysis
- Function Approximation in Numerical Analysis
- Minimization and Maximization Approximations and Least Squares Approximations in Numerical Analysis
- Chebyshev Expansion
- Chebyshev Nodes
- Numerical Integration
- Trapezoidal Rule
- Simpson's Rule
- Newton-Cotes Integration Formulas
- Gaussian Quadrature
- Tricks for Variable Substitution to Compute Improper Integrals Numerically
- Laguerre Polynomials
- Hermite Polynomials
- Gaussian Quadrature for Numerically Computing Improper Integrals
- Lipschitz Condition
- Euler Method in Numerical Analysis
- Strong Lipschitz Condition and Error of the Euler Method
- Initial Value Variation and the Error in the Euler Method
- Multistep Methods
- Consistency and Order of Convergence of Multistep Methods
- Convergence and Error of Multistep Methods
- Midpoint Method
- Parasitic Solution
- Trapezoidal Method
- Richardson Error Estimation
- Adams Method
- Multistep Methods' Root Condition
- Stability and Root Conditions of Consistent Multistep Methods
- Convergence and Root Condition of Consistent Multistep Methods
- A-Stable
- Fourth-order Runge-Kutta method
- Numerical Solution to the Initial Value Problem for the Heat Equation Given Dirichlet Boundary Conditions
- Explicit Runge-Kutta Methods
- Implicit Runge-Kutta Methods
- Finite Difference Method
- Numerical Solutions of Heat Equations: Finite Difference Method
- Derivation of Finite Difference Using Multiple Points