Numerical Analysis

No matter how great mathematics is, not all problems in the world can be solved with just paper and a pen. Numerical analysis is a methodology that uses machines to approximate solutions to problems, making it an indispensable tool in applied mathematical research.

Functions and Integration

Differentiation

• Finite Difference $\Delta$
• Finite Difference Method (FDM)
  • Derivation of Finite Differences Using Multiple Points
• 🔒(25/04/07) Complex-Step Approximation for Differentiation

Interpolation

• Divided Differences $f \left[ x_{0} , \cdots , x_{n} \right]$
  • Hermite-Genocchi Formula
• Interpolation
• Polynomial Interpolation
  • Derivation of the Lagrange Formula
  • Derivation of Newton’s Divided Difference Formula
• Hermite Interpolation
• Splines
  • B-Splines

Function Approximation

• Function Approximation
  • Minimax Approximation and Least Squares Approximation
• Chebyshev Polynomials of the First Kind $T_{n}$
• Chebyshev Polynomials of the Second Kind $U_{n}$
• Relationship Between First and Second Kind Chebyshev Polynomials
• Chebyshev Expansion
• Chebyshev Nodes

Numerical Integration

• Numerical Integration
• Trapezoidal Rule $I_{n}^{1}(f)$
• Simpson’s Rule $I_{n}^{2}(f)$
• Newton-Cotes Integration Formula $I_{n}^{p}(f)$
• Variable Substitution Trick for Evaluating Improper Integrals Numerically
• Gaussian Quadrature
• Laguerre Polynomials $L_{n}$
• Hermite Polynomials $H_{e_{n}}$, $H_{n}$
• Gaussian Quadrature for Evaluating Improper Integrals Numerically

Equations

Numerical Solutions of Algebraic Equations

• Rate of Convergence
• Bisection Method
• Newton-Raphson Method
  • Newton’s Method for Solving Nonlinear Systems
• Secant Method
• Müller’s Method

Numerical Solutions of Differential Equations

• What Is a Differential Equation Solver?
  • 🔒(25/06/28) Situations Where Implicit Methods Are Preferred Over Explicit Methods
  • 🔒(25/07/02) A Trick to Solve Ordinary Differential Equations Backwards in Time
• Lipschitz Condition
• Euler’s Method
  • Strong Lipschitz Condition and Euler’s Method Error
  • Euler’s Method Error Due to Slight Variations in Initial Values
• Multistep Methods
  • Consistency and Order of Convergence of Multistep Methods
  • Convergence and Error of Multistep Methods
• Parasitic Solutions
  • Midpoint Method
• Trapezoidal Method
• Richardson Error Estimation
• Adams Method
• Root Condition of Multistep Methods
  • Stability and Root Condition of Consistent Multistep Methods
  • Convergence and Root Condition of Consistent Multistep Methods
• A-Stability
• Explicit Runge-Kutta Methods (ERK)
  • Fourth-Order Runge-Kutta Method (RK4)
  • Determining Coefficients
• Implicit Runge-Kutta Methods (IRK)

Partial Differential Equations

• Heat Equation: Finite Difference Method (FDM)
• Heat Equation: The Method of Lines
• Wave Equation: Finite Difference Method (FDM)
• Wave Equation: $k$-Space Method
• Wave Equation: Absorbing Boundary Conditions

References

• Atkinson. (1989). An Introduction to Numerical Analysis(2nd Edition)