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

• Difference $\Delta$
• Finite Difference Method (FDM)
  • Finite Difference Derivation Using Multiple Points

Interpolation

• Divided Differences $f[x_0, \ldots, x_n]$
  • Hermite-Genocchi Formula
• Interpolation
• Polynomial Interpolation
  • Derivation of Lagrange Formula
  • Derivation of Newton 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 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

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

Numerical Solutions of Differential Equations

• Lipschitz Condition
• Euler Method
  • Strong Lipschitz Condition and Error of the Euler Method
  • Error of the Euler Method with Slightly Different 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 Methods
• 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)
  • 4th Order Runge-Kutta Method (RK4)
  • How to Choose Coefficients
• 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)