numericAnalysis

A repository that implements algorithms in the book "Numerical Analysis", Ninth Edition, by Richard L. Burden and J. Douglas Faires

To-Do List:

Next Step: Chapter 5

Chapter 2: Solutions of Equations in One Variable

Details

• Chapter 2.1 The Bisection Method
• Chapter 2.2 Fixed-Point Iteration
• Chapter 2.3 Newtons's Method and Its Extensions
• Chapter 2.4 Error Analysis for Iterative Methods
  • Modified Newton Method
• Chapter 2.5 Accelerating Convergence
  • Aitken's $\Delta^2$ Method
• Chapter 2.6 Zeros of Polynomials and Muller's Method
  • Horner's method incorperate with Newton's method to find the zeros of polynomial

Chapter 3: Interpolation and Polynomial Approximation

Details

• Chapter 3.1 Interpolation and the Lagrange Polynomial
• Chapter 3.2 Data Approximation and Neville's Method
• Chapter 3.3 Divided Differences
• Chapter 3.4 Hermite Interpolation
• Chapter 3.5 Cubic Spline Interpolation

Chapter 4: Numerical Differentiation and Integration

Details

• Chapter 4.1 Numerical Differentiation
• Chapter 4.2 Richardson's Extrapolation
  • Implement Richardson's Extrapolation algorithm for even order of $h$
• Chapter 4.3 Elements of Numerical Integration
  • Newton-Cotes Formula
• Chapter 4.4 Composite Numerical Integration
  • Visualization
  • General Composite Integral Algorithm
• Chapter 4.5 Romberg Integration
• Chapter 4.6 Adaptive Quadrature Methods
  • Adaptive Trapezoidal rule
  • Adaptive Closed Newton-Cotes
• Chapter 4.7 Gaussian Quadrature
  • Gaussian-Legendre Quadrature
• Chapter 4.8 Multiple Integrals
  • Double and Triple Closed Newton-Cotes

Chapter 5:

Details

• Chapter 5.2 Euler's Method
• Chapter 5.3 Higher-Order Taylor Methods
  • Taylor's method
• Chapter 5.4 Runge-Kutta Methods
  • More in detail about Runge-Kutta Methods
• Chapter 5.5 Error Control and the Runge-Kutta-Fehlberg Method
  • Implement Generalized Runge-Kutta Embedded
• Chapter 5.6 Multistep Method
  • Generalized Adams-Bashforth Algorithm
  • Milne-Simpson Predictor-Corrector
  • Generalized Predictor-Corrector Using Newton-Cotes Formulae
• Chapter 5.7 Variable Step-Size Multistep Method
  • Generalized Variable Step-Size MultiStep Method
• Chapter 5.8 Extrapolation
• Chapter 5.9 Higher-Order Equations and Systems of Differential Equations
• Chapter 5.11 Stiff Differential Equations

Chapter 6: Direct Method for Solving Linear Systems

Details

• Chapter 6.1 Linear Systems of Equations
• Chapter 6.3 Linear Algebra and Matrix Inversion
  • Algorithm to find the inverse of the matrix
• Chapter 6.4 The Determinant of a Matrix
  • Compute determinant using Gaussian Elimination
• Chapter 6.5 Matrix Factorization
  • PLU Decomposition
• Chapter 6.6 Special Types of Matrices
• PLDL' Decomposition (Refer to the book Matrix Computation)
• Remaining algorithm that solves tridiagonal linear system

Chapter 7: Iterative Techniques in Matrix Algebra

Details

• Chapter 7.3 The Jacobi and Gauss-Siedel Iterative Techniques
• Chapter 7.4 Relaxation Techniques for Solving Linear Systems
• Chaoter 7.6 The Conjugate Gradient Method
  • Conjugate Gradient Method
  • Biconjugate Gradient Method
  • Biconjuagte Gradient Stabilized Method
  • Minimal Residual Method
  • Generalized Minimal Residual Method

Chapter 8: Approximation Theory

Chapter 9: Approximating Eigenvalues

Chapter 10: Numerical Solutions of Nonlinear Systems of Equations

Chapter 11: Boundary-Value Problems for ODE

Chapter 12: Numerical Solutionns to PDE