Numerical methods

Tartalomjegyzék

• NUMERICAL METHODS
• COPYRIGHT PAGE
• 1. Introduction
• chevron_right2. Errors and Approximations
  • chevron_right2.1. Errors and Their Sources:
    • 2.1.1. Binary and Hexadecimal Numbers
    • 2.1.2. Floating Point and Rounddown Errors
    • 2.1.3. Absolute and Relative Errors
    • 2.1.4. Error Limits
    • 2.1.5. Error Propagation
    • 2.1.6. Subtracting Two Near Zero Numbers
  • chevron_right2.2. Iterative Methods
    • 2.2.1. Basic Iterative Method
    • 2.2.2. Convergence Rate
• chevron_right3. Single Variable Equations, System of Equations
  • chevron_right3.1. Solution of Single Variable Equations
    • 3.1.1. Bisection method
    • 3.1.2. Newton-Raphson Method
    • 3.1.3. Secant method
    • 3.1.4. Regula Falsi Method
    • 3.1.5. Fixed-point method
    • 3.1.6. Finding multiple roots
  • chevron_right3.2. Solution of System Equations
    • 3.2.1. Gauss elimination method
    • 3.2.2. LU factorization methods
    • chevron_right3.2.3. Iterative solutions
      • 3.2.3.1. Vector and matrix norms
      • 3.2.3.2. The Jacobi method
      • 3.2.3.3. Gauss-Seidel method
    • 3.2.4. Systems of nonlinear equations
• chevron_right4. Curve Fitting Methods
  • chevron_right4.1. Regression
    • 4.1.1. Linear Regression
    • chevron_right4.1.2. Linearization of Non-linear Input Data
      • 4.1.2.1. Exponential Function
      • 4.1.2.2. Power Function
      • 4.1.2.3. Saturation Function
    • 4.1.3. Polynomial Regression
  • chevron_right4.2. Finite difference
    • 4.2.1. Factorial Polynomials
    • 4.2.2. Anti-differentiation
  • chevron_right4.3. Interpolation
    • 4.3.1. Lagrange Interpolation
    • 4.3.2. Newton Divided-Difference Interpolation
    • chevron_right4.3.3. Spline Interpolation [3]
      • 4.3.3.1. Linear Splines
      • 4.3.3.2. Quadratic Splines
      • 4.3.3.3. Cubic Splines
      • 4.3.3.4. Constructing cubic splines with clamped boundary conditions
      • 4.3.3.5. Constructing Cubic splines with free boundary conditions
• chevron_right5. Numerical Differentiation
  • 5.1. 2-Point Backward Difference
  • 5.2. 2-Point Forward Difference
  • 5.3. 2-Point Central Difference
  • 5.4. 3-Points Backward Difference
  • 5.5. Numerical Difference Formulas
• chevron_right6. Numerical Integration
  • chevron_right6.1. Newton-Cotes Formulas
    • 6.1.1. Rectangular Rule
    • 6.1.2. Trapezoidal Rule
  • chevron_right6.2. Quadrature Formulas
    • 6.2.1. 1/3 Simpson Rule
    • 6.2.2. 3/8 Simpson Rule
  • 6.3. Romberg Integration
  • 6.4. Gaussian Quadrature
  • 6.5. Improper Integrals
  • 6.6. Numerical Integration of Bivariate Functions
• chevron_right7. Numerical Solution of Differential Equations – Initial Value Problems
  • chevron_right7.1. One-Step Methods
    • 7.1.1. Euler’s Method
    • 7.1.2. Second Order Taylor Method
    • 7.1.3. Runge-Kutta Methods
    • 7.1.4. Second Order Runge-Kutta Methods
    • 7.1.5. Improved Euler’s Method
    • 7.1.6. Heun’s Method
    • 7.1.7. Ralston’s Method
  • 7.2. Built in Functions in MATLAB
• chevron_right8. Partial Differential Equations
  • 8.1. Partial Differential Equations generally
  • chevron_right8.2. Elliptic Partial Differential Equations
    • chevron_right8.2.1. Dirichlet Problem
      • 8.2.1.1. Alternating Direction Implicit Methods
      • 8.2.1.2. Peaceman-Rachford Alternating Direction Implicit Method
    • 8.2.2. Neumann Problem
    • 8.2.3. Mixed problem
    • 8.2.4. Elliptic Partial Differential Equations in More Complex Regions
  • chevron_right8.3. Parabolic Partial Differential Equations
    • 8.3.1. Finite-Difference Method
    • 8.3.2. Crank-Nicolson Method
  • 8.4. Hyperbolic Partial Differential Equations
• References

Kiadó: Akadémiai Kiadó

Online megjelenés éve: 2019

ISBN: 978 963 454 283 4

DOI: 10.1556/9789634542834

This book on Numerical Methods is part of the Transportation and Vehicle Engineering Faculty’s curriculum at the Budapest University of Technology and Economics, created under the Stipendium Hungaricum Scholarship Programme. The book demonstrates the aim of developing and using numerical methods. Instead of relying on more complicated and expensive analytical solution methods, simpler arithmetical equations are used to approximate the accurate solution with the required level of accuracy. After discussing the various sources and propagation of solution errors, the book discusses the solution of single variable equations and system of equations. Following chapters describe the two main curve fitting methods; approximation and interpolation. These chapters are followed by the numerical differentiation and integration methods, then finally the numerical solutions for ordinary and partial differentiation equations are outlined. MATLAB is one of the most popular choice today for numerical computing, with its user friendly, yet powerful computing capability. It is practically an industry standard for prototyping and new developments, and there is great value for students to be introduced to solving problems with MATLAB. Many chapters in the notes show examples implemented in MATLAB, with actual working code available to use and practice.

Hivatkozás: https://mersz.hu/bicsak-numerical-methods//