Introduction to Programming in Excel

1. Introduction  Introduction to Programming in Excel  The Taylor Series  Error Propagation  Total Numerical Error  Blunders, Formulation Errors and Data Uncertainty

2. Roots of Equations – Bracketing Methods  Graphical Methods  The Bisection Method  The False-Position Method  Incremental Searches and Initial Guesses

3. Roots of Equations – Open Methods  Simple Fixed-Point Iteration  The Newton-Rapheson Method  The Secant Method, the Modified Secant Method  Multiple Roots

4. Roots of Equations – Polynomials  Computing with Polynomials  Conventional Methods  Muller’s Method  Bairstow’s Method

5. Linear Algebraic Equations – Gauss Elimination  Naïve Gauss Elimination  Pitfalls of Elimination Methods  Techniques for Improving Solutions  Gauss-Jordan

6. Linear Algebraic Equations – Decomposition and Matrix Inversion  LU Decomposition o Gauss Elimination o Crout Decomposition  The Matrix Inverse  Error Analysis and System Condition

7. Special Matrices  Special Matrices o Banded matrices o Tridiagonal systems (Thomas algorithm) o Cholesky Decomposition  Gauss-Seidel 8. One-Dimensional Unconstrained Optimization  Golden-Section Search  Quadratic Interpolation  Newton’s Method

9. Multidimensional Unconstrained Optimization  Direct Methods  Gradient Methods

10. Constrained Optimization  Linear Programming

11. Curve Fitting  Linear Regression  Polynomial Regression  Multiple Linear Regression  General Least Squares  Nonlinear Regression

12. Interpolation  Newton’s Divided-Difference Polynomials  Lagrange Interpolating Polynomials  Inverse Interpolation  Spline Interpolation

13. Fourier Approximation  Curve Fitting with Sinusoid Functions  Continuous Fourier Series  Frequency and Time Domains  Fourier Integral and Transform  Discrete Fourier Transform (DFT)  Fast Fourier Transform (FFT)  The Power Spectrum

14. Newton-Cotes Integration Formulas  The Trapezoidal Rule  Simpson’s Rules  Integration with Unequal Segments  Open Integration Formulas  Multiple Integrals 15. Integration of Equations  Newton-Cotes Formulas for Equations  Romberg Integration  Gauss Quadrature  Improper Integrals

16. Numerical Differentiation  High-Accuracy Differentiation Formulas  Richardson Extrapolation  Derivatives of Unequally-Spaced Data  Derivatives and Integrals of Data with Errors

17. ODE – Runge-Kutta Methods  Euler’s Method  Improvements to Euler’s Method  Runge-Kutta Methods  Systems of Equations  Adaptive Runge-Kutta Methods

18. ODE – Stiffness and Multistep Methods  Stiffness  Multistep Methods

19. ODE – Boundary-Value and Eigenvalue Problems  General Methods for Boundary-Value Problems  Eigenvalue Problems

20. PDE – Finite Difference: Elliptic Equations  The Laplace Equation  Solution Techniques  Boundary Conditions  The Control-Volume Approach

21. PDE – Finite Difference: Parabolic Equations  The Heat Conduction Equation  Explicit Methods  A Simple Implicit Method  The Crank-Nicholson Method  Parabolic Equations in Two Spatial Dimensions

22. PDE – Finite-Element Method  The General Approach  FEA in One-Dimension  Two-Dimensional Problems