Numerical analysis is the branch of mathematics that is used to find approximations to difficult problems such as finding the roots of non-linear equations, integration involving complex expressions and solving differential equations for which analytical solutions do not exist. It is applied to a wide variety of disciplines such as business, all fields of engineering, computer science, education, geology, meteorology, and others.
Years ago, high-speed computers did not exist, and if they did, the largest corporations could only afford them; consequently, the manual computation required lots of time and hard work. But now that computers have become indispensable for research work in science, engineering and other fields, numerical analysis has become a much easier and more pleasant task.
This book is written primarily for students/readers who have a good background of high-school
algebra, geometry, trigonometry, and the fundamentals of differential and integral calculus.* A prior knowledge of differential equations is desirable but not necessary; this topic is reviewed in Chapter 5.
One can use Fortran, Pascal, C, or Visual Basic or even a spreadsheet to solve a difficult problem. It is the opinion of this author that the best applications programs for solving engineering problems are 1) MATLAB which is capable of performing advanced mathematical and engineering computations, and 2) the Microsoft Excel spreadsheet since the versatility offered by spreadsheets have revolutionized the personal computer industry. We will assume that the reader has no prior knowledge of MATLAB and limited familiarity with Excel.
We intend to teach the student/reader how to use MATLAB via practical examples and for detailed explanations he/she will be referred to an Excel reference book or the MATLAB User’s Guide. The MATLAB commands, functions, and statements used in this text can be executed with either MATLAB Student Version 12 or later. Our discussions are based on a PC with Windows XP platforms but if you have another platform such as Macintosh, please refer to the appropriate sections of the MATLAB’s User Guide that also contains instructions for installation.
1. Introduction to MATLAB
2. Root Approximations
3. Sinusoids and Phasors
4. Matrices and Determinants
5. Differential Equations, State Variables, and State Equations
6. Fourier, Taylor, and Maclaurin Series
7. Finite Differences and Interpolation
8. Linear and Parabolic Regression
9. Solution of Differential Equations by Numerical Methods
10. Integration by Numerical Methods
11. Difference Equations
12. Partial Fraction Expansion
13. The Gamma and Beta Functions and Distributions
14. Orthogonal Functions and Matrix Factorizations
15. Bessel, Legendre, and Chebyshev Functions
16. Optimization Methods
A. Difference Equations in Discrete-Time Systems
B. Introduction to Simulink®
C. III-Conditioned Matrices
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