Numerical Analysis Using MATLAB and Spreadsheets - Steven T. Karris.pdf

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Numerical
Using MATLAB® and Spreadsheets
Second Edition
Steven T. Karris
Orchard Publications
www.orchardpublications.com
Analysis
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Table of Contents
Chapter 1
Introduction to MATLAB
Command Window....................................................................................................................... 1-1
Roots of Polynomials..................................................................................................................... 1-3
Polynomial Construction from Known Roots .............................................................................. 1-4
Evaluation of a Polynomial at Specified Values ........................................................................... 1-5
Rational Polynomials .................................................................................................................... 1-7
Using MATLAB to Make Plots.................................................................................................... 1-9
Subplots....................................................................................................................................... 1-18
Multiplication, Division and Exponentiation............................................................................. 1-18
Script and Function Files............................................................................................................ 1-25
Display Formats .......................................................................................................................... 1-29
Summary ..................................................................................................................................... 1-30
Exercises...................................................................................................................................... 1-35
Solutions to Exercises ................................................................................................................. 1-36
Chapter 2
Root Approximations
Newton’s Method for Root Approximation ................................................................................. 2-1
Approximations with Spreadsheets .............................................................................................. 2-7
The Bisection Method for Root Approximation ........................................................................ 2-19
Summary ..................................................................................................................................... 2-27
Exercises...................................................................................................................................... 2-28
Solutions to Exercises ................................................................................................................. 2-29
Chapter 3
Sinusoids and Phasors
Alternating Voltages and Currents .............................................................................................. 3-1
Characteristics of Sinusoids .......................................................................................................... 3-2
Inverse Trigonometric Functions ............................................................................................... 3-10
Phasors ........................................................................................................................................ 3-10
Addition and Subtraction of Phasors ......................................................................................... 3-11
Multiplication of Phasors ............................................................................................................ 3-12
Division of Phasors ..................................................................................................................... 3-12
Numerical Analysis Using MATLAB and Spreadsheets, Second Edition
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Exponential and Polar Forms of Phasors ....................................................................................3-13
Summary......................................................................................................................................3-18
Exercises ......................................................................................................................................3-21
Solutions to Exercises..................................................................................................................3-22
Chapter 4
Matrices and Determinants
Matrix Definition ......................................................................................................................... 4-1
Matrix Operations........................................................................................................................ 4-2
Special Forms of Matrices ............................................................................................................ 4-5
Determinants................................................................................................................................ 4-9
Minors and Cofactors................................................................................................................. 4-12
Cramer’s Rule ............................................................................................................................. 4-16
Gaussian Elimination Method ................................................................................................... 4-18
The Adjoint of a Matrix............................................................................................................. 4-19
Singular and Non-Singular Matrices ......................................................................................... 4-20
The Inverse of a Matrix.............................................................................................................. 4-21
Solution of Simultaneous Equations with Matrices................................................................... 4-23
Summary..................................................................................................................................... 4-29
Exercises ..................................................................................................................................... 4-33
Solutions to Exercises................................................................................................................. 4-35
Chapter 5
Differential Equations, State Variables, and State Equations
Simple Differential Equations .......................................................................................................5-1
Classification .................................................................................................................................5-2
Solutions of Ordinary Differential Equations (ODE) ...................................................................5-5
Solution of the Homogeneous ODE .............................................................................................5-8
Using the Method of Undetermined Coefficients for the Forced Response...............................5-10
Using the Method of Variation of Parameters for the Forced Response ....................................5-19
Expressing Differential Equations in State Equation Form ........................................................5-23
Solution of Single State Equations..............................................................................................5-27
The State Transition Matrix .......................................................................................................5-28
Computation of the State Transition Matrix..............................................................................5-30
Eigenvectors ................................................................................................................................5-37
Summary......................................................................................................................................5-41
Exercises ......................................................................................................................................5-46
Solutions to Exercises..................................................................................................................5-47
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Chapter 6
Fourier, Taylor, and Maclaurin Series
Wave Analysis .............................................................................................................................. 6-1
Evaluation of the Coefficients ...................................................................................................... 6-2
Symmetry ...................................................................................................................................... 6-7
Waveforms in Trigonometric Form of Fourier Series................................................................. 6-12
Alternate Forms of the Trigonometric Fourier Series ................................................................ 6-25
The Exponential Form of the Fourier Series .............................................................................. 6-28
Line Spectra ................................................................................................................................ 6-33
Numerical Evaluation of Fourier Coefficients ............................................................................ 6-36
Power Series Expansion of Functions ......................................................................................... 6-37
Taylor and Maclaurin Series....................................................................................................... 6-40
Summary ..................................................................................................................................... 6-47
Exercises...................................................................................................................................... 6-50
Solutions to Exercises ................................................................................................................. 6-52
Chapter 7
Finite Differences and Interpolation
Divided Differences ...................................................................................................................... 7-1
Factorial Polynomials.................................................................................................................... 7-6
Antidifferences............................................................................................................................ 7-11
Newton’s Divided Difference Interpolation Method ................................................................. 7-15
Lagrange’s Interpolation Method ............................................................................................... 7-18
Gregory-Newton Forward Interpolation Method....................................................................... 7-19
Gregory-Newton Backward Interpolation Method .................................................................... 7-20
Interpolation with MATLAB ..................................................................................................... 7-23
Summary ..................................................................................................................................... 7-37
Exercises...................................................................................................................................... 7-42
Solutions to Exercises ................................................................................................................. 7-43
Chapter 8
Linear and Parabolic Regression
Curve Fitting................................................................................................................................. 8-1
Linear Regression.......................................................................................................................... 8-2
Parabolic Regression ..................................................................................................................... 8-7
Regression with Power Series Approximations .......................................................................... 8-14
Summary ..................................................................................................................................... 8-24
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Exercises ..................................................................................................................................... 8-26
Solutions to Exercises................................................................................................................. 8-28
Chapter 9
Solution of Differential Equations by Numerical Methods
Taylor Series Method ................................................................................................................... 9-1
Runge-Kutta Method ................................................................................................................... 9-5
Adams’ Method.......................................................................................................................... 9-13
Milne’s Method .......................................................................................................................... 9-16
Summary..................................................................................................................................... 9-17
Exercises ..................................................................................................................................... 9-20
Solutions to Exercises................................................................................................................. 9-21
Chapter 10
Integration by Numerical Methods
The Trapezoidal Rule................................................................................................................. 10-1
Simpson’s Rule ........................................................................................................................... 10-6
Summary................................................................................................................................... 10-13
Exercises ................................................................................................................................... 10-15
Solution to Exercises ................................................................................................................ 10-16
Chapter 11
Difference Equations
Definition, Solutions, and Applications..................................................................................... 11-1
Fibonacci Numbers .................................................................................................................... 11-7
Summary................................................................................................................................... 11-10
Exercises ................................................................................................................................... 11-13
Solutions to Exercises............................................................................................................... 11-14
Chapter 12
Partial Fraction Expansion
Partial Fraction Expansion ......................................................................................................... 12-1
Alternate Method of Partial Fraction Expansion .................................................................... 12-13
Summary................................................................................................................................... 12-18
Exercises ................................................................................................................................... 12-21
Solutions to Exercises............................................................................................................... 12-22
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Numerical Analysis Using MATLAB and Spreadsheets, Second Edition
Orchard Publications
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