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Numerical Analysis using Matlab and Spreadsheet

Steven T. Karris

PDF, P.570, 9.23M

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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 ..................................................................................................................... 3-12Exponential 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..................................................................................................................5-47Chapter 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 .................................................................................................................................... 8-24Exercises .................................................................................................................................... 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...............................................................................................................12-22Chapter 13
The Gamma and Beta Functions and Distributions
The Gamma Function ................................................................................................................ 13-1
The Gamma Distribution ......................................................................................................... 13-15
The Beta Function.................................................................................................................... 13-17
The Beta Distribution............................................................................................................... 13-20
Summary .................................................................................................................................. 13-21
Exercises................................................................................................................................... 13-24
Solutions to Exercises ............................................................................................................... 13-25
Chapter 14
Orthogonal Functions and Matrix Factorizations
Orthogonal Functions ................................................................................................................14-1
Orthogonal Trajectories .............................................................................................................14-2
Orthogonal Vectors....................................................................................................................14-4
The Gram-Schmidt Orthogonalization Procedure .....................................................................14-7
The LU Factorization.................................................................................................................14-9
The Cholesky Factorization .....................................................................................................14-15
The QR Factorization...............................................................................................................14-17
Singular Value Decomposition ................................................................................................14-20
Summary.................................................................................................................................14-21
Exercises .................................................................................................................................14-23
Solutions to Exercises ..............................................................................................................14-25
Chapter 15
Bessel, Legendre, and Chebyshev Functions
The Bessel Function ................................................................................................................... 15-1
Legendre Functions .................................................................................................................. 15-10
Laguerre Polynomials................................................................................................................ 15-20
Chebyshev Polynomials ............................................................................................................ 15-21
Summary .................................................................................................................................. 15-26
Exercises................................................................................................................................... 15-32
Solutions to Exercises ............................................................................................................... 15-33Chapter 16
Optimization Methods
Linear Programming................................................................................................................... 16-1
Dynamic Programming............................................................................................................... 16-4
Network Analysis ..................................................................................................................... 16-14
Summary.................................................................................................................................. 16-19
Exercises .................................................................................................................................. 16-20
Solutions to Exercises............................................................................................................... 16-22

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