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Numerical analysis using MATLAB and Excel by Steven T. Karris

Table of Contents
1 Introduction to MATLAB 1−1
1.1 Command Window.................................................................................................1−1
1.2 Roots of Polynomials...............................................................................................1−3
1.3 Polynomial Construction from Known Roots ........................................................1−4
1.4 Evaluation of a Polynomial at Specified Values .....................................................1−5
1.5 Rational Polynomials ..............................................................................................1−8
1.6 Using MATLAB to Make Plots..............................................................................1−9
1.7 Subplots.................................................................................................................1−18
1.8 Multiplication, Division and Exponentiation.......................................................1−19
1.9 Script and Function Files......................................................................................1−26
1.10 Display Formats ....................................................................................................1−31
1.11 Summary ...............................................................................................................1−33
1.12 Exercises................................................................................................................1−37
1.13 Solutions to End−of−Chapter Exercises ...............................................................1−38
MATLAB Computations: Entire chapter
2 Root Approximations 2−1
2.1 Newton’s Method for Root Approximation...........................................................2−1
2.2 Approximations with Spreadsheets........................................................................2−7
2.3 The Bisection Method for Root Approximation .................................................2−19
2.4 Summary...............................................................................................................2−27
2.5 Exercises ...............................................................................................................2−28
2.6 Solutions to End−of−Chapter Exercises...............................................................2−29
MATLAB Computations: Pages 2−2 through 2−7, 2−14, 2−21 through 2−23,
2−29 through 2−34
Excel Computations: Pages 2−8 through 2−19, 2−24 through 2−26
3 Sinusoids and Phasors 3−1
3.1 Alternating Voltages and Currents ........................................................................3−1
3.2 Characteristics of Sinusoids....................................................................................3−2
3.3 Inverse Trigonometric Functions .........................................................................3−10
3.4 Phasors..................................................................................................................3−10
3.5 Addition and Subtraction of Phasors ...................................................................3−11
3.6 Multiplication of Phasors......................................................................................3−12
3.7 Division of Phasors ...............................................................................................3−13

ii Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications
3.8 Exponential and Polar Forms of Phasors ..............................................................3−13
3.9 Summary ...............................................................................................................3−24
3.10 Exercises................................................................................................................3−27
3.11 Solutions to End−of−Chapter Exercises................................................................3−28
MATLAB Computations: Pages 3−15 through 3−23, 3−28 through 3−31
Simulink Modeling: Pages 3−16 through 3−23
4 Matrices and Determinants 4−1
4.1 Matrix Definition.....................................................................................................4−1
4.2 Matrix Operations ...................................................................................................4−2
4.3 Special Forms of Matrices........................................................................................4−5
4.4 Determinants...........................................................................................................4−9
4.5 Minors and Cofactors ............................................................................................4−13
4.6 Cramer’s Rule ........................................................................................................4−18
4.7 Gaussian Elimination Method...............................................................................4−20
4.8 The Adjoint of a Matrix ........................................................................................4−22
4.9 Singular and Non−Singular Matrices ....................................................................4−22
4.10 The Inverse of a Matrix.........................................................................................4−23
4.11 Solution of Simultaneous Equations with Matrices ..............................................4−25
4.12 Summary................................................................................................................4−32
4.13 Exercises ................................................................................................................4−36
4.14 Solutions to End−of−Chapter Exercises ................................................................4−38
MATLAB Computations: Pages 4−3, 4−5 through 4−8, 4−10, 4−12, 4−3, 4−5, 4−19
 through 4−20, 4−24, 4−26, 4−28, 4−30, 4−38, 4−41, 4−43
Excel Computations: Pages 4−28 through 4−29, 4−42 through 4−43
5 Differential Equations, State Variables, and State Equations 5−1
5.1 Simple Differential Equations..................................................................................5−1
5.2 Classification............................................................................................................5−2
5.3 Solutions of Ordinary Differential Equations (ODE) .............................................5−6
5.4 Solution of the Homogeneous ODE ...................................................................... 5−8
5.5 Using the Method of Undetermined Coefficients for the Forced Response........ 5−10
5.6 Using the Method of Variation of Parameters for the Forced Response ............. 5−20
5.7 Expressing Differential Equations in State Equation Form.................................. 5−24
5.8 Solution of Single State Equations....................................................................... 5−27
5.9 The State Transition Matrix ................................................................................ 5−28
5.10 Computation of the State Transition Matrix...................................................... 5−30
5.11 Eigenvectors.......................................................................................................... 5−38
5.12 Summary.............................................................................................................. 5−42
Numerical Analysis Using MATLAB® and Excel®, Third Edition iii
Copyright © Orchard Publications
5.13 Exercises ............................................................................................................... 5−47
5.14 Solutions to End−of−Chapter Exercises............................................................... 5−49
MATLAB Computations: Pages 5−11, 5−13 through 5−14, 5−16 through 5−17,
5−19, 5−23, 5−33 through 5−35, 5−37,
5−49 through 5−53, 5−55
6 Fourier, Taylor, and Maclaurin Series 6−1
6.1 Wave Analysis ........................................................................................................6−1
6.2 Evaluation of the Coefficients ...............................................................................6−2
6.3 Symmetry ...............................................................................................................6−7
6.4 Waveforms in Trigonometric Form of Fourier Series .........................................6−12
6.5 Alternate Forms of the Trigonometric Fourier Series .........................................6−25
6.6 The Exponential Form of the Fourier Series .......................................................6−29
6.7 Line Spectra .........................................................................................................6−33
6.8 Numerical Evaluation of Fourier Coefficients .....................................................6−36
6.9 Power Series Expansion of Functions ..................................................................6−40
6.10 Taylor and Maclaurin Series ................................................................................6−41
6.11 Summary ..............................................................................................................6−48
6.12 Exercises ..............................................................................................................6−51
6.13 Solutions to End−of−Chapter Exercises ..............................................................6−53
MATLAB Computations: Pages 6−35, 6−45, 6−58 through 6−61
Excel Computations: Pages 6−37 through 6−39
7 Finite Differences and Interpolation 7−1
7.1 Divided Differences ...............................................................................................7−1
7.2 Factorial Polynomials .............................................................................................7−6
7.3 Antidifferences ...................................................................................................7−12
7.4 Newton’s Divided Difference Interpolation Method .........................................7−15
7.5 Lagrange’s Interpolation Method ........................................................................7−17
7.6 Gregory−Newton Forward Interpolation Method ..............................................7−19
7.7 Gregory−Newton Backward Interpolation Method ...........................................7−21
7.8 Interpolation with MATLAB .............................................................................7−24
7.9 Summary .............................................................................................................7−39
7.10 Exercises .............................................................................................................7−44
7.11 Solutions to End−of−Chapter Exercises .............................................................7−45
MATLAB Computations: Pages 7−8 through 7−9, 7−13 through 7−15,
7−26 through 7−38, 7−45 through 7−46,
7−48, 7−50, 7−52
Excel Computations: Pages 7−17 through 7−19, 7−22 through 7−25, 7−49, 7−52

iv Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications
8 Linear and Parabolic Regression 8−1
8.1 Curve Fitting ..........................................................................................................8−1
8.2 Linear Regression ...................................................................................................8−2
8.3 Parabolic Regression ..............................................................................................8−7
8.4 Regression with Power Series Approximations ....................................................8−14
8.5 Summary ..............................................................................................................8−24
8.6 Exercises ...............................................................................................................8−26
8.7 Solutions to End−of−Chapter Exercises ...............................................................8−28
MATLAB Computations: Pages 8−11 through 8−14, 8−17 through 8−23,
8−30 through 8−34
Excel Computations: Pages 8−5 through 8−10, 8−15 through 8−19, 8−28 through 8−32
9 Solution of Differential Equations by Numerical Methods 9−1
9.1 Taylor Series Method ............................................................................................ 9−1
9.2 Runge−Kutta Method ............................................................................................ 9−5
9.3 Adams’ Method ................................................................................................... 9−13
9.4 Milne’s Method .................................................................................................... 9−15
9.5 Summary .............................................................................................................. 9−17
9.6 Exercises .............................................................................................................. 9−20
9.7 Solutions to End−of−Chapter Exercises .............................................................. 9−21
MATLAB Computations: Pages 9−5, 9−9 through 9−12, 9−21 through 9−23
Excel Computations: Page 9−2, 9−14, 9−22 through 9−26
10 Integration by Numerical Methods 10−1
10.1 The Trapezoidal Rule .......................................................................................... 10−1
10.2 Simpson’s Rule ..................................................................................................... 10−6
10.3 Summary ............................................................................................................ 10−14
10.4 Exercises ............................................................................................................ 10−15
10.5 Solution to End−of−Chapter Exercises .............................................................. 10−16
MATLAB Computations: Pages 10−3 through 10−6, 10−9 through 10−13,
10−16, 10−18 through 10−21
Excel Computations: Pages 10−10, 10−19 through 10−21
11 Difference Equations 11−1
11.1 Introduction ......................................................................................................... 11−1
11.2 Definition, Solutions, and Applications .............................................................. 11−1
11.3 Fibonacci Numbers .............................................................................................. 11−7
Numerical Analysis Using MATLAB® and Excel®, Third Edition v
Copyright © Orchard Publications
11.4 Summary .............................................................................................................11−11
11.5 Exercises ............................................................................................................. 11−13
11.6 Solutions to End−of−Chapter Exercises .............................................................11−14
12 Partial Fraction Expansion 12−1
12.1 Partial Fraction Expansion ..................................................................................12−1
12.2 Alternate Method of Partial Fraction Expansion ..............................................12−13
12.3 Summary ............................................................................................................12−19
12.4 Exercises ............................................................................................................12−22
12.5 Solutions to End−of−Chapter Exercises ............................................................12−23
MATLAB Computations: Pages 12−3 through 12−5, 12−9 through 12−12,
12−16 through 12-18, 12−23 through 12−28
13 The Gamma and Beta Functions and Distributions 13−1
13.1 The Gamma Function .........................................................................................13−1
13.2 The Gamma Distribution ..................................................................................13−16
13.3 The Beta Function .............................................................................................13−17
13.4 The Beta Distribution ........................................................................................13−20
13.5 Summary ............................................................................................................13−22
13.6 Exercises ............................................................................................................13−24
13.7 Solutions to End−of−Chapter Exercises ............................................................13−25
MATLAB Computations: Pages 13−3, 13−5, 13−10, 13−19, 13−25
Excel Computations: Pages 13−5, 13−10, 13−16 through 13−17, 13−19, 13−21
14 Orthogonal Functions and Matrix Factorizations 14−1
14.1 Orthogonal Functions ......................................................................................14−1
14.2 Orthogonal Trajectories ...................................................................................14−2
14.3 Orthogonal Vectors ..........................................................................................14−4
14.4 The Gram−Schmidt Orthogonalization Procedure ..........................................14−7
14.5 The LU Factorization .......................................................................................14−9
14.6 The Cholesky Factorization ............................................................................14−23
14.7 The QR Factorization .....................................................................................14−25
14.8 Singular Value Decomposition .......................................................................14−28
14.9 Summary .........................................................................................................14−30
14.10 Exercises .........................................................................................................14−32
14.11 Solutions to End−of−Chapter Exercises .........................................................14−34
MATLAB Computations: Pages 14−8 through 14−9, 14−11 through 14−29,
14−36, 14−38 through 14−39

vi Numerical Analysis Using MATLAB® and Excel®, Third Edition
Copyright © Orchard Publications
15 Bessel, Legendre, and Chebyshev Functions 15−1
15.1 The Bessel Function ............................................................................................15−1
15.2 Legendre Functions ...........................................................................................15−10
15.3 Laguerre Polynomials .........................................................................................15−21
15.4 Chebyshev Polynomials .....................................................................................15−22
15.5 Summary ............................................................................................................15−27
15.6 Exercises .............................................................................................................15−32
15.7 Solutions to End−of−Chapter Exercises ............................................................15−33
MATLAB Computations: Pages 15−3 through 15−4, 15−6, 15−9,
14−19 through 15−22, 15−25, 15−33, 15−35 through 15−37
Excel Computations: Pages 15−5, 15−9
16 Optimization Methods 16−1
16.1 Linear Programming ........................................................................................... 16−1
16.2 Dynamic Programming ........................................................................................16−4
16.3 Network Analysis ...............................................................................................16−14
16.4 Summary ............................................................................................................16−19
16.5 Exercises .............................................................................................................15−20
16.6 Solutions to End−of−Chapter Exercises ............................................................15−22
MATLAB Computations: Pages 16−3
Excel Computations: Pages 16−4, 16−23, 16−25 through 16−27
A Difference Equations in Discrete−Time Systems A−1
A.1 Recursive Method for Solving Difference Equations........................................... A−1
A.2 Method of Undetermined Coefficients ................................................................A−1
MATLAB Computations: Pages A−4, A−7, A−9
B Introduction to Simulink® B−1
B.1 Simulink and its Relation to MATLAB ...............................................................B−1
B.2 Simulink Demos ..................................................................................................B−20
MATLAB Computations and Simulink Modeling: Entire Appendix B
C Ill-Conditioned Matrices C−1
C.1 The Norm of a Matrix ...........................................................................................C−1
C.2 Condition Number of a Matrix .............................................................................C−2
C.3 Hilbert Matrices ....................................................................................................C−3
Numerical Analysis Using MATLAB® and Excel®, Third Edition vii
Copyright © Orchard Publications
MATLAB Computations: Pages C−1, C−4 through C−5
References R−1
Index IN−1