Page 8 - Numerical Analysis Using MATLAB and Excel

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

ii Numerical Analysis Using MATLAB® and Excel®, Third Edition
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