Numerical Analysis

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Exponential and Polar Forms of Phasors ....................................................................................3-13Summary......................................................................................................................................3-18Exercises ......................................................................................................................................3-21Solutions to Exercises..................................................................................................................3-22Chapter 4Matrices and DeterminantsMatrix Definition ......................................................................................................................... 4-1Matrix Operations........................................................................................................................ 4-2Special Forms of Matrices ............................................................................................................ 4-5Determinants................................................................................................................................ 4-9Minors and Cofactors................................................................................................................. 4-12Cramer’s Rule............................................................................................................................. 4-16Gaussian Elimination Method ................................................................................................... 4-18The Adjoint of a Matrix............................................................................................................. 4-19Singular and Non-Singular Matrices ......................................................................................... 4-20The Inverse of a Matrix.............................................................................................................. 4-21Solution of Simultaneous Equations with Matrices................................................................... 4-23Summary..................................................................................................................................... 4-29Exercises ..................................................................................................................................... 4-33Solutions to Exercises................................................................................................................. 4-35Chapter 5Differential Equations, State Variables, and State EquationsSimple Differential Equations .......................................................................................................5-1Classification .................................................................................................................................5-2Solutions of Ordinary Differential Equations (ODE)...................................................................5-5Solution of the Homogeneous ODE .............................................................................................5-8Using the Method of Undetermined Coefficients for the Forced Response...............................5-10Using the Method of Variation of Parameters for the Forced Response....................................5-19Expressing Differential Equations in State Equation Form ........................................................5-23Solution of Single State Equations..............................................................................................5-27The State Transition Matrix.......................................................................................................5-28Computation of the State Transition Matrix..............................................................................5-30Eigenvectors ................................................................................................................................5-37Summary......................................................................................................................................5-41Exercises ......................................................................................................................................5-46Solutions to Exercises..................................................................................................................5-47iiNumerical Analysis Using MATLAB and Spreadsheets, Second EditionOrchard Publications
Chapter 6Fourier, Taylor, and Maclaurin SeriesWave Analysis .............................................................................................................................. 6-1Evaluation of the Coefficients ...................................................................................................... 6-2Symmetry ...................................................................................................................................... 6-7Waveforms in Trigonometric Form of Fourier Series................................................................. 6-12Alternate Forms of the Trigonometric Fourier Series................................................................ 6-25The Exponential Form of the Fourier Series .............................................................................. 6-28Line Spectra ................................................................................................................................ 6-33Numerical Evaluation of Fourier Coefficients............................................................................ 6-36Power Series Expansion of Functions......................................................................................... 6-37Taylor and Maclaurin Series....................................................................................................... 6-40Summary ..................................................................................................................................... 6-47Exercises...................................................................................................................................... 6-50Solutions to Exercises ................................................................................................................. 6-52Chapter 7Finite Differences and InterpolationDivided Differences ...................................................................................................................... 7-1Factorial Polynomials.................................................................................................................... 7-6Antidifferences............................................................................................................................ 7-11Newton’s Divided Difference Interpolation Method ................................................................. 7-15Lagrange’s Interpolation Method ............................................................................................... 7-18Gregory-Newton Forward Interpolation Method....................................................................... 7-19Gregory-Newton Backward Interpolation Method .................................................................... 7-20Interpolation with MATLAB ..................................................................................................... 7-23Summary ..................................................................................................................................... 7-37Exercises...................................................................................................................................... 7-42Solutions to Exercises ................................................................................................................. 7-43Chapter 8Linear and Parabolic RegressionCurve Fitting................................................................................................................................. 8-1Linear Regression.......................................................................................................................... 8-2Parabolic Regression ..................................................................................................................... 8-7Regression with Power Series Approximations .......................................................................... 8-14Summary ..................................................................................................................................... 8-24Numerical Analysis Using MATLAB and Spreadsheets, Second EditionOrchard Publicationsiii
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Numerical Analysis

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