Numerical Analysis

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