Foundations of Mathematical and ComputationalEconomics

Second Edition

Kamran Dadkhah

Foundations of Mathematicaland ComputationalEconomics

Second Edition

123

Kamran DadkhahNortheastern UniversityDepartment of EconomicsBostonUSAk.dadkhah@neu.edu

First edition published by Thomson South-Western 2007, ISBN 978-0324235838

ISBN 978-3-642-13747-1 e-ISBN 978-3-642-13748-8DOI 10.1007/978-3-642-13748-8Springer Heidelberg Dordrecht London New York

© Springer-Verlag Berlin Heidelberg 2007, 2011This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcastingreproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.

Cover design: WMXDesign, Heidelberg

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

To Karen and my daughter, Lara

Preface

Mathematics is both a language of its own and a way of thinking; applyingmathematics to economics reveals that mathematics is indeed inherent to eco-nomic life. The objective of this book is to teach mathematical knowledge andcomputational skills required for macro and microeconomic analysis, as well aseconometrics. In addition, I hope it conveys a deeper understanding and appreciationof mathematics.

Examples in the following chapters are chosen from all areas of economicsand econometrics. Some have very practical applications, such as determiningmonthly mortgage payments; others involve more abstract models, such as sys-tems of dynamic equations. Some examples are familiar in the study of micro andmacroeconomics; others involve less well-known and more recent models, such asreal business cycle theory.

Increasingly, economists need to make complicated calculations. Systems ofdynamic equations are used to forecast different economic variables several yearsinto the future. Such systems are used to assess the effects of alternative policies,such as different methods of financing Social Security over a few decades. Also,many theories in microeconomics, industrial organization, and macroeconomicsrequire modeling the behavior and interactions of many decision makers. Thesetypes of calculation require computational dexterity. Thus, this book provides anintroduction to numerical methods, computation, and programming with Excel andMatlab. In addition, because of the increasing use of computer software such asMaple and Mathematica, sections are included to introduce the student to differenti-ation, integration, and solving difference and differential equations using Maple andto the concept of computer-aided mathematical proof.

The second edition differs from the first in several respects. Parts of the bookare rearranged, some materials are deleted and some new topics and examples areadded. In the first edition most computational examples used Matlab and someExcel. In the present edition, Excel and Matlab are given equal weights. These aredone in the hope of making the book more reader friendly. Similarly, more useis made of the Maple program for solving non-numerical problems. Finally, manyerrors had crept into the first edition, which are corrected in the present edition. I amindebted to students in my math and stat classes for pointing out some of them.

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I would like to thank Barbara Fess of Springer-Verlag for her support in preparingthis new edition. I also would like to thank Saranya Baskar and her colleagues atIntegra for their excellent work in producing the book.

As always, my greatest appreciation is to Karen Challberg, who during the entireproject gave me support, encouragement, and love.

Contents

Part I Basic Concepts and Methods

1 Mathematics, Computation, and Economics . . . . . . . . . . . . 31.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Philosophies of Mathematics . . . . . . . . . . . . . . . . . . 91.3 Women in Mathematics . . . . . . . . . . . . . . . . . . . . . 111.4 Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Mathematics and Economics . . . . . . . . . . . . . . . . . . 141.6 Computation and Economics . . . . . . . . . . . . . . . . . . 14

2 Basic Mathematical Concepts and Methods . . . . . . . . . . . . . 172.1 Functions of Real Variables . . . . . . . . . . . . . . . . . . . 17

2.1.1 Variety of Economic Relationships . . . . . . . . . . 222.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Summation Notation � . . . . . . . . . . . . . . . . 242.2.2 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.3 Convergent and Divergent Series . . . . . . . . . . . 272.2.4 Arithmetic Progression . . . . . . . . . . . . . . . . 292.2.5 Geometric Progression . . . . . . . . . . . . . . . . . 312.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Permutations, Factorial, Combinations, and theBinomial Expansion . . . . . . . . . . . . . . . . . . . . . . . 352.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 37

2.4 Logarithm and Exponential Functions . . . . . . . . . . . . . 382.4.1 Logarithm . . . . . . . . . . . . . . . . . . . . . . . 382.4.2 Base of Natural Logarithm, e . . . . . . . . . . . . . 402.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 41

2.5 Mathematical Proof . . . . . . . . . . . . . . . . . . . . . . . 422.5.1 Deduction, Mathematical Induction, and

Proof by Contradiction . . . . . . . . . . . . . . . . . 422.5.2 Computer-Assisted Mathematical Proof . . . . . . . . 442.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 45

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2.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . 462.6.1 Cycles and Frequencies . . . . . . . . . . . . . . . . 502.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 51

2.7 Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . 512.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 56

3 Basic Concepts of Computation . . . . . . . . . . . . . . . . . . . 573.1 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . 57

3.1.1 Naming Cells in Excel . . . . . . . . . . . . . . . . . 603.2 Absolute and Relative Computation Errors . . . . . . . . . . . 613.3 Efficiency of Computation . . . . . . . . . . . . . . . . . . . 623.4 o and O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5 Solving an Equation . . . . . . . . . . . . . . . . . . . . . . . 663.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4 Basic Concepts and Methods of Probability Theory and Statistics 694.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.2 Random Variables and Probability Distributions . . . . . . . . 724.3 Marginal and Conditional Distributions . . . . . . . . . . . . 744.4 The Bayes Theorem . . . . . . . . . . . . . . . . . . . . . . . 794.5 The Law of Iterated Expectations . . . . . . . . . . . . . . . . 814.6 Continuous Random Variables . . . . . . . . . . . . . . . . . 824.7 Correlation and Regression . . . . . . . . . . . . . . . . . . . 854.8 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . 884.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

Part II Linear Algebra

5 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.1 Vectors and Vector Space . . . . . . . . . . . . . . . . . . . . 96

5.1.1 Vector Space . . . . . . . . . . . . . . . . . . . . . . 1005.1.2 Norm of a Vector . . . . . . . . . . . . . . . . . . . . 1025.1.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . 1045.1.4 Angle Between Two Vectors

and the Cauchy-Schwarz Theorem . . . . . . . . . . 1055.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2 Orthogonal Vectors . . . . . . . . . . . . . . . . . . . . . . . 1095.2.1 Gramm-Schmidt Algorithm . . . . . . . . . . . . . . 1095.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 111

6 Matrices and Matrix Algebra . . . . . . . . . . . . . . . . . . . . 1136.1 Basic Definitions and Operations . . . . . . . . . . . . . . . . 113

6.1.1 Systems of Linear Equations . . . . . . . . . . . . . 1186.1.2 Computation with Matrices . . . . . . . . . . . . . . 1216.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2 The Inverse of a Matrix . . . . . . . . . . . . . . . . . . . . . 1236.2.1 A Number Called the Determinant . . . . . . . . . . 127

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6.2.2 Rank and Trace of a Matrix . . . . . . . . . . . . . . 1326.2.3 Another Way to Find the Inverse of a Matrix . . . . . 1336.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 135

6.3 Solving Systems of Linear Equations Using Matrix Algebra . . 1376.3.1 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . 1396.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 142

7 Advanced Topics in Matrix Algebra . . . . . . . . . . . . . . . . . 1437.1 Quadratic Forms and Positive and Negative

Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . 1437.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2 Generalized Inverse of a Matrix . . . . . . . . . . . . . . . . . 1477.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 150

7.3 Orthogonal Matrices . . . . . . . . . . . . . . . . . . . . . . 1507.3.1 Orthogonal Projection . . . . . . . . . . . . . . . . . 1507.3.2 Orthogonal Complement of a Matrix . . . . . . . . . 1527.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 153

7.4 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . 1537.4.1 Complex Eigenvalues . . . . . . . . . . . . . . . . . 1597.4.2 Repeated Eigenvalues . . . . . . . . . . . . . . . . . 1607.4.3 Eigenvalues and the Determinant and Trace of

a Matrix . . . . . . . . . . . . . . . . . . . . . . . . 1647.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 166

7.5 Factorization of Symmetric Matrices . . . . . . . . . . . . . . 1677.5.1 Some Interesting Properties of Symmetric Matrices . 1677.5.2 Factorization of Matrix with Real Distinct Roots . . . 1707.5.3 Factorization of a Positive Definite Matrix . . . . . . 1727.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 176

7.6 LU Factorization of a Square Matrix . . . . . . . . . . . . . . 1767.6.1 Cholesky Factorization . . . . . . . . . . . . . . . . 1817.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 182

7.7 Kronecker Product and Vec Operator . . . . . . . . . . . . . . 1837.7.1 Vectorization of a Matrix . . . . . . . . . . . . . . . 1857.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 185

Part III Calculus

8 Differentiation: Functions of One Variable . . . . . . . . . . . . . 1898.1 Marginal Analysis in Economics . . . . . . . . . . . . . . . . 189

8.1.1 Marginal Concepts and Derivatives . . . . . . . . . . 1908.1.2 Comparative Static Analysis . . . . . . . . . . . . . . 1928.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 193

8.2 Limit and Continuity . . . . . . . . . . . . . . . . . . . . . . 1948.2.1 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 1948.2.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . 1968.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 198

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8.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 1988.3.1 Geometric Representation of Derivative . . . . . . . . 2008.3.2 Differentiability . . . . . . . . . . . . . . . . . . . . 2018.3.3 Rules of Differentiation . . . . . . . . . . . . . . . . 2048.3.4 Properties of Derivatives . . . . . . . . . . . . . . . . 2078.3.5 l’Hôpital’s Rule . . . . . . . . . . . . . . . . . . . . 2148.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 215

8.4 Monotonic Functions and the Inverse Rule . . . . . . . . . . . 2168.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 219

8.5 Second- and Higher-Order Derivatives . . . . . . . . . . . . . 2208.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 221

8.6 Differential . . . . . . . . . . . . . . . . . . . . . . . . . . . 2218.6.1 Second- and Higher-Order Differentials . . . . . . . . 2238.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 224

8.7 Computer and Numerical Differentiation . . . . . . . . . . . . 2248.7.1 Computer Differentiation . . . . . . . . . . . . . . . 2248.7.2 Numerical Differentiation . . . . . . . . . . . . . . . 2258.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 226

9 Differentiation: Functions of Several Variables . . . . . . . . . . . 2279.1 Partial Differentiation . . . . . . . . . . . . . . . . . . . . . . 227

9.1.1 Second-Order Partial Derivatives . . . . . . . . . . . 2309.1.2 Differentiation of Functions of Several

Variables Using Computer . . . . . . . . . . . . . . . 2329.1.3 The Gradient and Hessian . . . . . . . . . . . . . . . 2329.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 234

9.2 Differential and Total Derivative . . . . . . . . . . . . . . . . 2359.2.1 Differential . . . . . . . . . . . . . . . . . . . . . . . 2359.2.2 Total Derivative . . . . . . . . . . . . . . . . . . . . 2379.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 240

9.3 Homogeneous Functions and the Euler Theorem . . . . . . . . 2409.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 243

9.4 Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 2449.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 248

9.5 Differentiating Systems of Equations . . . . . . . . . . . . . . 2489.5.1 The Jacobian and Independence of Nonlinear Functions 2489.5.2 Differentiating Several Functions . . . . . . . . . . . 2509.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 255

10 The Taylor Series and Its Applications . . . . . . . . . . . . . . . 25710.1 The Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . 257

10.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 26610.2 The Remainder and the Precision of Approximation . . . . . . 267

10.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 27010.3 Finding the Roots of an Equation . . . . . . . . . . . . . . . . 270

10.3.1 Iterative Methods . . . . . . . . . . . . . . . . . . . 270

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10.3.2 The Bisection Method . . . . . . . . . . . . . . . . . 27210.3.3 Newton’s Method . . . . . . . . . . . . . . . . . . . 27310.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 276

10.4 Taylor Expansion of Functions of Several Variables . . . . . . 27610.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 279

11 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28111.1 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . 282

11.1.1 Rules of Integration . . . . . . . . . . . . . . . . . . 28311.1.2 Change of Variable . . . . . . . . . . . . . . . . . . . 28511.1.3 Integration by Parts . . . . . . . . . . . . . . . . . . 28711.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 289

11.2 The Definite Integral . . . . . . . . . . . . . . . . . . . . . . 28911.2.1 Properties of Definite Integrals . . . . . . . . . . . . 29411.2.2 Rules of Integration for the Definite Integral . . . . . 29811.2.3 Change of Variable . . . . . . . . . . . . . . . . . . . 29911.2.4 Integration by Parts . . . . . . . . . . . . . . . . . . 30011.2.5 Riemann-Stieltjes Integral . . . . . . . . . . . . . . . 30411.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 306

11.3 Computer and Numerical Integration . . . . . . . . . . . . . . 30611.3.1 Computer Integration . . . . . . . . . . . . . . . . . 306

11.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . 30711.4.1 The Trapezoid Method . . . . . . . . . . . . . . . . . 30811.4.2 The Lagrange Interpolation Formula . . . . . . . . . 31011.4.3 Newton-Cotes Method . . . . . . . . . . . . . . . . . 31211.4.4 Simpson’s Method . . . . . . . . . . . . . . . . . . . 31311.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 315

11.5 Special Functions . . . . . . . . . . . . . . . . . . . . . . . . 31611.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 316

11.6 The Derivative of an Integral . . . . . . . . . . . . . . . . . . 31711.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 320

Part IV Optimization

12 Static Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 32312.1 Maxima and Minima of Functions of One Variable . . . . . . 324

12.1.1 Inflection Point . . . . . . . . . . . . . . . . . . . . . 33112.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 333

12.2 Unconstrained Optima of Functions of Several Variables . . . 33412.2.1 Convex and Concave Functions . . . . . . . . . . . . 33812.2.2 Quasi-convex and Quasi-concave Functions . . . . . 34112.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 342

12.3 Numerical Optimization . . . . . . . . . . . . . . . . . . . . . 34312.3.1 Steepest Descent . . . . . . . . . . . . . . . . . . . . 34312.3.2 Golden Section Method . . . . . . . . . . . . . . . . 34412.3.3 Newton Method . . . . . . . . . . . . . . . . . . . . 344

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12.3.4 Matlab Functions . . . . . . . . . . . . . . . . . . . 34512.3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 346

13 Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . 34713.1 Optimization with Equality Constraints . . . . . . . . . . . . 347

13.1.1 The Nature of Constrained Optimaand the Significance of λ . . . . . . . . . . . . . . . . 354

13.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 35413.2 Value Function . . . . . . . . . . . . . . . . . . . . . . . . . 355

13.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 36213.3 Second-Order Conditions and Comparative Static . . . . . . . 362

13.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 36813.4 Inequality Constraints and Karush-Kuhn-Tucker Conditions . . 368

13.4.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . 37213.4.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 375

14 Dynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . 37714.1 Dynamic Analysis in Economics . . . . . . . . . . . . . . . . 37714.2 The Control Problem . . . . . . . . . . . . . . . . . . . . . . 379

14.2.1 The Functional and Its Derivative . . . . . . . . . . . 38114.3 Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . 384

14.3.1 The Euler Equation . . . . . . . . . . . . . . . . . . 38614.3.2 Second-Order Conditions . . . . . . . . . . . . . . . 38914.3.3 Generalizing the Calculus of Variations . . . . . . . . 39014.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 391

14.4 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . 39114.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 396

14.5 The Maximum Principle . . . . . . . . . . . . . . . . . . . . 39714.5.1 Necessary and Sufficient Conditions . . . . . . . . . 40514.5.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 405

Part V Differential and Difference Equations

15 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 40915.1 Examples of Continuous Time Dynamic Economic Models . . 40915.2 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 412

15.2.1 Initial Value Problem . . . . . . . . . . . . . . . . . 41415.2.2 Existence and Uniqueness of Solutions . . . . . . . . 41615.2.3 Equilibrium and Stability . . . . . . . . . . . . . . . 417

15.3 First-Order Linear Differential Equations . . . . . . . . . . . . 41815.3.1 Variable Coefficient Equations . . . . . . . . . . . . 42015.3.2 Particular Integral, the Method

of Undetermined Coefficients . . . . . . . . . . . . . 42115.3.3 Separable Equations . . . . . . . . . . . . . . . . . . 42415.3.4 Exact Differential Equations . . . . . . . . . . . . . . 42615.3.5 Integrating Factor . . . . . . . . . . . . . . . . . . . 43015.3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 433

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15.4 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . 43315.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 436

15.5 Second-Order Linear Differential Equations . . . . . . . . . . 43615.5.1 Two Distinct Real Roots . . . . . . . . . . . . . . . . 43715.5.2 Repeated Root . . . . . . . . . . . . . . . . . . . . . 43915.5.3 Complex Roots . . . . . . . . . . . . . . . . . . . . . 44215.5.4 Particular Integral . . . . . . . . . . . . . . . . . . . 44315.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 446

15.6 Computer Solution of Differential Equations . . . . . . . . . . 44715.7 Numerical Analysis of Differential Equations . . . . . . . . . 448

15.7.1 The Euler Method . . . . . . . . . . . . . . . . . . . 44815.7.2 Runge-Kutta Methods . . . . . . . . . . . . . . . . . 45015.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 452

16 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . . . 45316.1 An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 45416.2 Examples of Discrete Dynamic Economic Models . . . . . . . 457

16.2.1 Adaptive Expectations . . . . . . . . . . . . . . . . . 45816.2.2 Partial Adjustment . . . . . . . . . . . . . . . . . . . 45916.2.3 Hall’s Consumption Function . . . . . . . . . . . . . 46016.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 461

16.3 First-Order Linear Difference Equations . . . . . . . . . . . . 46216.3.1 Solution of First-Order Linear Homogeneous

Difference Equations . . . . . . . . . . . . . . . . . . 46216.3.2 Solution of First-Order Nonhomogeneous Equations . 46516.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 472

16.4 Second-Order Linear Difference Equations . . . . . . . . . . . 47216.4.1 Solution of Second-Order Linear

Homogeneous Difference Equations . . . . . . . . . 47316.4.2 Behavior of the Solution of Second-Order Equation . 47516.4.3 Computer Solution of Difference Equations . . . . . . 48116.4.4 The Lag Operator . . . . . . . . . . . . . . . . . . . 48216.4.5 Solution of Second-Order Nonhomogeneous

Difference Equations . . . . . . . . . . . . . . . . . . 48516.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 490

16.5 n-th-Order Difference Equations . . . . . . . . . . . . . . . . 49116.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 493

17 Dynamic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 49517.1 Systems of Differential Equations . . . . . . . . . . . . . . . 495

17.1.1 Equivalence of a Second-Order LinearDifferential Equation and a System of TwoFirst-Order Linear Equations . . . . . . . . . . . . . 497

17.1.2 Linearizing Nonlinear Systems of Differential Equations 50017.2 The Jordan Canonical Form . . . . . . . . . . . . . . . . . . . 502

17.2.1 Diagonalization of a Matrix with DistinctReal Eigenvalues . . . . . . . . . . . . . . . . . . . . 502

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17.2.2 Block Diagonal Form of a Matrixwith Complex Eigenvalues . . . . . . . . . . . . . . 504

17.2.3 An Alternative Form for a Matrix withComplex Roots . . . . . . . . . . . . . . . . . . . . . 505

17.2.4 Decomposition of a Matrix with Repeated Roots . . . 50617.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 508

17.3 Exponential of a Matrix . . . . . . . . . . . . . . . . . . . . . 50917.3.1 Real Distinct Roots . . . . . . . . . . . . . . . . . . 51017.3.2 Complex Roots . . . . . . . . . . . . . . . . . . . . . 51117.3.3 Repeated Roots . . . . . . . . . . . . . . . . . . . . 51317.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 515

17.4 Solution of Systems of Linear Differential Equations . . . . . 51517.4.1 Decoupled Systems . . . . . . . . . . . . . . . . . . 51617.4.2 Systems with Real and Distinct Roots . . . . . . . . . 51817.4.3 Systems with Complex Roots . . . . . . . . . . . . . 51917.4.4 Systems with Repeated Roots . . . . . . . . . . . . . 52017.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 521

17.5 Numerical Analysis of Systems of Differential Equations . . . 52117.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 523

17.6 The Phase Portrait . . . . . . . . . . . . . . . . . . . . . . . . 52317.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 527

Solutions to Selected Problems . . . . . . . . . . . . . . . . . . . . . . . 529

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539