
Difference Equations
From Rabbits to Chaos
By:Â Paul Cull, Mary Flahive, Robby Robson
Hardcover | 12 April 2005
At a Glance
408 Pages
23.5 x 15.88 x 2.54
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In this new text, designed for sophomores studying mathematics and computer science, the authors cover the basics of difference equations and some of their applications in computing and in population biology. Each chapter leads to techniques that can be applied by hand to small examples or programmed for larger problems. Along the way, the reader will use linear algebra and graph theory, develop formal power series, solve combinatorial problems, visit Perron-Frobenius theory, discuss pseudorandom number generation and integer factorization, and apply the Fast Fourier Transform to multiply polynomials quickly.
The book contains many worked examples and over 250 exercises. While these exercises are accessible to students and have been class-tested, they also suggest further problems and possible research topics.
Industry Reviews
From the reviews:
"The book under review is written for advanced undergraduate majors. ... This book is well written and easy to read, with a large number of exercises ... . Difference equations: from rabbits to chaos is a good text for those who want to teach an introductory course in recurrences from a collection of various classical points of view. ... anyone interested in difference equations ... will want Cull, Flahive, and Robson as a reference because of its interesting and unique collection of topics." (Shandelle M. Henson, Mathematical Reviews, Issue 2006 j)
"This book is a nice addition to an already nice group of books on difference equations ... . would be especially appealing to students who like to compute. ... This book contains many interesting examples ... . There are over 250 exercises. This book could be used for an upper level undergraduate course or for a beginning graduate level course. ... it would be nice to have in one's personal library." (Allan Peterson, SIAM Review, Vol. 48 (1), 2006)
"Recognizing the increasing importance of difference equation models ... the authors aimed to write a book that provides a solid foundation in the field and is accessible to undergraduates. ... this text is written in an informal style and with an 'algorithmic spirit'. ... There are many worked examples and the exercise sets are good ... . it provides an accessible introduction to the material and a firm foundation for applications in various scientific fields." (Henry Ricardo, MathDL, September, 2005)
"As the authors point out in their introduction this is a textbook for undergraduate students taking introductory courses on discrete mathematics. The text is quite elementary, written for an audience with a preference for computational aspects of mathematics and modelling real world phenomena. This is also reflected by the exercises at the end of each chapter. Starting with simple examples ... the reader is lead to basic notions and theoremsof basic linear algebra and analysis. Proofs are worked out accurately ... ." (Stefan Hilger, Zentralblatt MATH, Vol. 1085, 2006)
"This book is specifically designed for American undergraduate students studying mathematics or computer science; it is part of Springer's undergraduate texts in mathematics. ... It will be useful to have on the bookshelf, and when there is a specific difference equation in front of me it will be my first port of call." (Tony Crilly, The Mathematical Gazette, Vol. 91 (521), 2007)
| Preface | p. v |
| Computation | p. vii |
| Notational Preliminaries | p. viii |
| Fibonacci Numbers | p. 1 |
| The Rabbit Problem | p. 1 |
| The Fibonacci Sequence | p. 2 |
| Computing Fibonacci numbers | p. 4 |
| A formula for the Fibonacci numbers | p. 5 |
| Further Fibonacci facts | p. 6 |
| Notation for Asymptotic Analysis | p. 6 |
| Exercises | p. 7 |
| Homogeneous Linear Recurrence Relations | p. 11 |
| The Solution Space of (HL) | p. 12 |
| The Matrix Form | p. 15 |
| A Simpler Basis for the Solution Space | p. 17 |
| Distinct eigenvalues | p. 19 |
| Repeated eigenvalues | p. 21 |
| The Asymptotic Behavior of Solutions | p. 25 |
| Exercises | p. 28 |
| Finite Difference Equations | p. 33 |
| Linear Difference Equations | p. 33 |
| First-order equations | p. 34 |
| General and Particular Solutions | p. 36 |
| Finding a particular solution via summation | p. 39 |
| A Special Class of Linear Recurrences | p. 41 |
| Operator Notation | p. 45 |
| The Shift Operator on the Space of Sequences | p. 47 |
| Formal Power Series | p. 50 |
| Formal differentiation | p. 55 |
| An application of formal power series | p. 56 |
| Exercises | p. 58 |
| Generating Functions | p. 67 |
| Counting Strings with Some Restrictions | p. 67 |
| An Overview of the Generating Function Technique | p. 70 |
| Rational representation | p. 75 |
| A Review of Partial Fractions | p. 76 |
| Examples of the Generating Function Technique | p. 82 |
| The Catalan numbers | p. 83 |
| Stirling numbers of the second kind | p. 85 |
| Reversion of Generating Functions | p. 87 |
| Using the Fourier Transform | p. 91 |
| Exercises | p. 94 |
| Nonnegative Difference Equations | p. 101 |
| Nonnegative Polynomials | p. 102 |
| The dominant root | p. 102 |
| When are integer solutions rounded powers of an eigenvalue? | p. 106 |
| Using the Rounding Theorem | p. 110 |
| Estimation of the Roots | p. 113 |
| Estimation of the dominant root | p. 113 |
| Estimation of the second root | p. 113 |
| Calculation of the Roots | p. 116 |
| The rate of convergence in Newton's method | p. 121 |
| Asymptotic Size of Solutions | p. 125 |
| Homogeneous nonnegative recurrences | p. 125 |
| Nonhomogeneous nonnegative equations | p. 127 |
| Exercises | p. 132 |
| Leslie's Population Matrix Model | p. 137 |
| Leslie's Model | p. 137 |
| How to tell whether a Leslie matrix is primitive | p. 141 |
| Leslie's Convergence Theorem | p. 142 |
| Imprimitive Leslie Matrices | p. 144 |
| A simple example | p. 144 |
| A special case: Only one positive fertility rate | p. 145 |
| Asymptotically periodic Leslie matrices | p. 145 |
| Companion Matrices | p. 147 |
| Matrices with repeated eigenvalues | p. 155 |
| Nonnegative Companion Matrices | p. 157 |
| Periodic nonnegative companion matrices | p. 159 |
| Back to Leslie Matrices | p. 164 |
| Periodic Leslie matrices | p. 165 |
| Averaging | p. 168 |
| The Limiting Effect of L on Nonnegative Vectors | p. 169 |
| The period of the total population | p. 171 |
| Afterword | p. 173 |
| Exercises | p. 174 |
| Matrix Difference Equations | p. 179 |
| Homogeneous Matrix Equations | p. 179 |
| Nonnegative Matrix Equations | p. 186 |
| Applications to Markov chains | p. 187 |
| Graphs and Matrices | p. 189 |
| Next node representation | p. 193 |
| Comments on imprimitivity | p. 194 |
| Algorithms for Primitivity | p. 198 |
| Algorithm I | p. 198 |
| Algorithm II | p. 202 |
| Matrix Difference Equations with Input | p. 206 |
| Reduction to one dimension | p. 207 |
| Reduction to homogeneous form | p. 211 |
| Exercises | p. 212 |
| Modular Recurrences | p. 217 |
| Periodicity | p. 218 |
| Periodicity of linear modular recurrences | p. 221 |
| Fast modular computations | p. 224 |
| Finite Fields | p. 225 |
| Periods of First-Order Modular Recurrences | p. 227 |
| First-order modular recurrences with maximal period | p. 230 |
| Periodic Second-Order Modular Recurrences | p. 232 |
| Periods of modular Fibonacci sequences | p. 233 |
| Applications | p. 238 |
| Application 1: Pseudorandom number generation | p. 238 |
| Application 2: Integer factorization | p. 242 |
| Exercises | p. 246 |
| Computational Complexity | p. 253 |
| Analysis of Algorithms | p. 254 |
| Measuring run time | p. 254 |
| An example: The Towers of Hanoi puzzle | p. 256 |
| Computer Arithmetic | p. 261 |
| Addition and subtraction | p. 262 |
| Multiplication and division | p. 262 |
| An Introduction to Divide-and-Conquer | p. 263 |
| Example: Polynomial multiplication | p. 264 |
| Simple Divide-and-Conquer Algorithms | p. 268 |
| Example 1: A return to polynomial multiplication | p. 270 |
| Example 2: Matrix multiplication | p. 271 |
| Example 3: MERGESORT | p. 272 |
| Example 4: Applications of Newton's method | p. 273 |
| The Fast Fourier Transform | p. 274 |
| The general form of the Fast Fourier Transform | p. 276 |
| The FFT when n = 2[superscript k] | p. 277 |
| Fast evaluation and fast interpolation | p. 280 |
| The fast polynomial multiplication algorithm | p. 281 |
| Average Case Analysis | p. 284 |
| The LARGETWO algorithm | p. 284 |
| The QUICKSORT algorithm | p. 286 |
| Exercises | p. 289 |
| Some Nonlinear Recurrences | p. 297 |
| Some Examples | p. 297 |
| Nonlinear Systems | p. 299 |
| Sarkovskii's Theorem | p. 302 |
| Chaos | p. 303 |
| A simple chaotic system | p. 303 |
| Local Stability | p. 307 |
| Local stability of a fixed point | p. 307 |
| Local stability of a cycle | p. 308 |
| Local stability in two dimensions | p. 310 |
| Global Stability | p. 313 |
| Staircase convergence | p. 314 |
| Nonmonotonic convergence | p. 315 |
| Linear Fractional Recurrences | p. 317 |
| Asymptotic behavior | p. 318 |
| Rational coefficients and periodicity | p. 322 |
| Chaotic-like behavior | p. 324 |
| Invariant distributions | p. 326 |
| Proving global stability | p. 330 |
| Summary | p. 331 |
| Conclusion | p. 332 |
| Exercises | p. 333 |
| Worked Examples | p. 337 |
| All Simple Roots | p. 337 |
| One Multiple Root | p. 342 |
| One Multiple Root, Several Simple Roots | p. 345 |
| The Input is [gamma superscript n subscript 1]p[subscript 1](n) + [gamma superscript n subscript 2]p[subscript 2](n) | p. 346 |
| Complex Numbers | p. 347 |
| Highlights of Linear Algebra | p. 353 |
| Vector Spaces and Subspaces | p. 353 |
| Linear Independence and Basis | p. 354 |
| Linear Transformations | p. 355 |
| Eigenvectors | p. 356 |
| Characteristic and Minimal Polynomials | p. 358 |
| Exercises | p. 359 |
| Roots in the Unit Circle | p. 361 |
| Marden's Method | p. 362 |
| Exercises | p. 367 |
| References | p. 369 |
| Table of Contents provided by Ingram. All Rights Reserved. |
ISBN: 9780387232331
ISBN-10: 0387232338
Series: Springer Undergraduate Texts in Mathematics and Technology
Published: 12th April 2005
Format: Hardcover
Language: English
Number of Pages: 408
Audience: College, Tertiary and University
Publisher: Springer Nature B.V.
Country of Publication: US
Dimensions (cm): 23.5 x 15.88 x 2.54
Weight (kg): 0.74
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