


Introduction to the Numerical Solution of Markov Chains 
A cornerstone of applied probability, Markov chains can be used to help model how plants grow, chemicals react, and atoms diffuseand applications are increasingly being found in such areas as engineering, computer science, economics, and education. To apply the techniques to real problems, however, it is necessary to understand how Markov chains can be solved numerically. In this book, the first to offer a systematic and detailed treatment of the numerical solution of Markov chains, William Stewart provides scientists on many levels with the power to put this theory to use in the actual world, where it has applications in areas as diverse as engineering, economics, and education. His efforts make for essential reading in a rapidly growing field. Here Stewart explores all aspects of numerically computing solutions of Markov chains, especially when the state is huge. He provides extensive background to both discretetime and continuoustime Markov chains and examines many different numerical computing methodsdirect, singleand multivector iterative, and projection methods. More specifically, he considers recursive methods often used when the structure of the Markov chain is upper Hessenberg, iterative aggregation/disaggregation methods that are particularly appropriate when it is NCD (nearly completely decomposable), and reduced schemes for cases in which the chain is periodic. There are chapters on methods for computing transient solutions, on stochastic automata networks, and, finally, on currently available software. Throughout Stewart draws on numerous examples and comparisons among the methods he so thoroughly explains. "The book contains very rich material which is the result of longterm research in this field. No other book is known to the reviewer that treats this subject in such detail. . . . The book excellently reflects the great experience that the author has in the theory of Markov chains, matrix algebra, numerics and informatics. He . . . richly illustrates the book with numerous examples, flowcharts, pictures and even computer screen copies."Mathematical Reviews Endorsement: "The big attraction of this book is its timeliness: many engineers and scientists are currently becoming interested in iterative methods for solving large linear systems and eigenvalue problems. The book assembles together in a nicely presented form a large set of numerical techniques, including the most recently developed ones. It offers comparisons that will be very helpful to the specialist as well as the beginner. On the whole, this is an excellent text."Yousef Saad, University of Minnesota "I know of no other book that has the same breadth of coverage as this one by William Stewart. Because it is both comprehensive and wellorganized, this work will be valuable as a reference book and for use in the classroom."Richard Muntz, University of California, Los Angeles
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