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A cornerstone of applied probability, Markov chains can be used to help model how plants grow, chemicals react, and atoms diffuse--and 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 discrete-time and continuous-time Markov chains and examines many different numerical computing methods--direct, single-and multi-vector 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.

Markov processes --- Markov-processen. --- Numerical solutions. --- Absorbing state. --- Arnoldi method. --- Arnoldi process. --- Asymptotic convergence rate. --- Block Gauss-Seidel. --- Block splitting. --- Block triangular matrix. --- Chebyshev approximation. --- Coefficient matrix. --- Conditional probabilities. --- Conditioning. --- Connected vertices. --- Consistently ordered matrix. --- Convergence factors. --- Convergence properties. --- Convergence. --- Courtois problem. --- Cyclic tendencies. --- Decompositional methods. --- Diagonal blocks. --- Direct methods. --- Directed graph. --- Discretization parameter. --- Eigenvalues. --- Eigenvectors. --- Ephemeral states. --- Erlang law. --- Euler methods. --- Euler polygon. --- Explicit method. --- Family of solution curves. --- Full pivoting. --- Galerkin condition. --- Gauss transformation. --- Gaussian elimination. --- Gerschgorin theorem. --- Guard vectors. --- Hermitian matrices. --- Hessenberg Matrices. --- Indicator function. --- Iterative process. --- Jacobi method. --- Krylov subspace methods. --- Lanczos methods. --- Load-dependent. --- Low-stepping transitions. --- Lower Hessenberg matrix. --- Machine epsilon. --- Marca ordering. --- Markov chains. --- Memoryless property. --- Near-breakdown. --- Optimality property. --- Orthogonal complement. --- Overrelaxation.

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The description for this book, Contributions to the Theory of Nonlinear Oscillations (AM-36), Volume III, will be forthcoming.

Oscillations. --- Addition. --- Almost periodic function. --- Analytic function. --- Analytic manifold. --- Asymptote. --- Asymptotic analysis. --- Banach space. --- Basis (linear algebra). --- Betti number. --- Big O notation. --- Boundary (topology). --- Boundary value problem. --- Boundedness. --- Calculation. --- Cartesian coordinate system. --- Characteristic equation. --- Characteristic exponent. --- Coefficient matrix. --- Coefficient. --- Combination. --- Complex number. --- Complex space. --- Connected space. --- Continuous function. --- Counterexample. --- Curve. --- Degeneracy (mathematics). --- Degrees of freedom (statistics). --- Derivative. --- Determinant. --- Differentiable function. --- Differential equation. --- Dissipative system. --- Eigenvalues and eigenvectors. --- Equation. --- Existence theorem. --- Existential quantification. --- Exterior (topology). --- First variation. --- Fixed-point theorem. --- Fundamental theorem. --- Geometry. --- Half-space (geometry). --- Homeomorphism. --- Homotopy. --- Hyperbolic sector. --- Identity matrix. --- Imaginary number. --- Implicit function. --- Infimum and supremum. --- Integral curve. --- Interior (topology). --- Intersection (set theory). --- Interval (mathematics). --- Invertible matrix. --- Jacobian matrix and determinant. --- Jordan curve theorem. --- Limit cycle. --- Limit point. --- Limit set. --- Line at infinity. --- Linear approximation. --- Linear differential equation. --- Linear equation. --- Linear map. --- Lipschitz continuity. --- Matrix (mathematics). --- Monotonic function. --- N-vector. --- Nonlinear system. --- Ordinary differential equation. --- Parameter. --- Parametric equation. --- Parametrization. --- Partial derivative. --- Periodic function. --- Phase plane. --- Phase space. --- Point at infinity. --- Polynomial. --- Projective plane. --- Quantity. --- Saddle point. --- Scientific notation. --- Second derivative. --- Separatrix (mathematics). --- Sign (mathematics). --- Simultaneous equations. --- Singular perturbation. --- Special case. --- Submanifold. --- Summation. --- Tangent. --- Taylor series. --- Theorem. --- Theory. --- Topology. --- Vector field. --- Velocity. --- Zero of a function.

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The description for this book, Lectures on Differential Equations. (AM-14), Volume 14, will be forthcoming.

Differential equations. --- Abscissa. --- Absolute value. --- Addition. --- Adjoint. --- Algebraic topology. --- Antiderivative. --- Approximation. --- Canonical form. --- Cartesian coordinate system. --- Characteristic equation. --- Characteristic exponent. --- Circumference. --- Coefficient matrix. --- Coefficient. --- Compact space. --- Complex number. --- Complex plane. --- Complex-valued function. --- Condition index. --- Conformal map. --- Connected space. --- Conservation of energy. --- Continuous function. --- Convex hull. --- Coordinate system. --- Corollary. --- Degrees of freedom (statistics). --- Derivative. --- Determinant. --- Diagram (category theory). --- Differentiable function. --- Differential equation. --- Dimension. --- Dissipation. --- Eigenvalues and eigenvectors. --- Empty set. --- Entire function. --- Equation. --- Euclidean geometry. --- Euclidean space. --- Existence theorem. --- Existential quantification. --- Exterior (topology). --- Holomorphic function. --- Homogeneous polynomial. --- Inflection point. --- Initial point. --- Integer. --- Intersection (set theory). --- Jacobian matrix and determinant. --- Jordan curve theorem. --- Limit point. --- Limit set. --- Line at infinity. --- Linear differential equation. --- Linear map. --- Lipschitz continuity. --- Lyapunov stability. --- Mathematical physics. --- Mathematician. --- Matrix function. --- Maximal set. --- Monotonic function. --- Nonlinear system. --- Notation. --- Open set. --- Parameter. --- Parametric equation. --- Partial derivative. --- Periodic function. --- Phase space. --- Polar coordinate system. --- Polynomial. --- Power series. --- Projective geometry. --- Projective plane. --- Quadratic. --- Radius of convergence. --- Rectangle. --- Regular representation. --- Saddle point. --- Separatrix (mathematics). --- Set theory. --- Simple polygon. --- Solomon Lefschetz. --- Special case. --- Spherical cap. --- Stereographic projection. --- Subset. --- Summation. --- Theorem. --- Theory. --- Topological property. --- Toroid. --- Two-dimensional space. --- Uniform convergence. --- Upper and lower bounds. --- Variable (mathematics). --- Vector space. --- Zero of a function.