Convergence Estimates of Krylov Subspace Methods for the Approximation of Matrix Functions Using Tools from Potential Theory
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چکیده
File List 97 Notation 98 References 100 Index 103 2 Preface This work is about the numerical evaluation of the expression f (A)b, where A ∈ C N ×N is an arbitrary square matrix, b ∈ C N is a vector and f is a suitable matrix function. This task is of very high importance in all applied sciences since it is a generalization of the following problems, to name just a few: • Solve the linear system of equations Ax = b. The solution is x = f (A)b, where f (z) = 1/z. • Solve an ordinary differential equation y (t) = Ay (t) with given initial value y (0) = b. The solution is y (t) = f (tA)b, where f (z) = exp(z). • Solve identification problems in stochastic semigroups. Here one needs to compute f (A)b with f (z) = log(z) (see Singer, Spilermann [29]). • Simulate Brownian motion of molecules. Here one needs to determine f (A)b with f (z) = √ z (see Ericsson [9]). In the first chapter we will define the term f (A). There are different equivalent approaches. A constructive one is to involve the Jordan canonical form of the matrix A. Later we shall see that f (A) = p f,A (A), where p f,A is a polynomial of degree ≤ N − 1 that interpolates f at the eigenvalues of A. In practical applications N is very large and the spectrum of A is not known. Therefore we will determine an f-interpolating polynomial p f,m of low degree m − 1 N and hope that p f,m (A)b ≈ f (A)b. The resulting methods are called Krylov subspace methods or polynomial methods and they are considered in Chapter 2. 3 Preface The choice of the interpolation nodes for p f,m is an important issue. If the interpolation nodes are uniformly distributed on a compact subset of C, we may analyze the asymptotic convergence behavior of the arising methods using theory of interpolation and best approximation. This is done in Chapter 3. Another very popular choice of interpolation nodes are Ritz values. The resulting Arnoldi approximations converge in many cases very fast to f (A)b. To explain this, it is necessary to describe the behavior of Ritz values. In Chapter 4 we will present a theory on the convergence of Ritz values, which was mainly developed by Beckermann and …
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تاریخ انتشار 2015