Computing the Zeros of Analytic Functions, Edizione 1727Springer Science & Business Media, 27 mar 2000 - 111 pagine Computing all the zeros of an analytic function and their respective multiplicities, locating clusters of zeros and analytic fuctions, computing zeros and poles of meromorphic functions, and solving systems of analytic equations are problems in computational complex analysis that lead to a rich blend of mathematics and numerical analysis. This book treats these four problems in a unified way. It contains not only theoretical results (based on formal orthogonal polynomials or rational interpolation) but also numerical analysis and algorithmic aspects, implementation heuristics, and polished software (the package ZEAL) that is available via the CPC Program Library. Graduate studets and researchers in numerical mathematics will find this book very readable. |
Sommario
1 Zeros of analytic functions | 1 |
111 Computing the total number of zeros with certainty | 4 |
112 An overview of other approaches | 8 |
12 Formal orthogonal polynomials | 10 |
13 An accurate algorithm to compute zeros of FOPs | 21 |
14 Numerical examples | 26 |
15 The software package ZEAL | 34 |
151 The approach taken by ZEAL | 35 |
22 A numerical example | 65 |
23 Rational interpolation at roots of unity | 69 |
24 More numerical examples | 79 |
3 Zeros and poles of meromorphic functions | 83 |
32 Theoretical considerations and numerical algorithm | 85 |
33 A numerical example | 88 |
4 Systems of analytic equations | 91 |
41 Introduction | 92 |
152 The structure of ZEAL | 37 |
153 ZEALs user interface | 40 |
154 A few examples of how to use ZEAL | 43 |
155 Concluding remarks | 50 |
16 A derivative free approach | 51 |
2 Clusters of zeros of analytic functions | 61 |
42 A multidimensional logarithmic residue formula | 93 |
43 The algorithm | 97 |
44 Numerical examples | 101 |
| 105 | |
Altre edizioni - Visualizza tutto
Computing the Zeros of Analytic Functions Peter Kravanja,Marc Van Barel Anteprima non disponibile - 2014 |
Parole e frasi comuni
algorithm allsmall analytic functions approach arithmetic mean calculated coefficients computed approximations computing zeros corresponding defined Delves and Lyness dth components eigenvalue problem equations FOP of degree formal orthogonal polynomials fr(z ƒ that lie given Hankel matrix ill-conditioned implies inner polynomial inside this box integrand interior interpolation conditions 2.3 interpolation problem lie inside logarithmic derivative logarithmic residue Math meromorphic functions monic polynomial monomial basis mutually distinct zeros Newton's method nonsingular number of mutually number of zeros numerical examples numerical integration obtain ordinary moments pencil polynomial of degree problem of computing Proof proves the theorem qr(z quadrature rational interpolation regular FOP regular index respective multiplicities scaled counterparts Schur decomposition solving standard monomial basis subroutine Suppose symmetric bilinear form total number v₁ Vandermonde matrix Vandermonde system ZEAL zeros and poles zeros inside zeros of analytic zeros of ƒ zeros z1
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