## Complex Interval Arithmetic and Its ApplicationsThe aim of this book is to present formulas and methods developed using complex interval arithmetic. While most of numerical methods described in the literature deal with real intervals and real vectors, there is no systematic study of methods in complex interval arithmetic. The book fills this gap. Several main subjects are considered: outer estimates for the range of complex functions, especially complex centered forms, the best approximations of elementary complex functions by disks, iterative methods for the inclusion by polynomial zeros including their implementation on parallel computers, the analysis of numerical stability of iterative methods by using complex interval arithmetic and numerical computation of curvilinear integrals with error bounds. Mainly new methods are presented developed over the last years, including a lot of very recent results by the authors some of which have not been published before. |

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### Contents

Introduction | 9 |

Interval Arithmetic | 15 |

Circular Complex Inclusion Forms | 27 |

Best Approximations by Disks | 52 |

Inclusion of Polynomial Zeros | 90 |

Simultaneous Inclusion of Complex Zeros | 137 |

Improved Inclusion Methods | 180 |

Parallel Implementation of Inclusion Methods | 219 |

Numerical Stability of Iterative Processes | 236 |

Numerical Computation of Curvilinear Integrals | 258 |

266 | |

278 | |

### Common terms and phrases

According algorithm Algorithms 4.5 analytic functions applied assume calculated circular approximations circular arithmetic circular centered form circular complex circular interval combined method complex approximation complex functions complex interval arithmetic complex numbers complex plane complex zeros considered convergence order defined denoted diametrical disk disk containing equation error bounds estimate evaluation exact range Example fixed point relation given Halley-like Halley's method Herzberger Horner scheme inclusion disks inclusion method inequality initial disks intersection Interval Analysis interval mathematics interval method interval slope method inversion iteration index iteration step iterative formula iterative function Lemma M. S. Petkovic Math Newton's correction numerical numerical stability obtain order of convergence pc(Z polynomial P(z polynomial zeros procedure processor proof proved quadratic convergence radii radius range f(Z rational function Ratschek rectangle rectangular arithmetic Rokne rounding errors secant method Section sequence simple zeros simultaneous inclusion Theorem Weierstrass