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. |
Contents
Introduction | 9 |
Interval Arithmetic | 15 |
References | 50 |
52 | 121 |
Simultaneous Inclusion of Complex Zeros 137 I | 137 |
70 | 160 |
Improved Inclusion Methods | 180 |
Common terms and phrases
According Alefeld algorithm analytic functions applied assume asynchronous method Börsch-Supan calculation Carstensen circular arithmetic circular complex combined method complex functions complex interval arithmetic complex numbers complex zeros considered convergence order defined denoted diametrical disk equation error bounds estimate evaluation Example fixed point relations Gargantini given Halley-like Halley's Halley's method Herzberger inclusion disks inclusion isotonicity inequality initial disks interval arithmetic interval slope inversion iteration formula iteration index iteration steps iterative methods Lemma M. S. Petković Math Newton's correction Newton's method numerical numerical stability obtain order of convergence P(zi parallel Parallel Computing polynomial zeros procedure processors Proof proved quadratic convergence R-matrix R-order R₁ radii radius rational function rectangle rectangular arithmetic Rokne root rounding errors secant method Section sequence simple zeros simultaneous methods slope method Theorem W₁ Weierstrass Z₁ zeros $1