Lecture Notes in Mathematics

By: Miodrag Petkovic

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Paperback | 3 June 2008

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The problem of solving nonlinear equations and systems of equations ranks among the most signi?cant in the theory and practice, not only of applied mathematicsbutalsoofmanybranchesofengineeringsciences, physics, c- puter science, astronomy, ?nance, and so on. A glance at the bibliography and the list of great mathematicians who have worked on this topic points to a high level of contemporary interest. Although the rapid development of digital computers led to the e?ective implementation of many numerical methods, in practical realization, it is necessary to solve various problems such as computational e?ciency based on the total central processor unit time, the construction of iterative methods which possess a fast convergence in the presence of multiplicity (or clusters) of a desired solution, the control of rounding errors, information about error bounds of obtained approximate solution, stating computationally veri?able initial conditions that ensure a safe convergence, etc. It is the solution of these challenging problems that was the principal motivation for the present study. In this book, we are mainly concerned with the statement and study of initial conditions that provide the guaranteed convergence of an iterative method for solving equations of the form f(z) = 0. The traditional approach to this problem is mainly based on asymptotic convergence analysis using some strong hypotheses on di?erentiability and derivative bounds in a rather wide domain.

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Non-Fiction Mathematics Algebra Calculus & Mathematical Analysis Numerical Analysis

Basic Concepts	p. 1
Simultaneous Methods for Finding Polynomial Zeros	p. 1
Localization of Polynomial Zeros	p. 26
Complex Interval Arithmetic	p. 32
Iterative Processes and Point Estimation Theory	p. 35
Newton's Method Estimates	p. 35
Approximate Zeros of Quadratically Convergent Methods	p. 44
Approximate Zeros of Higher-Order Methods	p. 48
Improvement of Smale's Result	p. 51
Point Estimation of Durand-Kerner's Method	p. 59
Point Estimation of Simultaneous Methods	p. 67
Point Estimation and Polynomial Equations	p. 67
Guaranteed Convergence: Correction Approach	p. 71
Guaranteed Convergence: Sequence Approach	p. 97
A Posteriori Error Bound Methods	p. 118
Families of Simultaneous Methods of Higher Order: Part I	p. 129
Hansen-Patrick's Family of Root Finding Methods	p. 129
Derivative-Free Family of Simultaneous Methods	p. 132
Family of Simultaneous Methods with Derivatives	p. 146
Numerical Examples	p. 156
Families of Simultaneous Methods of Higher Order: Part II	p. 161
One-Parameter Family for Simple Zeros	p. 161
Family of Methods with Corrections	p. 174
Family of Interval Methods	p. 186
References	p. 197
Glossary	p. 205
Index	p. 207
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Lecture Notes in Mathematics

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