Methods for Numerical Solving the Single Nonlinear Equations
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Stanisław Rosłoniec
Fundamental Numerical Methods for Electrical Engineering
123
Prof. Dr. Hab. Ing. Stanisław Rosłoniec Institute of Radioelectronics Warsaw University of Technology Nowowiejska 15/19 00-665 Warsaw Poland [email protected]
ISBN: 978-3-540-79518-6
e-ISBN: 978-3-540-79519-3
Library of Congress Control Number: 2008927874 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: eStudio Calamar S.L. Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 Methods for Numerical Solution of Linear Equations . . . . . . . . . . . . . . . . 1.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Gauss Elimination Method . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Gauss–Jordan Elimination Method . . . . . . . . . . . . . . . . . . . 1.1.3 The LU Matrix Decomposition Method . . . . . . . . . . . . . . . . . . . 1.1.4 The Method of Inverse Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Indirect or Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 The Direct Iteration Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Jacobi and Gauss–Seidel Methods . . . . . . . . . . . . . . . . . . . . . . . 1.3 Examples of Applications in Electrical Engineering . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 5 5 9 11 14 17 17 18 23 27
2 Methods for Numerical Solving the Single Nonlinear Equations . . . . . . 2.1 Determination of the Complex Roots of Polynomial Equations by Using the Lin’s and Bairstow’s Methods . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Lin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Bairstow’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Laguerre Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Iterative Methods Used for S
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