Solution of Equations and Systems of Equations

Pure and Applied Mathematics: A Series of Monographs and Textbooks, Vol. 9

A. M. Ostrowski, Paul A. Smith, Samuel Eilenberg

• Publisher: Academic Press
• Year: 2016
• ISBN:
  9781483223643
• Edition: 2
• Topic: Mathematik
• Subtopic: Funktionsanalyse

Solution of Equations and Systems of Equations, Second Edition deals with the Laguerre iteration, interpolating polynomials, method of steepest descent, and the theory of divided differences. The book reviews the formula for confluent divided differences, Newton's interpolation formula, general interpolation problems, and the triangular schemes for computing divided differences. The text explains the method of False Position (Regula Falsi) and cites examples of computation using the Regula Falsi. The book discusses iterations by monotonic iterating functions and analyzes the connection of the Regula Falsi with the theory of iteration. The text also explains the idea of the Newton-Raphson method and compares it with the Regula Falsi. The book also cites asymptotic behavior of errors in the Regula Falsi iteration, as well as the theorem on the error of the Taylor approximation to the root. The method of steepest descent or gradient method proposed by Cauchy ensures "global convergence" in very general conditions.

This book is suitable for mathematicians, students, and professor of calculus, and advanced mathematics.

Table of contents

• Front Cover
• Solution of Equations and Systems of Equations
• Copyright Page
• Table of Contents
• Preface to the First Edition
• Preface to the Second Edition
• Chapter 1. Divided Differences
• Chapter 2. Inverse Interpolation. Derivatives of the Inverse Function. One Interpolation Point
• Chapter 3. Method of False Position (Regula Falsi)
• Chapter 4. Iteration
• Chapter 5. Further Discussion of Iterations. Multiple Zeros
• Chapter 6. Newton-Raphson Method
• Chapter 7. Fundamental Existence Theorems for Newton-Raphson Iteration
• Chapter 8. An Analog of the Newton-Raphson Method for Multiple Roots
• Chapter 9. Fourier Bounds for Newton-Raphson Iteration
• Chapter 10. Dandelin Bounds for Newton-Raphson Iteration
• Chapter 11. Three Interpolation Points
• Chapter 12. Linear Difference Equations
• Chapter 13. n Distinct Points of Interpolation
• Chapter 14. n + 1 Coincident Points of Interpolation and Taylor Development of the Root
• Chapter 15. The Square Root Iteration
• Chapter 16. Further Discussion of Square Root Iteration
• Chapter 17. A General Theorem on Zeros of Interpolating Polynomials
• Chapter 18. Approximation of Equations by Algebraic Equations of a Given Degree. Asymptotic Errors for Simple Roots
• Chapter 19. Norms of Vectors and Matrices
• Chapter 20. Two Theorems on Convergence of Products of Matrices
• Chapter 21. A Theorem on Divergence of Products of Matrices
• Chapter 22. Characterization of Points of Attraction and Repulsion for Iterations with Several Variables
• Chapter 23. Further Discussion of Norms of Matrices. ∆q(A)
• Chapter 24. An Existence Theorem for Systems of Equations
• Chapter 25. n-Dimensional Generalization of the Newton- Raphson Method. Statement of the Theorems
• Chapter 26. n-Dimensional Generalization of the Newton- Raphson Method. Proofs of the Theorems
• Chapter 27. Method of Steepest Descent. Convergence of the Procedure
• Chapter 28. Method of Steepest Descent. Weakly Linear Convergence of the εμ
• Chapter 29. Method of Steepest Descent. Linear Convergence of the εμ
• Appendices
• BIBLIOGRAPHICAL NOTES
• Index