Solution of Equations and Systems of Equations
Pure and Applied Mathematics: A Series of Monographs and Textbooks, Vol. 9
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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
Preface to the First Edition
Preface to the Second Edition
1. Divided Differences
Divided Differences for Distinct Arguments
Symmetry
Integral Representation
Mean Value Formulas
Divided Differences with Repeated Arguments
A Formula for Confluent Divided Differences
Newton's Interpolation Formula
General Interpolation Problem
Polynomial Interpolation
The Remainder for a General Interpolating Function
Triangular Schemes for Computing Divided Differences
2. Inverse Interpolation. Derivatives of the Inverse Function. One Interpolation Point
The Concept of Inverse Interpolation
Darboux's Theorem on Values of f(x)
Derivatives of the Inverse Function
One Interpolation Point
A Development of a Zero of f(x)
3. Method of False Position (regula Falsi)
Definition of the Regula Falsi
Use of Inverse Interpolation
Geometric Interpretation (Fourier's Conditions)
Iteration with Successive Adjacent Points
Homer Units and Efficiency Index
The Rounding-Off Rule
Locating the Zero with the Regula Falsi
Examples of Computation by the Regula Falsi
4. Iteration
A Convergence Criterion for an Iteration
Points of Attraction and Repulsion
Improving the Convergence
5. Further Discussion of Iterations. Multiple Zeros
Iterations by Monotonic Iterating Functions
Multiple Zeros
Connection of the Regula Falsi with the Theory of Iteration
6. Newton-Raphson Method
The Idea of the Newton-Raphson Method
The Use of Inverse Interpolation
Comparison of Regula Falsi and Newton-Raphson Method
7. Fundamental Existence Theorems for Newton-Raphson Iteration
Error Estimates a Priori and a Posteriori
Fundamental Existence Theorems
8. An Analog of the Newton-Raphson Method for Multiple Roots
9. Fourier Bounds for Newton-Raphson Iteration
10. Dandelin Bounds for Newton-Raphson Iteration
11. Three Interpolation Points
Interpolation by Linear Fractions
Two Coincident Interpolation Points
Error Estimates
Use in Iteration Procedure
12. Linear Difference Equations
Inhomogeneous and Homogeneous Difference Equations
General Solution of the Homogeneous Equation
Lemma on Division of Power Series
Asymptotic Behavior of Solutions of (12.1)
Asymptotic Behavior of Errors in the Regula Falsi Iteration
A Theorem on Roots of Certain Equations
13. n Distinct Points of Interpolation
Error Estimates
Iteration with n Distinct Points of Interpolation
Discussion of the Roots of Some Special Equations
14. n + 1 Coincident Points of Interpolation and Taylor Development of the Root
Statement of the Problem
A Theorem on Inverse Functions and Conformal Mapping
Theorem on the Error of the Taylor Approximation to the Root
Discussion of the Conditions of the Theorem
15. The Square Root Iteration
16. Further Discussion of Square Root Iteration
17. A General Theorem on Zeros of interpolating Polynomials
18. Approximation of Equations by Algebraic Equations of a Given Degree. Asymptotic Errors for Simple Roots
19. Norms of Vectors and Matrices
20. Two Theorems on Convergence of Products of Matrices
21. A Theorem on Divergence of Products of Matrices
22. Characterization of Points of Attraction and Repulsion for Iterations with Several Variables
An Example
23. Further Discussion of Norms Matrices △q(A)
Triangle Inequality
Bilinear and Quadratic Forms of Symmetric Matrices
Estimate of △p(ABC)
Variation of △p(A-l)
Length of Arc in the |ξ|p Metric
24. An Existence Theorem for system of Equations
Formulation of the Theorem
Proof of Theorem 24.1
A Uniqueness Theorem
Example
25. n-Dimensional Generalization of the Newton-Raphson Method. Statement of the Theorems
Variation of the Jacobian Matrix
Statement of the n-Dimensional Analog of the Newton-Raphson Method
26. n-Dimensional Generalization of the Newton-Raphson Methods. Proofs of the Theorems
27. Method of Steepest Descent. Convergence of the Procedure
Idea of the Method
Convergence of the Procedure
Application to |f(x + iy)|2
28. Method of Steepest Descent. Weakly Linear Convergence of the ξu
The Derived Set at the ξu
Weakly Linear Convergence
Condition for the Regular Minimum of the Function (27.3)
Algebraic Equations with One Unknown
29. Method of Steepest Descent. Linear Convergence of the ξu
Example
Appendices
A. Continuity of the Roots of Algebraic Equations
B. Relative Continuity of the Roots of Algebraic Equations
C. An Explicit Formula for the nth Derivative of the Inverse Function
D. Analog of the Regula Falsi for Two Equations with Two Unknowns
E. Steffensen's Improved Iteration Rule
F. The Newton-Raphson Algorithm for Quadratic Polynomials
G. Some Modifications and Improvements of the Newton-Raphson Method
H. Rounding Off in Inverse Interpolation
I. Accelerating Iterations with Superlinear Convergence
J. Roots of f(z) = 0 in Terms of the Coefficients of the Development of 1/f(z)
K. Continuity of the Fundamental Roots as Functions of the Elements of the Matrix
L. The Determinantal Formulas for Divided Differences
M. Remainder Terms in Interpolation Formulas
N. Generalization of Schroder's Series to the Case of Multiple Roots
O. Laguerre Iterations
P. Approximation of Equations by Algebraic Equations of a Given Degree. Asymptotic Errors for Multiple Zeros
Bibliographical Notes
Index
Product details
- No. of pages: 352
- Language: English
- Copyright: © Academic Press 1966
- Published: January 1, 1966
- Imprint: Academic Press
- eBook ISBN: 9781483223643
About the Author
A. M. Ostrowski
About the Editors
Paul A. Smith
Samuel Eilenberg
Affiliations and Expertise
Columbia University
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