# 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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