Solution of Equations and Systems of Equations

Solution of Equations and Systems of Equations

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

2nd Edition - January 1, 1966

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  • Author: A. M. Ostrowski
  • eBook ISBN: 9781483223643

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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


    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


    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



    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


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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