Polynomial Operator Equations in Abstract Spaces and Applications / Edition 1

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Overview

Polynomial operators are a natural generalization of linear operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations. Often the polynomial nature of many nonlinear problems goes unrecognized by researchers. This is more likely due to the fact that polynomial operators - unlike polynomials in a single variable - have received little attention. Consequently, this comprehensive presentation is needed, benefiting those working in the field as well as those seeking information about specific results or techniques.
Polynomial Operator Equations in Abstract Spaces and Applications - an outgrowth of fifteen years of the author's research work - presents new and traditional results about polynomial equations as well as analyzes current iterative methods for their numerical solution in various general space settings.
Topics include:

  • Special cases of nonlinear operator equations
  • Solution of polynomial operator equations of positive integer degree n
  • Results on global existence theorems not related with contractions
  • Galois theory
  • Polynomial integral and polynomial differential equations appearing in radiative transfer, heat transfer, neutron transport, electromechanical networks, elasticity, and other areas
  • Results on the various Chandrasekhar equations
  • Weierstrass theorem
  • Matrix representations
  • Lagrange and Hermite interpolation
  • Bounds of polynomial equations in Banach space, Banach algebra, and Hilbert space
    The materials discussed can be used for the following'studies
  • Advanced numerical analysis
  • Numerical functional analysis
  • Functional analysis
  • Approximation theory
  • Integral and differential equations
    Tables include
  • Numerical solutions for Chandrasekhar's equation I to VI
  • Error bounds comparison
  • Accelerations schemes I and II for Newton's method
  • Newton's method
  • Secant method
    The self-contained text thoroughly details results, adds exercises for each chapter, and includes several applications for the solution of integral and differential equations throughout every chapter.

"...includes applied exercises and examples, surveys theoretic results of polynomial equations in abstract spaces, profiles solutions using numerical techniques, and much more."

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Editorial Reviews

Booknews
In addition to presenting new results, and selected old results, about polynomial equations, Argyros (mathematics, Cameron U., Lawton, Oklahoma) analyzes new iterative methods for the numerical solution of polynomial equations in various space settings. His discussion of special cases of nonlinear operator equations focuses on the solution of polynomial operator equations of positive integer degree . Other topics include results on the Chandrasekhar equations, matrix presentations, and completely continuous multilinear operators. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

  • ISBN-13: 9780849387029
  • Publisher: CRC Press
  • Publication date: 3/25/1998
  • Edition number: 1
  • Pages: 592
  • Product dimensions: 6.25 (w) x 9.50 (h) x 1.50 (d)

Table of Contents

Introduction
Quadratic Equations and Perturbation Theory
Algebraic Theory of Quadratic Operators
Perturbation Theory
Chandrasekhar's Integral Equation
Anselone and Moore's Equation
Other Perturbation Theorems
More Methods for Solving Quadratic Equations
Banach Algebras
The Majorant Method
Compact Quadratic Equations 83
Finite Rank Equations
Noncontractive Solutions
On a Class of Quadratic Integral Equations with Perturbation
Polynomial Equations in Banach Space
Polynomial Equations
Noncontractive Results
Solving Polynomial Operator Equations in Ordered Banach Spaces
Integral and Differential Equations
Equations of Hammerstein Type
Radiative Transfer Equations
Differential Equations
Integrals on a Separable Hilbert Space
Approximation of Solutions of Some Quadratic Integral Equations in Transport Theory
Multipower Equations
Uniformly Contractive Systems and Quadratic Equations in Banach Space
Polynomial Operators in Linear Spaces
A Weierstrass Theorem
Matrix Representations
Lagrange and Hermite Interpolation
Bounds of Polynomial Equations
Representations of Multilinear and Polynomial Operators on Vector Spaces
Completely Continuous and Related Multilinear Operators
General Methods for Solving Nonlinear Equations
Accessibility of Solutions of Equations by Newton-Like Methods and Applications
The Super-Halley Method
Convergence Rates for Inexact Newton-Like Methods at Singular Points
A Newton-Mysovskii-Type Theorem with Applications to Inexact Newton-Like Methods and Their Discretizations
Convergence Domains for Some Iterative Processes in Banach Spaces Using Outer and Generalized Inverses
Convergence of Inexact Newton Methods on Banach Spaces with a Convergence Structure
References
Glossary of Symbols
Subject Index

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