Iterative Methods without Inversion

Anatoly Galperin

• Publisher: Chapman and Hall/CRC
• Year: 2016
• Print ISBN: 9781498758925
• eBook ISBN: 9781315350745
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Iterative Methods without Inversion presents the iterative methods for solving operator equations f(x) = 0 in Banach and/or Hilbert spaces. It covers methods that do not require inversions of f (or solving linearized subproblems). The typical representatives of the class of methods discussed are Ulm's and Broyden's methods. Convergence analyses of the methods considered are based on Kantorovich's majorization principle which avoids unnecessary simplifying assumptions like differentiability of the operator or solvability of the equation. These analyses are carried out under a more general assumption about degree of continuity of the operator than traditional Lipschitz continuity: regular continuity.

Key Features

• The methods discussed are analyzed under the assumption of regular continuity of divided difference operator, which is more general and more flexible than the traditional Lipschitz continuity.
• An attention is given to criterions for comparison of merits of various methods and to the related concept of optimality of a method of certain class.
• Many publications on methods for solving nonlinear operator equations discuss methods that involve inversion of linearization of the operator, which task is highly problematic in infinite dimensions.
• Accessible for anyone with minimal exposure to nonlinear functional analysis.

Chapter 1

Some useful tools of the trade

This introductory chapter gives some known facts that will be needed later. The reader can skip it on the first reading and return to it only after being prompted by a reference.

1.1 Banach’s lemma on perturbations

Lemma 1.1. 1° A linear bounded operator A acting from one Banach space 𝕏 into another 𝕐 is boundedly invertible if and only if

$l\left(A\right):=\underset{\Vert x\Vert =1}{\min }\Vert Ax\Vert >0.$

(1.1)

In this case,

$\Vert A^{-1}\Vert =\frac{1}{l\left(A\right)}.$

(1.2)

2° For every two linear operators A and B,

$\left|l\left(A\right)-l\left(B\right)\right|\le \Vert A-B\Vert .$

Proof. 1° Let A be boundedly invertible. If l(A) = 0, then there exists a sequence x_n with $\Vert x_n\Vert ...$

Table of contents

• Cover
• Title Page
• Copyright Page
• Table of Contents
• INTRODUCTION
• 1 Tools of the trade
• 2 Ulm’s method
• 3 Ulm’s method without derivatives
• 4 Broyden’s method
• 5 Optimal secant updates of low rank
• 6 Optimal secant-type methods
• 7 Majorant generators and their convergence domains
• References
• Index