Hilbert Spaces and Its Applications 1536189839, 9781536189834

Table of contents :
HILBERT SPACESAND ITS APPLICATIONS
HILBERT SPACESAND ITS APPLICATIONS
CONTENTS
PREFACE
Chapter 1A NEWTON-TRAUB-LIKE FIFTHCONVERGENCE ORDER METHOD INHILBERT SPACE
Abstract
1. INTRODUCTION
2. LOCAL CONVERGENCE
REFERENCES
Chapter 2CORRECTING AND EXTENDING THEAPPLICABILITY OF TWO FAST ALGORITHMSFOR SOLVING SYSTEMS
Abstract
1. INTRODUCTION
2. SEMILOCAL CONVERGENCE
REFERENCES
Chapter 3 EXTENDED DIRECTIONAL NEWTON-TYPE METHODS
Abstract
1. INTRODUCTION
2. SEMI-LOCAL CONVERGENCE ANALYSIS OF(TSDNM) USING RECURRENT SEQUENCES
3. CONVERGENCE ANALYSIS USING RECURRENTFUNCTIONS
REFERENCES
Chapter 4 EXTENDED KANTOROVICH THEOREM FOR GENERALIZED EQUATIONS AND VARIATIONAL INEQUALITIES
Abstract
1. INTRODUCTION
2. THE MAIN RESULTS
REFERENCES
Chapter 5 EXTENDED THE APPLICABILITH OF NEWTON’S METHOD FOR EQUATIONSWITH MONOTONE OPERATOR
Abstract
1. INTRODUCTION
2. LOCAL CONVERGENCE
REFERENCES
Chapter 6IMPROVED LOCAL CONVERGENCE FORA PROXIMAL GAUSS-NEWTON SOLVER
Abstract
1. INTRODUCTION
2. LOCAL CONVERGENCE
REFERENCES
Chapter 7IMPROVED ERROR ESTIMATESFOR SOME NEWTON-TYPE METHODS
Abstract
1. INTRODUCTION
2. SEMI-LOCAL CONVERGENCE
3. ERROR BOUNDS OF NEWTON-LIKE METHODFOR POLYNOMIALS
REFERENCES
Chapter 8TWO NON CLASSICAL QUANTUM LOGICSOF PROJECTIONS IN HILBERT SPACEAND THEIR MEASURES
Abstract
1. INTRODUCTION
2. REAL-ORTHOGONAL PROJECTIONS
3. A PARTIAL PSEUDO ORDERING ONR-ORTHOGONAL PROJECTIONS
4. ALGEBRAIC PROPERTIES OF THE STRONGR-ORTHOGONAL PROJECTIONS WITH APURELY DISCRETE SPECTRUM
5. ABOUT PARTIAL ORDER ON STRONGREAL-ORTHOGONAL PROJECTIONS
6. MEASURE ON STRONG REAL-ORTHOGONALPROJECTIONS
7. APPENDIX
8. MEASURES IN HILBERT SPACE WITHINDEFINITE METRIC
9. INDEFINITE MEASURE ON P(GLEASON’S ANALOGY)
REFERENCES
Chapter 9EXTENDED FOURTH ORDER NEWTON-LIKEMETHOD UNDER !-CONTINUITYFOR SOLVING EQUATIONS
Abstract
1. INTRODUCTION
2. LOCAL CONVERGENCE
REFERENCES
Chapter 10ON THE SEMI-LOCAL CONVERGENCE OF HALLEY'S METHOD: AN EXTENSION
Abstract
1. INTRODUCTION
2. CONVERGENCE OF HA
REFERENCES
Chapter 11SEMI LOCAL CONVERGENCE CRITERIONOF NEWTON’S ALGORITHM FOR SINGULARSYSTEMS UNDER CONSTANT RANKDERIVATIVES: AN EXTENSION
Abstract
1. INTRODUCTION
2. CONVERGENCE OF NA
REFERENCES
Chapter 12EXTENDING THEGAUSS-NEWTON-ALGORITHM UNDERl−AVERAGE CONTINUITY CONDITIONS
Abstract
1. INTRODUCTION
2. SEMI-LOCAL CONVERGENCE
3. LOCAL CONVERGENCE
REFERENCES
Chapter 13ON THE SOLUTION OF GENERALIZEDEQUATIONS IN HILBERT SPACE
Abstract
1. INTRODUCTION
2. CONVERGENCE
3. NUMERICAL EXAMPLES
CONCLUSION
REFERENCES
Chapter 14NEWTON’S ALGORITHM ON RIEMANNIANMANIFOLDS: EXTENDED KANTOROVICH’STHEOREM
Abstract
1. INTRODUCTION
2. CONVERGENCE
REFERENCES
Chapter 15EXTENDED GAUSS-NEWTON-KURCHATOVALGORITHM FOR LEAST SQUARESPROBLEMS
Abstract
1. INTRODUCTION
2. CONVERGENCE OF GNKA
REFERENCES
Chapter 16EXTENDED GAUSS-NEWTON ALGORITHMFOR CONVEX COMPOSITE OPTIMIZATION
Abstract
1. INTRODUCTION
2. CONVERGENCE OF GNA
REFERENCES
Chapter 17EXTENDED LOCAL CONVERGENCE OFNEWTON’S ALGORITHM ON RIEMANNIANMANIFOLDS
Abstract
1. INTRODUCTION
2. CONVERGENCE
REFERENCES
Chapter 18UNIQUENESS OF THE SOLUTION OFEQUATIONS IN HILBERT SPACE: I
Abstract
1. INTRODUCTION
2. CONVERGENCE
REFERENCES
Chapter 19UNIQUENESS OF THE SOLUTION OFEQUATIONS IN HILBERT SPACE: II
Abstract
1. INTRODUCTION
2. CONVERGENCE
REFERENCES
Chapter 20EXTENDED NEWTON’S ALGORITHMON RIEMANNIAN MANIFOLDSWITH VALUES IN A CONE
Abstract
1. INTRODUCTION
2. SEMI-LOCAL CONVERGENCE
REFERENCES
Chapter 21EXTENDED GAUSS-NEWTON ALGORITHMON RIEMANNIAN MANIFOLDS UNDERL-AVERAGE LIPSCHITZ CONDITIONS
Abstract
1. INTRODUCTION
2. SEMI-LOCAL CONVERGENCE
REFERENCES
Chapter 22NEW RESULTS ON BEREZIN NUMBERINEQUALITIES IN REPRODUCING KERNELHILBERT SPACE
Abstract
1. INTRODUCTION
2. GENERALIZED BEREZIN NUMBER INEQUALITIES
3. GENERALIZED EUCLIDEAN BEREZIN NUMBERINEQUALITIES
4. BEREZIN NUMBER INEQUALITIES FOR2 × 2 OPERATOR MATRIX
REFERENCES
GLOSSARY OF SYMBOLS
ABOUT THE EDITORS
INDEX
Blank Page
Blank Page

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MATHEMATICS RESEARCH DEVELOPMENTS

HILBERT SPACES AND ITS APPLICATIONS

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MATHEMATICS RESEARCH DEVELOPMENTS Additional books and e-books in this series can be found on Nova’s website under the Series tab.

MATHEMATICS RESEARCH DEVELOPMENTS

HILBERT SPACES AND ITS APPLICATIONS

MICHAEL ARGYROS, IOANNIS K. ARGYROS AND

SAMUNDRA REGMI EDITORS

Copyright © 2021 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the Publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data ISBN: H%RRN

Published by Nova Science Publishers, Inc. † New York

The first editor dedicates this book to his beloved grandparents, Anastasia, Yolanda, Konstantinos and Michalakis. The second editor dedicates this book to his beloved wife, Diana. The third editor dedicates this book to his beloved parents, Moti Ram and Madhu Kumari.

CONTENTS Preface

xi

Chapter 1

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space Ioannis K. Argyros

Chapter 2

Correcting and Extending the Applicability of Two Fast Algorithms for Solving Systems Samundra Regmi

15

Chapter 3

Extended Directional Newton-Type Methods Michael I. Argyros

25

Chapter 4

Extended Kantorovich Theorem for Generalized Equations and Variational Inequalities Gus I. Argyros

39

Chapter 5

Extended the Applicability of Newton's Method for Equations with Monotone Operator Michael I. Argyros

51

Chapter 6

Improved Local Convergence for a Proximal GaussNewton Solver Samundra Regmi

59

Chapter 7

Improved Error Estimates for Some Newton-Type Methods Samundra Regmi

69

1

viii

Contents

Chapter 8

Two Non Classical Quantum Logics of Projections in Hilbert Space and Their Measures Marjan Matvejchuk and Elena Vladova

77

Chapter 9

Extended Fourth Order Newton-Like Method under  -Continuity for Solving Equations Ioannis K. Argyros

123

Chapter 10

On the Semi-Local Convergence of Halley's Method: An Extension Michael I. Argyros

131

Chapter 11

Semi Local Convergence Criterion of Newton's Algorithm for Singular Systems under Constant Rank Derivatives: An Extension Samundra Regmi

139

Chapter 12

Extending the Gauss-Newton-Algorithm under l-Average Continuity Conditions Gus I. Argyros

147

Chapter 13

On the Solution of Generalized Equations in Hilbert Space Samundra Regmi

157

Chapter 14

Newton's Algorithm on Riemannian Manifolds: Extended Kantorovich's Theorem Michael I. Argyros, Gus I. Argyros and Ioannis K. Argyros

165

Chapter 15

Extended Gauss-Newton-Kurchatov Algorithm for Least Squares Problems Gus I. Argyros, Michael I. Argyros and Ioannis K. Argyros

171

Chapter 16

Extended Gauss-Newton Algorithm for Convex Composite Optimization Michael I. Argyros, Gus I. Argyros, Samundra Regmi and Ioannis K. Argyros

177

Contents

ix

Chapter 17

Extended Local Convergence of Newton's Algorithm on Riemannian Manifolds Michael I. Argyros and Gus I. Argyros

185

Chapter 18

Uniqueness of the Solution of Equations in Hilbert Space: I Gus I. Argyros and Michael Argyros

193

Chapter 19

Uniqueness of the Solution of Equations in Hilbert Space: II Michael I. Argyros and Ioannis K. Argyros

199

Chapter 20

Extended Newton's Algorithm on Riemannian Manifolds with Values in a Cone Samundra Regmi

203

Chapter 21

Extended Gauss-Newton Algorithm on Riemannian Manifolds under L-Average Lipschitz Conditions Gus I. Argyros and Ioannis K. Argyros

211

Chapter 22

New Results on Berezin Number Inequalities in Reproducing Kernel Hilbert Space Satyajit Sahoo

219

Glossary of Symbols

239

About the Editors

241

Index

243

PREFACE This book contains a plethora of selected contemporary topics primarily in Hilbert space although related extended material in Banach space and Riemannian manifolds is also included. Numerous concrete problems from diverse disciplines such as Applied Mathematics; Mathematical Biology; Chemistry; Economics; Physics; Scientific Computing, and Engineering to mention a few reduce to solving an equation on the aforementioned spaces. The solution of such equations can be found in closed form only in special cases. That forces researchers and practitioners to the development of iterative method used to generate a sequence converging to the solutions provided that some convergence criteria depending on the initial data are satisfied. Due to the exponential development of technology, new iterative methods should be found to improve existing and create faster and more efficient computers. This is out motivation for writing this book. We have cooperated with the department of Computing and Technology although we work for the Department of Mathematical Sciences at Cameron University. The book contains 22 chapters. All chapters suggest or contain applications in the aforementioned areas including local and semi-local convergence analysis of the iterative methods. In the case, we revisit old method, then we extend their applicability by introducing new ideas and techniques, so general that they can be used on method not investigated in this book too along the same lines. For new methods, we use these ideas and technique or more fitting to take advantage of the particular structure of the method. The book also contains Chapter 8 and Chapter 22 which answer to important open theoretical problems. The book

xii

Michael Argyros, Ioannis K. Argyros and Samundra Regmi

contains problems not solved yet. The results reported in these chapters are new. Chapter 1: Motivated by the popularity of Newton’s as well as Traub’s method, a Newton-Traub-like method is developed of convergence order five. But the convergence is shown using the sixth derivative not appearing on the method. The usage of high order derivatives not appearing on the method significantly reduce their applicability. That is why we develop a local convergence analysis using only the first derivative that actually appears on the method. That is how we extend the applicability of this method. Moreover, we present computable error bounds on the distances involved as well as uniqueness of the solution results. Numerical examples complete this chapter. Chapter 2: The applicability of two fast algorithms is discussed with benefits as stated in Chapter 1. Chapter 3: Our idea of the restricted convergence domain is employed to extend the semi-local convergence of directional Newton-type methods. The advantages are weaker sufficient convergence criteria (so more initial points become available to be used for the convergence of the method) and fewer iterates are needed to achieve a predetermined error tolerance on the distances involved. Chapter 4: The celebrated Kantorovich Theorem is extended for solving variational inequalities. Chapter 5: Newton’s method is extended to solve equations involving monotone operator. Chapter 6: Contemporary convergence results are provided for the proximal Gauss-Newton method. Chapter 7: New and tighter estimates are presented for a certain class of Newton-type methods. Chapter 8: Are there logics or projections in Hilbert space other than orthogonal projections, which allow the construction of theory of quantum mechanics as productive as the logic of orthogonal projections in a complex Hilbert space setting? Chapter 9: The concept of w-continuity is used to extend the applicability of a fourth order Newton-like method with advantages similar to the one as already stated in Chapter 1. Chapter 10: The semi-local convergence of Halley’s method is extended with advantages similar to the ones in Chapter 3. Chapter 11: We extend the applicability of Newton algorithms for solving singular systems with constant rank derivatives.

Preface

xiii

Chapter 12: L-average conditions are used to extend the Gauss-Newton algorithm. Chapter 13: Generalized equations are solved in a Hilbert space setting. Chapter 14: New results are presented using Newton’s algorithm on Riemannian manifolds and an extended version of the Kantorovich’s theorem. Chapter 15: We develop a Gauss-Newton-Kurchatov method to solve least squares problems Chapter 16: A gauss-Newton algorithm is developed to solve convex compositive optimization problems. Chapter 17: The local convergence of Newton’s algorithm on Riemannian manifolds is extended using our new idea of the restricted convergence domain. Chapter 18 and Chapter 19: The uniqueness of the solution equations in Hilbert space is discussed. Several extensions are provided. Chapter 20: The applicability of Newton’s algorithms on Riemannian manifolds with values in cone is extended. Chapter 21: Modified L-average conditions are used to extend the GaussNewton algorithm. Chapter 22: Berezin number inequalities play a significant role in many areas of mathematics. In particular, refinements of generalized Euclidean Berezin numbers inequalities are presented. Moreover, an estimate of a Berezin number of 2 x 2 operator matrix is given.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 1

A N EWTON -T RAUB -L IKE F IFTH C ONVERGENCE O RDER M ETHOD IN H ILBERT S PACE Ioannis K. Argyros∗ Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma, US

Abstract Higher convergence order methods have been developed to approximate solutions of nonlinear Hilbert Space valued equations. In particular, we present an extended local convergence analysis for the celebrated Newton-Traub-like method.

AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, local convergence, iterative methods, algorithm, convergence criteria, Newton method

1.

INTRODUCTION

We solve equation F (x) = 0, ∗

Corresponding Author’s Email: [email protected].

(1.1)

2

Ioannis K. Argyros

where F : Ω ⊂ H1 → H2 is Frchet-differentiable, H1 , H2 are Hilbert spaces and Ω is open and convex. To find a solution x∗ ∈ Ω, we resort to the NewtonTraub-Like three step method of convergence order five defined for x0 ∈ Ω, and all n = 0, 1, 2, . . . by yn = xn − αF 0 (xn )−1 F (xn ) 1 [I − F 0 (xn )−1 F 0 (yn )]F 0 (xn )−1 F (xn ) 2α (1.2) 1 = zn − F 0 (xn )−1 F (zn ) − (I − F 0 (xn )−1 F 0 (yn ))F 0 (xn )−1 F (zn ), α

zn = xn − F 0 (xn )−1 F (xn ) −

xn+1

where α ∈ R − {0} or α ∈ C − {0}. Method (1.2) was studied in [32]. In the special case when H1 = H2 = Ri using Taylor expansions and derivatives up to the order six not appearing in method (1.2). Moreover, no computable error bounds on kxn − x∗ k or uniqueness of the solution x∗ results. But  these problems limit the applicability of method (1.2). For example, let Ω = − 12 , 32 . Define f on Ω as ( s3 log s2 + s5 − s4 , s 6= 0 f (s) = 0, s = 0. Then, x∗ = 1 and f 000 (s) = 6 log s2 +60s2 −24s+22. But f 000 (s) is unbounded on Ω. Hence the results in [32] cannot apply. In this chapter, we only use the first derivative that actually appears on method (1.2). Hence, we extend the applicability of the method. Our technique can be used to extend the applicability of other methods [1-34].

2.

L OCAL C ONVERGENCE

We shall use some real functions and parameters in the local convergence of method (1.2). Set S = [0, ∞). Suppose that there exists function w0 : S → S continuous and nondecreasing such that equation w0 (t) − 1 = 0,

(2.1)

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space 3 has a minimal positive solution denoted by %0 . Set S0 = [0, %0). Suppose that there exists functions w : S0 → S, w1 : S0 → S continuous and nondecreasing such that for

g1 (t) =

R1 0

R1 w((1 − θ)t)dθ + |1 − α| w1 (θt)dθ 0

1 − w0 (t)

and g1 (t) = g1 (t) − 1 equation g1 (t) = 0

(2.2)

has a minimal positive solution %1 ∈ (0, %0). Suppose that for functions g2 , g2 defined on S0 by

g2 (t) =

R1

w((1 − θ)t)dθ

0

1 − w0 (t)

+

1 2|α|

R1 (w0 (t) + w0 (g1 (t)t)) w1 (θt)dθ 0

(1 − w0 (t))2

and g 2 (t) = g2 (t) − 1, equation g2 (t) = 0

(2.3)

has a minimal solution %2 ∈ (0, %0). Suppose that equation w0 (g2 (t)t) − 1 = 0

(2.4)

has a minimal solution %3 ∈ (0, %), where % = min{%1 , %2}. Set S1 = [0, %).

4

Ioannis K. Argyros

Suppose that for functions g3 , g3 defined on S1 by

g3 (t) =

R1 " w((1 − θ)g2 (t)t) 0

1 − w0 (g2 (t)t)

+

1 |α|

+

R1 (w0 (t) + w0 (g2 (t)t)) w1 (θg2 (t)t)dθ 0

(1 − w0 (t))(1 − w0 (g2 (t)t))

R1 (w0(t) + w0 (g1 (t)t)) w1 (θg2 (t)t)dθ # 0

(1 − w0 (t))2

g2 (t)

and g 3 (t) = g3 (t) − 1, equation g3 (t) = 0

(2.5)

has a least solution % ∈ (0, %). We shall show that % is a radius of convergence for method (1.2). these definitions it follows that

By

0 ≤ w0 (t) < 1

(2.6)

0 ≤ w0 (g2 (t)t) < 1

(2.7)

0 ≤ gi (t) < 1

(2.8)

and

for each t ∈ [0, %) and i = 1, 2, 3. Define U (x∗ , r), U(x∗ , γ) to be the open and closed balls in H1 , respectively with center x∗ and of radius γ > 0. The following conditions (A) shall be used: (A1 ) There exists a simple solution x∗ of equation F (x) = 0. (A2 ) There exists w0 : S → S continuous and nondecreasing such that for each x ∈ Ω kF 0 (x∗ )−1 (F 0 (x) − F 0 (x∗ ))k ≤ w0 (kx − x∗ k). Set Ω0 = Ω ∩ U (x∗ , %0).

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space 5 (A3 ) There exist functions w : S0 → S, w1 : S0 → S continuous and nondecreasing such that for each x, y ∈ Ω0 kF 0 (x∗ )−1 (F 0 (y) − F 0 (x))k ≤ w(ky − xk) and kF 0 (x∗ )−1 F 0 (x)k ≤ w1 (kx − x∗ k). (A4 ) U (x∗ , %) ⊆ Ω (A5 ) There exists %∗ ≥ % such that Z1

w0 (θ%∗ )dθ < 1.

0

Set Ω1 = Ω ∩ U (x∗ , %∗ ). Next, the local convergence of method (1.2) is shown under the conditions (A) and the developed notation. Theorem 2.1. Under the conditions (A) further suppose that starter x0 ∈ U (x∗ , %) − {x∗ }. Then, sequence {xn } generated by method (1.2) is well defined in U (x∗ , %), remains in U (x∗ , %) and converges to x∗ . Moreover, for en = kxn − x∗ k the following error estimates hold kyn − x∗ k ≤ g1 (en )en ≤ en < %,

(2.9)

kzn − x∗ k ≤ g2 (en )en ≤ en

(2.10)

en+1 ≤ g3 (en )en ≤ en ,

(2.11)

and

where the gi, % are defined previously. Furthermore, x∗ is the only solution of equation F (x) = 0 in the set Ω1 given in (A5 ). Proof. Let v ∈ U (x∗ , %) − {x∗ }. Then, by (A2 ), we get that kF 0 (x∗ )−1 (F 0 (v) − F 0 (x∗ ))k ≤ w0 (kv − x∗ k) ≤ w0 (%) < 1.

(2.12)

6

Ioannis K. Argyros

Then, it follows by (2.12) and the Banach lemma on invertible operators [22] that F 0 (v) ∈ L(H2 , H1 ) and kF 0 (v)−1 F 0 (x∗ )k ≤

1 . 1 − w0 (kv − x∗ k)

(2.13)

We also have that y0 , z0 , and x1 are well defined by method (1.2) for n = 0. By method (1.2), and (A1 ) we can write y0 − x∗ = x0 − x∗ − F 0 (x0 )−1 F (x0 ) + (1 − α)F 0 (x0 )−1 F (x0 ).

(2.14)

Next, by (2.13) for v = x0 , (A3 ), (2.8) (for i = 1) and (2.14) we have in turn that ky0 − x∗ k ≤ kF 0 (x0 )−1 F 0 (x∗ )k× " Z1

F 0 (x∗ )−1 (F 0 (x∗ + θ(x0 − x∗ )) − F 0 (x0 ))dθ(x0 − x1 )k

k

0

+k

Z1

F 0 (x∗ )−1 F 0 (x∗ + θ(x0 − x∗ ))dθ(x0 − x∗ )k

0

≤

R1

#

R1 ( w((1 − θ)e0 )dθ + |1 − α| w1 (θe0 )dθ)e0 0

1 − w0 (e0 ) ≤ g1 (e0 )e0 ≤ e0 < %

0

(2.15)

showing (2.9) for n = 0 and y0 ∈ U (x∗ , %). Moreover, by the second sub-step of method (1.2) for n = 0, we similarly have but using (2.8) ( for i = 2) kz0 − x∗ k = kx0 − x∗ − F 0 (x0 )−1 F (x0 ) 1 0 − F (x0 )−1 (F 0 (x0 ) − F 0 (y0 ))F 0 (x0 )−1 F (x0 )k 2α 21 3 R R1 w((1 − θ)e )dθ (w (e ) + w (ky − x k)) w (θe )dθ 0 0 0 0 0 ∗ 1 0 6 7 1 60 7 0 + ≤6 7 e0 2 1 − w0 (e0 ) 2|α| (1 − w0 (e0 )) 4 5 ≤ g2 (e0 )e0 ≤ e0 ,

(2.16)

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space 7 showing (2.10) for n = 0 and z0 ∈ U (x∗ , %). In view of the last sub-step of method (1.2) for n = 0, (2.7), (2.13) (for v = z0 ), (2.8) (for i = 3) and (2.16), we obtain in turn kx1 − x∗ k = kz0 − x∗ − F 0 (z0 )−1 F (z0 )

≤

+ F 0 (x0 )−1 (F 0 (z0 ) − F 0 (x0 ))F 0 (z0 )−1 F (z0 ) 1 + F 0 (x0 )−1 (F 0 (x0 ) − F 0 (y0 ))F 0 (x0 )−1 F (z0 )k α R1 " w((1 − θ)kz0 − x∗ k)dθ 0

1 − w0 (kz0 − x∗ k)

+

R1 ((w0 (e0 ) + w0 (kz0 − x∗ k)) w1 (θkz0 − x∗k)dθ 0

(1 − w0 (e0 ))(1 − w0 (kz0 − x∗ k)) R1 (w0 (e0 ) + w0 (ky0 − x∗ k)) w1 (θkz0 − x∗ k)dθ # 1 0 kz0 − x∗ k + |α| (1 − w0 (e0 ))2 ≤ g3 (e0 )e0 ≤ e0 ,

(2.17)

showing (2.11) for n = 0 and x1 ∈ U (x∗ , %). Hence, we verified estimates (2.9) - (2.11) for n = 0. Suppose these estimates hold for m = 0, 1, 2, . . ., n − 1. Then, by replacing x0 , y0 , z0 , x1 by xm , ym , zm , xm+1 in the preceding calculations, we show (2.9) - (2.11) holds for all n. It then follows from the estimate em+1 = cem < %,

c = g3 (e0 ) ∈ [0, 1)

that lim xm = x∗

m→∞

and xm+1 ∈ U (x∗ , %).

Consider q ∈ Ω1 with F (q) = 0, and set C =

R1 0

F 0 (x∗ + θ(q − x∗ ))dθ. Then,

8

Ioannis K. Argyros

using (A2 ) and (A5 ), we get in turn that 0

kF (x∗ )

−1

0

(C − F (x∗ )k ≤

Z1

w0 (θkq − x∗ k)dθ

Z1

w0 (θ%∗ )dθ < 1,

0

≤

0

so x∗ = q by the inevitability of C and the identity 0 = F (q) − F (x∗ ) = C(q − x∗ ).

Remark 2.2. 1. By (A2 ), and the estimate kF 0 (x∗ )−1 F 0 (x)k = kF 0 (x∗ )−1 (F 0 (x) − F 0 (x∗ )) + Ik ≤ 1 + kF 0 (x∗ )−1 (F 0 (x) − F 0 (x∗ ))k ≤ 1 + w0 (kx − x∗ k) second condition in (A3 ) can be dropped, and w1 can be defined as w1 (t) = 1 + w0 (t). Notice that, if w1 (t) < 1 + w0 (t), then R1 can be larger than %. 2. The results obtained here can be used for operator G satisfying autonomous differential equations [2,5] of the form F 0 (x) = T (F (x)) where T is a continuous operator. Then, since F 0 (x∗ ) = T (F (x∗ )) = T (0), we can apply the results without actually knowing x∗ . For example let F (x) = ex − 1. Then, we can choose: T (x) = x + 1.

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space 9 3. The local results obtained here can be used for projection algorithms such as the Arnoldi’s algorithm, the generalized minimum residual algorithm(GMRES), the generalized conjugate algorithm (GCR) for combined Newton/finite projection algorithms and in connection to the mesh independence principle can be used to develop the cheapest and most efficient mesh refinement strategies [1-7]. 4. Let w0 (t) = L0 t, and w(t) = Lt. The parameter rA = 2L02+L was shown by us to the convergence radius of Newton’s algorithm xn+1 = xn − F 0 (xn )−1 F (xn ) for each n = 0, 1, 2, . . .. under the conditions (A1 ) − (A3 ) (w1 is not used). It follows that the convergence radius R of algorithm (1.2) cannot be larger than the convergence radius rA of the second order Newton’s algorithm. As already noted in [1,2] rA is the least as large as the convergence ball given by Rheinboldt [27] 2 rT R = , 3L1 where L1 is the Lipschitz constant on D, L0 ≤ L1 and L ≤ L1 . In particular, for L0 < L1 or L < L1 , we have that rT R < rA and

1 rT R → rA 3

as

L0 → 0. L1

That is our convergence ball rA is at most three times larger than Rheinboldt’s. The same value for rT R was given by Traub [33]. 5. It is worth noticing that algorithm (1.2) is not changing, when we use the conditions (A) of Theorem 2.1 instead of stronger conditions used in [32]. Moreover, we compute the computational order of convergence (COC) defined by   −x∗ k ln kxkxn+1 n −x∗ k  µ=  kxn−x∗ k ln kxn−1 −x∗ k

10

Ioannis K. Argyros or the approximate computational order of convergence   kx −xn k ln kxn+1 n −xn−1 k . µ1 =  kxn −xn−1 k ln kxn−1 −xn−2 k

This way we obtain in practice the order of convergence in a way that avoids the bounds given in [32] involving estimates up to the seventh Frchet derivative of F . 6. Clearly, if g3 is dropped, zn is xn+1 and r = min{r1, r2 },

(2.18)

we obtain from Theorem 2.1, the corresponding results for the corresponding three step algorithm.

R EFERENCES [1] Argyros, I.K., Computational Theory of Iterative Methods, Studies in Computational Mathematics, Elsevier Publ. Comp., New York, 15, 2007. [2] Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer Verlag, Berlin, Germany, 2008. [3] Argyros, I.K., Quadratic equations and applications to Chandrasekhar’s and related equations, Bull. Aust. Math. Soc., 32, 1985, 275-292. [4] Argyros, I.K and George, S., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications: Volume-III, Nova Science Publisher, New York, 2019. [5] Argyros, I.K and Hilout, S., Weaker conditions for the convergence of Newton’s method, J. Complexity, 28, 3, 2012, 364 - 387. [6] Argyros, I.K and Magren, A.A., A Contemporary Study of Iterative Methods, Academic Press, Elsevier, San Diego, CA, USA, 2018. [7] Argyros, I.K. and Regmi, S., Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces, Nova Science Publisher, New York, USA, 2019.

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space 11 [8] Chandrasekhar, S., Radiative Transfer, Dover, New York, 1960. [9] Cordero, A. and Torregrosa, J.R., Variants of Newton’s method for functions of several variables, Appl. Math. Comput., 183, 2006, 199-208. [10] Cordero, A. and Torregrosa, J.R., Variants of Newton’s method using fifthorder quadrature formulas, Appl. Math. Comput., 190, 2007, 686-698. [11] Cordero, A., Hueso, J.L., Martnez, E. and Torregrosa, J.R., A modified Newton-Jaratt’s composition, Numer. Algor., 55, 2010, 87-99. [12] Cordero, A., Hueso, J.L., Martnez, J.R. and Torrgrosa, J.R., Increasing the convergence order of an iterative method for nonlinear systems, Appl. Math. Lett., 25, 2012, 2369-2374. [13] Cordero, A., Soleymani, F. and Torregrosa, J.R., Dynamical analysis of iterative methods for nonlinear systems or how to deal with the dimension?, Appl. Math. Comput., 244, 2014, 398-412. [14] Darvishi, M.T. and Barati, A., A third-order Newton-type method to solve system of nonlinear equations, Appl. Math. Comput., 187, 2007, 630-635. [15] Fousse, L., Hanrot, G., Lefvre, V., Plissier, P., Zimmermann, P., MPFR: a multiple-precision binary floating-point library with correct rounding, ACM Trans. Math. Softw., 33, 2, 2007, 15. [16] Frontini, M. and Sormani, E., Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math. Comput., 149, 2004, 771-782. [17] Grau, M.S., Grau, A. and Noguero, M., On the computational efficiency index and some iterative methods for solving systems of nonlinear equations, J. Comput. Appl. Math., 236, 2011, 1259-1266. [18] Grau, M.S., Noguera, M. and Amat. S., On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. Appl. Math., 237, 2013, 363-372. [19] Homeier, H.H., A modified Newton method with cubic convergence: the multivariable case, J. Comput. Appl. Math., 169, 2004, 161-169.

12

Ioannis K. Argyros

[20] Jaratt, P., Some fourth order multipoint iterative methods for solving equations, Math. Comput., 20, 1996, 434-437. [21] Jay, L.O., A note on Q-order of convergence, BIT, 41, 2001, 422-429. [22] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [23] Kelley, C.T., Solving Nonlinear Equations with Newton’s Method, SIAM, Philadelphia, 2003. [24] Magren, .A., A new tool to study real dynamics: the convergence plane, Appl. Math. Comput., 248, 2014, 215-224. [25] Magren, A.A. and Argyros, I.K., Iterative Methods and Their Dynamics with Applications, CRC Press, New York, USA, 2018. [26] Noor, M.A. and Waseem, M., Some iterative methods for solving a system of nonlinear equations, Comput. Math. Appl., 57, 2009, 101-106. [27] Ortega, J.M. and Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [28] Petkovi, M.S., On a general class of multipoint root-finding methods of high computational efficiency, SIAM J. Numer. Anal., 49, 2011, 13171319. [29] Petkovi, M.S., Neta, B., Petkovi, L.D., and Duni, J., Multipoint Methods for Solving Nonlinear Equations, Elsevier, Boston, 2013. [30] Sharma, J.R. and Gupta, P., An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl., 67, 2014, 591-601. [31] Sharma, J.R., Guha, R.K. and Sharma, R., An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algor., 62, 2013, 307-323. [32] Sharma, J.R., Sharma, R., Kalra, N., A novel family of composite NewtonTraub method for solving systems of nonlinear equations, Appl. Math. Comput., 269, 2015, 520-535.

A Newton-Traub-Like Fifth Convergence Order Method in Hilbert Space 13 [33] Traub, J.F., Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964. [34] Wolfram, S., The Mathematica Book, fifth edition, Wolfram Media, 2003.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 2

C ORRECTING AND E XTENDING THE A PPLICABILITY OF T WO FAST A LGORITHMS FOR S OLVING S YSTEMS Samundra Regmi∗ Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma, US

Abstract It turns out that the sufficient semi-local convergence criteria of two fast algorithms for solving systems with nonlinear equations given in several articles are false. In this work, we present the corrected version. Moreover, we develop a new technique for extending the applicability of algorithms.

AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, semi-local convergence, Iterative Methods, algorithm, convergence criteria

1.

INTRODUCTION

Let F : Ω ⊆ Cm −→ Cm be a G-continuously differentiable mapping defined on an open and convex set Ω. ∗

Corresponding Author’s Email: [email protected].

16

Samundra Regmi

A plethora of problems from diverse disciplines reduce to finding a solution x∗ ∈ Ω of equation F (x) = 0. (1.1) For example in many discretization studies of nonlinear differential equations, we end up solving a system like (1.1) [1–16]. The solution x∗ can be found in closed form only in special cases. That forces researchers and practitioners to develop algorithms generating sequences approximating x∗ . Recently, there is a surge in the development of algorithms faster than Newton’s [6]. These algorithms share common setbacks limiting their applicability such as: the convergence region is small and the estimates on the error distances pessimistic in general. The information on the location of x∗ is not the best either. Motivated by these setbacks, we develop a technique extending the applicability of algorithms without adding conditions. In particular, we find a subset of Ω (on which the original Lipschitz parameters are defined) which also contains the iterates. But on this subset the parameters are special cases of the earlier ones and at least as tight. Therefore, they can replace the old ones in the proofs, leading to a finer convergence analysis. In section 2, we demonstrate our technique to extend the applicability of NHSS [13] and INHSS [3] algorithms. However, this technique is so general that it can be used to extend the applicability of other algorithms in an analogous way [1–16]. Next, we reintroduce NHSS and INHSS. Notice that the semi-local convergence analysis of these Algorithms given in [3] and [13], respectively are false, since they both rely on a false semi-local convergence criterion (see Remark 2.3). We reported this problem to the journal of Linear Algebra Applications, especially since many other related publications have appeared relying on the same false criterion (see (2.2)) without checking (see [3, 13] and the references therein). But unfortunately, they refused to publish the corrections, we submitted. That simply means among other things that this misinformation will not stop, since many other authors will use this false criterion in future works. This is our second motivation for presenting this work (with first being the introduction of a new technique). For brievity and to avoid repetitions, we refer the reader to [3,13] for the benefits and applications out of using these fast algorithms, and we only concern ourselves with the presentation of the correct version of the results. Following the notation of [3, 13], we have:

Correcting and Extending the Applicability of Two Fast Algorithms ... 17 NHSS Algorithm [13] Input: x(0) and tol. For n = 1, 2, . . . until kF (x(n) )k ≤ tolkF (x(0))k do: (n)

1 For hn ∈ [0, 1) find e1 such that (n)

kF (x(n) ) + F 0 (x(n) )e1 k < hn kF (x(n))k (n+1)

2 Set x∗

(n)

= x(n) + e1 .

(n)

3 Find e2 such that (n+1)

kF (x∗

(n+1)

4 Set x(n+1) = x∗

(n)

(n+1)

) + F 0 (x(n) )e2 k < hn kF (x∗

)k

(n)

+ e2 .

End INHSS Algorithm [3] 0 ∞ Input: w (0) and tol, a and positive integer sequences {µn }∞ n=0 , {µn}n=0 . (n) (0) For n = 1, 2, . . . until kF (w )k ≤ tolkF (w )k do: 1 set e1n,0 = 0. 2 For l = 0, 1, 2, . . .µn − 1 until kF (w (n)) + F 0 (w (n))e1n,µn k < hn kF (w (n))k apply the HSS algorithm: (aI + H(w (n))e1n,l+ 1 = (aI − S(w (n)))e1n,l − F (w (n) ) 2

(aI + S(w (n))e1n,l+1 = (aI − H(w (n)))e1n,l+ 1 − F (w (n) ) 2

(n+1)

3 Set w∗

= w (n) + e1n,ln .

4 Set e2n,0 = 0.

18

Samundra Regmi 5 For l 0 = 0, 1, 2, . . . ln0 − 1 until (n+1)

kF (w∗

(n+1)

) + F 0 (w (n) )e2n,l0n k < hn kF (w∗

)k

apply the HSS algorithm: (aI + H(w (n))e2n,l0 + 1 1 = (aI − S(w (n)))e2n,l − F (w (n+1) ) 2

(aI + S(w (n))e2n,l0 +1 = (aI − H(w (n)))e2n,l0+ 1 − F (w (n+1) ) 2

(n+1)

6 Ser w (n+1) = w∗

+ d2n,l0 . n

End

2.

SEMILOCAL C ONVERGENCE

Pick x(0) ∈ Cm and consider F : Ω ⊂ Cm −→ Cm to be a G-differentiable function on an open set Ω on which F 0 (x) is continuous and positive definite. Suppose that F 0 (x) = H(x) + S(x), with H(x) = 12 (F 0 (x) + F 0 (x)∗ ) and S(x) = 12 (F 0 (x) − F 0 (x)∗ ) being the Hermitian and Skew-Hermitian part of the Jacobian matrix F 0 (x), respectively. The semi-local convergence is based on the following conditions. (a1) There exist positive constants b, c and d such taht max{kH(x(0)k, kS(x(0))k} ≤ b, kF 0 (x(0))−1 k ≤ c, kF (x(0))k ≤ c. (a2) There exist nonnegative parameters `0h and L0s such that for all x ∈ U (x(0), r) ⊆ Ω, kH(x) − H(x(0))k ≤ `0h kx − x(0)k, kS(x) − S(x(0))k ≤ `0s kx − x(0)k, Set `0 = `0h + `0s and r0 =

1 `0 .

Correcting and Extending the Applicability of Two Fast Algorithms ... 19 (a3) (Lipschitz condition)There exist nonnegative constants `k and `s such that for all x, y ∈ U (x(0), `10 ) ∩ Ω, kH(x) − H(y)k ≤ `h kx − yk, kS(x) − S(y)k ≤ `s kx − yk, Set ` = `h + `s . By applying Banach’s lemma, the next result holds, improving the corresponding one in [13]. Lemma 2.1. Suppose conditions (a1)-(a3) hold. Then, we have (1) kF 0 (x) − F 0 (y)k ≤ `kx − yk, (2) kF 0 (x)k ≤ `0 kx − x(0)k + 2b, (3) If r ≤

1 c`0 ,

then F 0 (x) is nonsingular and satisfies kF 0 (x)−1 k ≤

c . 1 − c`0 kx − x(0)k

The following semi-local convergence result corrects the corresponding ones in [13, Theorem 1]. Theorem 2.2. Suppose conditions (a1)-(a3), so that dc2 ` ≤ η0 =

−η 3

(1 − η)2 , + 2η 2 + 3η + 4

(2.1)

where η = max{ηk } < 1, r = min{ρ1 , ρ2} with s ! a+b 2aξτ ρ1 = 1+ −1 , ` (2c + cξτ )(a + b)2 ρ2 = α=

β−

p β 2 − 2αγ , α

c`(1 + η) , β = 1 − η, γ = 2cd 1 + 2c2 d`η ln(η)

and with `∗ = limk−→∞ inf `k satisfying `∗ > b ln((ξ+1)τ c, ξ ∈ (0, 1−τ τ ) and τ = τ (a; x(0)) = kM (a; x(0))k,

20

Samundra Regmi

where bvc is the largest integer smaller than or equal to v. Then, the iteration sequence {x(k)}∞ k=0 generated by NHSS algorithm is well defined and converges to x∗ , with F (x∗ ) = 0. Next, we state and prove the extension of this result for INHSS method. We first introduce s0 = 0, sk+1 = sk −

f (sk ) , k = 0, 1, 2, . . . f0 (sk )

where f (s) = 12 αs2 − βs + γ and f0 (s) = αs − 1. It is shown in, [13] that the above sequence converges to ρ2 monotone increasingly and f (sk ) ≤ 0. So, we get sk < sk+1 < ρ2 and sk −→ s∗ (= ρ2 ). Theorem 2.3. Suppose that the tolerance in INHSS algorithm is smaller than 18 η, and conditions (a1)-(a3) hold for constants defined in Theorem 2.2 and condition (a1) is max{kH(w (0))k, kS(w (0))k} ≤ b, kF 0 (w (0))k ≤ c0 , kF (w (0))k ≤

d , 4

for an initial guess w (0). Moreover, µ∗ = ln η min{limn−→∞ inf µn , limn−→∞ inf µ0n }, satisfying µ∗ > b ln(ξ+1)τ c, ξ ∈ (0, 1−τ τ ) and τ = τ (a; w (0)) = kM (a; w (0))k < 1. Then, the iteration sequence {w (n) }∞ n=0 generated by INHSS algorithm is well defined and converges to w∗ with F (w∗ ) = 0. Further, sequence {w (n) }∞ n=0 hold the following relations 1 (1) kw∗ − w (0)k ≤ (s1 − s0 ), 4 (n)

kw∗ − w (n−1) k ≤

1 2n+3

(s2n−1 − sn−1 ), n = 2, 3, . . .

and also for n = 1, 2, . . ., we have (n)

kF (w∗ )k ≤

1 1 − cµs2n−1 (s2n − sn ), c(1 + c)

2n+3

w (n) − w (n−1) k ≤

1 (s2n − sn−1 ), 2n+2

Correcting and Extending the Applicability of Two Fast Algorithms ... 21 (n)

kF (w∗ )k ≤

1 1 − cµs2n (s2n+1 − sn ), 2n+2 c(1 + c)

1 r2 , 2 1 (n) kw∗ − w (0)k ≤ r2 , 4

kw (n) − w (0)k ≤

(n+1)

where c = 4c0 , w∗ = w (n) − F 0 (w (n) )−1 F (w (n) ), ρ2 is defined as in Theorem 2 and {sn } is defined previously. Remark 2.4. (a) The following stronger (than (a3)) condition was used in [3, 13] (a2’) kH(x) − H(y)k ≤ `¯h kx − yk kS(x) − S(y)k ≤ `¯s kx − yk for all x, y ∈ Ω0 . But, we have Ω1 ⊆ Ω0 , so

`h ≤ `¯h `s ≤ `¯s

and

` ≤ `¯ = `¯h + `¯s .

¯ Ω0, respectively in all the results Hence, `h , `s , `, Ω1 can replace `¯h , `¯s, `, in [3, 13]. Moreover, Lemma (2.1) reduces to Lemma 1 in [3, 13], if `h = `¯h and `s = `¯s . Otherwise it constitutes an improvement. That is how, we obtain the advantages already stated in the introduction. (b) On the other hand, we correct the semi-local convergence criterion δγ 2 `¯ ≤ η¯0 :=

1−η 2(1 + η 2 )

(2.2)

given in [3, 13] and other publications based on it. Indeed, if (2.2) exists ¯ then that does not necessarily imply b2 − 4ac ≥ 0 (with ` replaced by `),

22

Samundra Regmi so ρ1 and ρ2 may not exist. Then, there is no guarantee that the algorithms NHSS or INHSS converge to the solution. Notice also that η0 ≤ η¯0 . However, our condition (2.1) guarantees the existence of ρ1 and ρ2 . Hence, all the proofs of the results in [3, 13] break down at this point.

R EFERENCES [1] Amat, S., Bermudez, C, Hernandez, M. A., Martinez, E., On an efficient k−step iterative method for nonlinear equations, J. Comput. Appl. Math., 302, 2016, 258-271. [2] Amat, S.,Argyros, I. K., Busquier, S., Hern´andez, M. A., On two highorder families of frozen Newton-type methods, Numer. Linear. Algebra Appl., 25, 2018, e2126, 1-13. [3] Amiri, A, Cordero, A., Darvishi, M. T, Torregrosa, J.R., A fast algorithm to solve systems of nonlinear equations, J. Comput. Appl. Math., 354, 201), 242-258. [4] An, H. B., Bai, Z. Z., A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations, Appl. Numer. Math., 57, 2007, 235-252. [5] Argyros, I. K., Computational theory of iterative solvers. Series: Studies in Computational Mathematics, 15, Editors: C.K. Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A, 2007. [6] Argyros, I. K., Hilout, S., Weaker conditions for the convergence of Newton’s method. J. Complexity, 28, 2012, 364-387. [7] Argyros, I. K., Magr´en˜ an, A. A., A contemporary study of iterative methods, Elsevier (Academic Press), New York, 2018. [8] Argyros, I. K., Magr´en˜ an, A. A., Iterative methods and their dynamics with applications, CRC Press, New York, USA, 2017.

Correcting and Extending the Applicability of Two Fast Algorithms ... 23 [9] Argyros, I. K., Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics, 15, Editors: Chui C.K. and Wuytack L. Elsevier Publ. Company, New York, 2007. [10] Argyros, I. K., George, S., Mathematical modeling for the solution of equations and systems of equations with applications, Volume-IV, Nova Publishes, NY, 2020. [11] Argyros, I. K., George, S., On the complexity of extending the convergence region for Traub’s method, Journal of Complexity, 56, 2020, 101423, https://doi.org/10.1016/j.jco.2019.101423 [12] Bai, Z. Z., Yang, X., On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math., 59, 2009, 2923-2936. [13] Guo, X.P., Duff, I.S., Semilocal and global convergence of the NewtonHSS method for systems of nonlinear equations, Linear Algebra Appl., 18, 2010, 299-315. [14] Magre˜na´ n, A. A., Cordero, A., Guti´errez,J. M., Torregrosa, J. R., Real qualitative behavior of a fourth-order family of iterative methods by using the convergence plane, Mathematics and Computers in Simulation, 105, 2014, 49-61. [15] Magre˜na´ n, A. A., Argyros, I. K., Two-step Newton methods. Journal of Complexity, 30, 4, 2014, 533-553. [16] Ortega, J.M., Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 3

E XTENDED D IRECTIONAL N EWTON -T YPE M ETHODS Michael I. Argyros∗ Department of Computing and Technology, Cameron University, Lawton, Oklahoma, US

Abstract Directional Newton-type methods have been used extensively to generate sequences converging to solutions of Hilbert space valued equations. But the convergence domain is very small in general limiting their applicability. That is why we developed a technique to increase the convergence domain without additional hypotheses.

AMS AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, Local convergence, Iterative Methods, Algorithm, Convergence Criteria, Newton method

1.

INTRODUCTION

In this chapter, we are concerned with the problem of approximating a zero x∗ of a differentiable function F defined on a convex subset D of a Hilbert space H, with values in R. ∗

Corresponding Author’s Email: [email protected].

26

Michael I. Argyros

A large number of problems in applied mathematics and also in engineering are solved by finding the solution of certain equations. More specifically, when it comes to computer graphics, we often need to compute and display the intersection C = A ∩ B of two surfaces A and B, where H = R3 [6,11]. If the two surfaces are explicitly given by A = {(u, v, w)T : w = F1 (u, v)} and B = {(u, v, w)T : w = F2 (u, v)}, then, the solution x∗ = (u∗ , v ∗ , w ∗ )T ∈ C must satisfy the nonlinear equation F1 (u∗ , v ∗ ) = F2 (u∗ , v ∗) and w ∗ = F1 (u∗ , v ∗ ). Hence, we must solve a nonlinear equation in two variable x = (u, v)T of the form F (x) = F1 (x) − F2 (x) = 0. The marching method can be used to compute the intersection C. In this method, we first need to compute a starting point x0 = (u0 , v0 , w0 )T ∈ C, and then compute the succeeding intersection points by successive updating. In mathematical programming [11,12,15], for an equality-constraint optimization problem, e.g., min ψ(x) s.t. F (x) = 0 where, ψ, F : D ⊆ H = Ri → R are nonlinear functions, we need a feasible point to start a numerical algorithm. That is, we must compute a solution of equation F (x) = 0. In the case of a system of nonlinear equations G(x) = 0, with G : D ⊆ H = Ri → Ri , we may solve instead kG(x)k2 = 0, if the zero of function G is isolated or locally isolated and if the rounding error is neglected [11,15]. The directional Newton Method (DNM) [1,6,11] given by xn+1 = xn −

F (xn ) dn ∇F (xn ) · dn

(n ≥ 0)

Extended Directional Newton-Type Methods

27

is used to generate a sequence {xn } converging to x∗ , when H = Ri . Let us explain how (DN M ) is conceived. We start with an initial guess x0 ∈ D, where F is differentiable and a direction vector d0 . Then, we restrict F on the line A = {x0 + θd0 , θ ∈ R}, where it is a function of one variable f (θ) = F (x0 + θd0 ). Set θ0 = 0 to obtain the Newton iteration for f , that is the next point: v1 = −

f (0) . f 0 (0)

The corresponding iteration for F is x1 = x0 −

F (x0 ) d0 . ∇F (x0 ) · d0

Note that f (0) = F (x0 ) and f 0 (0) is the directional derivative f 0 (0) = F 0 (x0 , d0 ) = ∇F (x0 ) · d0 . by repeating this process we arrive at (DNM), If i = 1, (DNM) reduces to the classical Newton method [10]. Remark 1.1. A semi-local convergence analysis for (DNM) was provided in the elegant work by Levi and Ben-Isreal in [11]. The convergence of the method was established for directions dn sufficiently close to the gradients ∇F (xn ), and under standard Newton-Kantorovich-type hypotheses [10]. In particular, the following conditions were used: (A1 ) |F (x0 )| ≤ λ; (A2 ) |∇F (x0 ) · d0 | ≥ a; (A3 ) sup kF 00 (x)k = M1 ; x∈D

(A4 ) hk = λM1 a−2 ≤ In view of (A3 ), there exist M0 , such that:

1 . 2

28

Michael I. Argyros

(A5 ) k∇F (x) − ∇F (x0 )k ≤ M0 kx − x0 k,

x ∈ D.

Clearly, M0 ≤ M1

(1.1)

1 hold in general, and M M0 can be arbitrarily large [4]. Note that (A5 ) is not an additional hypothesis, since in practice the computation of M , requires that of M0 .

Using hypotheses (A1 ), (A2), (A3), (A5 ), and replacing (A4 ) by (A∗4 )

1 hA = λM a−2 ≤ , 2

where,

q 1 M = (M1 + 4M0 + M12 + 8M0 M1 ), 8 and Lipschitz conditions (along the lines of works in [2-5]), Argyros [1], provided a semi-local convergence analysis with the following advantages over the work in [11]: 1. Weaker hypotheses; 2. Larger convergence domain for (DNM); 3. Finer error bounds on the distances kxn+1 − xn k, kxn − x∗ k n ≥ 0; 4. An at least as precise information on the location of the zero x∗ .

The main idea used the needed (A5 ) instead of (A3 ) for the computation of the bounds on k∇F (xn )k. This modification leads to more precise estimates which in turn provide the advantages as mentioned above. Note that

1 1 ⇒ hA ≤ (1.2) 2 2 but not necessarily vice versa (unless if M1 = M0 ). Hence, the applicability of (DNM) is extended under the same computational cost, provided that M0 < hk ≤

Extended Directional Newton-Type Methods

29

M 1 . Theseresults can be extended even further, if we defined D0 = D ∩ 1 U x0 , aM and replace (A3 ) by the tighter 0 sup kF 00 (x)k = M. x∈D0

It follows that M ≤ M1 , since D0 ⊂ D. From now on, we assume that M0 ≤ M. Otherwise the following results hold with M0 replacing M . Hence, the proofs in [11] can be rewritten with M replacing M1 used there. It is worth noticing that the iterates xn ∈ D0 which is a tighter domain than D used in [11]. Therefore, again we obtain 1-4 listed previously. It is also important to see that these advantages are obtained under the same computational effort, since in practice the computation of M1 requires that of M0 and M . Our technique can be used to extend the applicability of other iterative methods along the same lines. Next, we demonstrate it on a more general method too. The proofs are omitted. We provide an extended semi-local convergence analysis of the Two Step Direction Newton Method (TSDNM) for starting x0 ∈ D: yn = xn − Γn F (xn )dn;

1 , F (x) · dn p ∈ (0, 1], Γn =

zn = xn + p(yn − xn ),  1  xn+1 = xn − 2 Γn (p2 + p + 1)F (xn ) + F (zn ) dn . p (TSDNM) is the Hilbert space analog of the cubically convergence two step Newton method defined on a Banach space X, which values in a Banach space Y , and given by yn = xn − F 0 (xn )−1 F (xn ); zn = xn + p(yn − xn ), p ∈ (0, 1],   1 xn+1 = xn − 2 F 0 (xn )−1 (p2 + p + 1)F (xn ) + F (zn ) . p

30

Michael I. Argyros

where F 0 (x)−1 ∈ L(Y, X) stands for the Frchet-derivative of the operator F [10]. We provide two different and competing technique to generate the sufficient semi-local convergence conditions as well as the corresponding error bounds for the cubically convergence (TSDNM). The first technique (see Section 2) uses the concept of recurrent sequences. The second technique (see Section 3) uses our idea of recurrent functions already employed in a Banach space setting [7]. Moreover, we compare the two techniques.

2.

SEMI -L OCAL C ONVERGENCE A NALYSIS OF (TSDNM) U SING R ECURRENT SEQUENCES

It is convenient for us to define functions: 1
1 − αt 1 + 12 t t g(t) = (8 + 4t + t2 ), 8 h(t) = g(t)f (t)2 − 1, f (t) =

and h1 (t) = h(t) + 1, where , α ∈ [0, 1]. Denote by r= the only positive zero of function

2 p α + α(α + 2)

f1 (t) =

1 . f (t)

We shall provided some properties on the above functions. Lemma 2.1. Let functions f, f1 , g, h, h1, and number γ, be as defined in previously. Then, the following hold: (a) Function h has a minimal positive zero in (0, γ) denote by ξ. For t ∈ (0, ξ):

Extended Directional Newton-Type Methods

31

(b) f1 is decreasing, and f1 (t) ∈ (0, 1); (c) g, h, and h1 are increasing, and g(t), h1(t) ∈ (0, 1); (d) f1 (θt) ≥ f1 (t), g(θt) ≤ θg(t), h1 (θt) ≥ θh1 (t) for all θ ∈ [0, 1]. Note that Lemma 2.1 reduces to the corresponding one in [6] provided that α = 1. We suppose the following conditions (C) hold for x0 ∈ D: (C1 ) |∇F (x0 )| ≥ β > 0; (C2 ) 0 < |F (x0 )| ≤ λ; (C3 )0 k∇F (x) − ∇F (x0 )k ≤ M0 kx − x0 k for all x ∈ D.   1 . Set D0 = D ∩ U x0 , βM 0 (C3 ) k∇F (x) − ∇F (y)k ≤ M kx − yk,

for all x, y ∈ D0 ;

(C4 ) kdn k = 1 (n ≥ 0), and there exists α ∈ (0, 1], such that |∇F (xn ) · dn | ≥ αk∇F (xn )k (n ≥ 0); (C5 ) For a0 =

Mλ , (αβ)2

and R=

1+

a0 2




λ βα

1 − f (a0 )g(a0)

we have U (x0 , R) ⊆ D;

,

32

Michael I. Argyros

(C6 ) a0 < ξ. The angle between two element x, y ∈ D denoted by ∠(x, y) is given by ∠(u, v) = arccos

u·v , kuk · kvk

u 6= 0, v 6= 0.

Condition (C4 ) is equivalent to ∠(∇F (xn ), dn) ≤ arccos α. We need the following result relating iterates {xn }. Lemma 2.2. Assume iterate {xn } generated by (TSDNM) are well defined for all n ≥ 0. Then, the following identities hold for all n ≥ 0: F (zn ) = (1 − p)F (xn ) + p

Z1

(∇F (xn + pθ(yn − xn )) − ∇F (xn ))(yn − xn )dθ,

0

F (xn) =

1 p

Z1

+

Z1

(∇F (xn) − ∇F (xn + pθ(yn − xn )))(yn − xn )dθ

0

(∇F (xn + θ(xn+1 − xn )) − ∇F (xn ))(xn+1 − xn )dθ,

0

and 1 xn+1 − yn = − Γn p

Z1

(∇F (xn + pθ(yn − xn )) − ∇F (xn ))(yn − xn )dθ.

0

Lemma 2.3. Under the (C) conditions, the following items hold for all n ≥ 1: (I) k∇F (xn )k = 6 0 and k∇F (xn )k−1 ≤ f (an1 )k∇F (xn−1 )k−1 ; (II) kyn − xn k ≤ f (an−1 )g(an−1 )kyn−1 − xn−1 k λ ≤ (f (a0 )g(a0 ))nky0 − x0 k < ; αβ (III) k∇F (xn )k−1 kyn − xn k ≤

an ; M

Extended Directional Newton-Type Methods (IV) kxn−1 − yn k ≤ (V)

33

an kyn − xn k; 2

 an  kyn − xn k; kxn+1 − xn k ≤ 1 + 2

(VI)  an  1 − (f (a0 )g(a0 ))n+1 kxn+1 − x0 k ≤ 1 + ky0 − x0 k < R. 2 1 − f (a0 )g(a0) We can show the main semi-local convergence result for (TSDNM). Theorem 2.4. Let F : D ⊆ H → R be a differentiable function on an nonempty open convex domain D of a Hilbert space H, with values in R, and let x0 ∈ D. Assume the (C) conditions hold. Then, sequence {xn } generated by (TSDNM) is well defined, remain in U (x0 , R) for all n ≥ 0, and converges to a zero x∗ of function in U (x0 , R). Moreover, k∇F (x∗ )k = 6 0, unless if kx∗ − x0 k = R.

3.

C ONVERGENCE A NALYSIS U SING R ECURRENT F UNCTIONS

It is convenient for us to define some parameters, and sequences. Definition 3.1. Let α > 0, β > 0, L0 > 0, L > 0, and λ ≥ 0, be given constants. Define parameters: λ , αβ Ls2 t1 = s0 + 0 , 2 η = s0 =

0 < δ1 =

2L p < 1, 2L0 + L + (2L0 + L)2 + 8L0 L

is the unique positive root of polynomial G1 given by:

G1 (w) = 2L0 w 2 + (2L0 + L)w − L.

34

Michael I. Argyros

δ2 ∈ (0, 1) is the unique positive root of polynomial G2 given by G2 (w) = (2L0 + L)w 3 + (L + 2L0 )w 2 − 2L, Ls0 , δ3 = δ3 (η) = 2 L((s0 − t0 )2 + (t1 − t0 )2 )(s0 − t0 )−1 δ4 = δ4 (η) = , 2(1 − L0 t1 ) 1 − L0 η , w∞ = 1 + L0 η µ = max{δi , i = 2, 3, 4}, and scalar sequence {tn }, {sn} by: t0 = 0,

s0 = η,

L(sn − tn )2 , 2(1 − L0 tn ) L((sn − tn )2 + (tn+1 − tn )2 ) = tn+1 + 2(1 − L0 tn+1 )

tn+1 = sn + sn+1 Note that

G2 (w) − G1 (w) = ((2L0 + L)w + L)(w 2 − 1). In particular, we have G2 (δ2 ) − G1 (δ2 ) = ((2L0 + L)δ2 + L)(δ22 − 1) < 0, which implies G1 (δ2 ) > 0, since G2 (δ2 ) = 0, and δ2 < 1. That is we conclude δ1 < δ2 . We need the following result on majorizing sequences for (T SDN M ): Lemma 3.2. Assume: µ ≤ w∞ . Set η∗ =

1 ; L + L0 η < η ∗.

Extended Directional Newton-Type Methods

35

Then, sequence {tn }, {sn }(n ≥ 0) are well defined, nondecreasing, bounded above by:   2δ ∗∗ t = 1+ η, 2−δ and converge to their common and unique least upper bound t∗ ∈ [0, t∗∗]. Moreover, the following estimates hold for all n ≥ 0: tn ≤ sn ≤ tn+1 ≤ sn+1 , and 0 ≤ sn+1 − tn+1

δ ≤ (sn − tn ) ≤ 2

 n+1 δ η, 2

where δ = 2µ. Let us drop condition (C3 ) in the set of conditions of Lemma 2.2. Denote by (C ∗ ), the resulting set of conditions together with (C3∗ ). Note is not an additional condition, since in practice the computation of M requires that constant M0 . We can state the main semi-local convergence result for (TSDNM) using recurrent functions. Theorem 3.3. Let F : D ⊆ H → R be a continuously differentiable function, and let x0 ∈ D. Assume hypotheses of C ∗ hold for M0 M λ L0 = , L= , η= . β αβ αβ Furthermore, assume U (x0 , t∗ ) ⊆ D. Then, sequence {xn } generated by (TSDNM) is well defined, remains in U (x0 , t∗ ) for all n ≥ 0, and converges to a zero function F . Moreover, the following estimates hold for all n ≥ 0: kxn+1 − yn k ≤ tn+1 − sn , kyn+1 − xn+1 k ≤ sn+1 − tn+1 , kyn+1 − yn k ≤ sn+1 − sn , kxn+1 − xn k ≤ tn+1 − tn , and kxn+1 − x∗ k ≤ t∗ − tn ,

36

Michael I. Argyros

where, sequence {tk }, {sn }, and t∗ are given in Definition 3.1.

R EFERENCES [1] Argyros, I.K., A semilocal convergence analysis for directional Newton methods, Mathematics of Computation, AMS, 50, 273, 2011, 327-342. [2] Argyros, I.K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298, 2004, 374-397. [3] Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer Verlag, Berlin, Germany, 2008. [4] Argyors, I.K., On the Newton-Kantorovich hypothesis for solving equations, J. Comput. Appl. Math., 169, 2004, 315-332. [5] Argyros, I.K. and George, S., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications: Volume-III, Nova Science Publisher, New York, 2019. [6] Argyros, I.K. and Hilout, S., Directional Newton methods, Math. Comput., 2004. [7] Argyros, I.K. and Hilout, S., Weaker conditions for the convergence of Newton’s method, J. Complexity, 28, 3, 2012, 364 - 387. [8] Argyros, I.K. and Magren, A.A., A Contemporary Study of Iterative Methods, Academic Press, Elsevier, San Diego, CA, USA, 2018. [9] Argyros, I.K. and Regmi, S., Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces, Nova Science Publisher, New York, USA, 2019. [10] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [11] Levin, Y. and Ben, A.I., Directional Newton methods in n variables, Mathematics of Computation, A.M.S., 71, 2002, 251-262.

Extended Directional Newton-Type Methods

37

[12] Lukcs, G., The generalized inverse matrix and the surface-surface intersection problem, Theory and practice of geometric modeling, Springer, Berlin, 1989, 167-185. [13] Magren, A.A. and Argyros, I.K., Iterative Methods and Their Dynamics with Applications, CRC Press, New York, USA, 2018. [14] Ortega, J.M. and Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970. [15] Pereyra, V., Iterative methods for solving nonlinear least square problems, SIAM J. Numer. Anal., 4, 1967, 27-36. [16] Polyak, B.T., Introduction to optimization, Optimization Software Inc., Publications Division, New York, 1987. [17] Ralston, A. and Rabinowitz, P., A first course in numerical analysis, 2nd Edition, McGraw-Hill, 1978. [18] Stoer, J. and Bulirsch, K., Introduction to numerical analysis, SpringerVerlag, 1980. [19] Walker, H.F. and Watson, L.T., Least-change secant update methods, SIAM J. Numer. Anal., 27, 1990, 1227-1262.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 4

E XTENDED K ANTOROVICH T HEOREM FOR G ENERALIZED E QUATIONS AND VARIATIONAL I NEQUALITIES Gus I. Argyros∗ Department of Computing and Technology, Cameron University, Lawton, Oklahoma, US

Abstract The Kantorovich theorem is a very useful tool for determining existence and uniqueness results for Hilbert space valued equations. We present an extended version of this theorem to find solutions in cases not possible before.

AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, Kantorovich theorem, Iterative Methods, algorithm, convergence criteria, error estimates

1.

INTRODUCTION

Let H be a Hilbert space, let C be a closed convex subset of H with non-empty interior D, let f : C → H be a continuous function that is Frchet-differentiable ∗

Corresponding Author’s Email: [email protected].

40

Gus I. Argyros

on D, and let g be a non-empty maximal monotone operator defined on H × H, fixed all through this chapter. Then there exists α ≥ 0 (the monotonicity modulus of g) such that [x1 , y1 ] ∈ g and [x2 , y2 ] ∈ g ⇒ (y2 − y1 , x2 − x1 ) ≥ αkx1 − x2 k2 .

(1.1)

It is well known [17] that g is closed in the sense that [xm , ym ] ∈ g,

lim xm = x and lim ym = y ⇒ [x, y] ∈ g.

m→+∞

m→∞

(1.2)

In the sequel, we will regard the statements [x, y] ∈ g, g(x) 3 y, −y + g(x) 3 0 and y ∈ g(x) as synonymous. Given any u0 ∈ D and r > 0, B[u0 , r] will designate the closed ball {x ∈ H : kx − u0 k ≤ r} and B(u0 , r) will designate the corresponding open wall. We are interested in the solvability of the generalized equation f (u) + g(u) 3 0.

(1.3)

Our goal in this chapter is to prove a weak Kantorovich theorem that generalizes the Kantoroich theorem for generalized equations proved previously by Uko [20,21] and a Hilbert space version of the classical Kantorovich theorem [13,14] on the solvability of nonlinear operator equations. The Kantorovich theorem is a fundamental tool in nonlinear analysis for proving the existence and uniqueness of solutions of equations arising in various fields. An important extension of the Kantorovich theorem was obtained by Argyros [3,5] who used a combination of Lipschitz and center-Lipschitz conditions in place of the Lipscthiz conditions used by Kantorovich. In this chapter, we will formulate and prove an extension of the Kantorovich theorem for the generalized equation (1.3). The depth and scope of this theorem is such that when we specialize it to nonlinear operator equations we get results that are weaker than the Kantorovich theorem and the Argyros extension of it. Our approach will be iterative, and the solution of problem (1.3) will be obtained as the limit of the solution of the generalized Newton subproblems f 0 (um )um+1 + g(um+1 ) 3 f 0 (um )um − f (um ),

m = 0, 1, . . .

(1.4)

A well known example [17] of a maximal monotone operator is obtained on setting g(x) = ∂φ(x) := {v ∈ H : φ(x) − φ(y) ≤ (v, x − y) ∀y ∈ H}

41

Extended Kantorovich Theorem for Generalized Equations ...

Where φ : H → (−∞, ∞) is a proper lower semicontinuous convex function. In this case, problem (1.3) becomes the variational inequality f (u) + ∂φ(u) 3 0.

(1.5)

Such problems were introduced in the early sixties by Stampacchia [18] and have found important applications in the physical and engineering sciences and in many other fields. In the next section, we will use the method of majorant sequences to formulate and prove a weak Kantorovich-type solvability theorem for problem (1.3). The scope of this theorem is such that when we specialize it to nonlinear operator equations, variational inequalities and nonlinear complementarity problems in Corollaries 1-2, we obtain novel results for these problems as well.

2.

T HE MAIN R ESULTS

In the next results, we define the majorant scalar sequence that will be used in the sequel and give its main properties. Proposition 2.1. Let a, α, b, M and M0 be nonnegative constants. Let c0 > b −α, a = c0 +α and 0 ≤ M0 ≤ M and suppose that (4M0 + M +

p M 2 + 8M M0 )a ≤ 4(c0 + α);

(2.1)

The Inequality (2.1) is strict if M0 = 0. Then the sequence t0 = 0, t1 = a, tm+1 := tm +

M (tm − tm−1 )2 2(c0 + α − M0 tm )

(2.2)

is well defined and converges to a real number t∗ that satisfied the inequality M0 t∗ ≤ c0 + α.

(2.3)

Proof. If M0 = 0, then (2.3) holds trivially. In this case, an induction argument  2k Ma 0 +α) shows that tk+1 − tk = 2(cM for k = 0, 1, . . ., and therefore 2(c0 +α) m

tm+1 = t1 + (t2 − t1 ) + · · · + (tm+1

2(c0 + α) X − tm ) = M

k=0



Ma 2(c0 + α)

2k

42

Gus I. Argyros

So, in this case, we have t∗ = lim tm m→∞

 2k m  2(c0 + α) X Ma = M 2(c0 + α) k=0

We observe that this series converges since k ≤ 2k and M2a < c0 + α and that is bounded above by the number k ∞  Ma 4(c0 + α)2 2(c0 + α) X = . M 2(c0 + α) M [2(c0 + α) − M a] k=0

In the rest of the proof, we assume that M ≥ M0 > 0. If we set θ = √ 4M , then it is easy to see that 1 ≤ θ < 2 and 2 M+

M +8M M0

M0 θ2 + M θ = 2M.

(2.4)

Moreover, it is easy to see that (2.1) can be rewritten in the form: (M + M0 θ)a ≤ θ(c0 + α).

(2.5)

We show by induction that the inequalities (M + M0 θ)tm − M tm−1 ≤ θ(c0 + α) (2.6) m    M (2.7) M0 tm ≤ (c0 + α) 1 − (M + M0 θ) hold from all m ≥ 1. It follows from (2.5) that they hold when m = 1. Suppose, by induction, that these inequalities hold for a certain value of m. Then, it follows from (2.6) that M (tm − tm−1 ) θ ≤ . 2(c0 + α − M0 tm ) 2

(2.8)

Hence, by invoking (2.2), (2.4), (2.8) and the induction hypotheses, we see that (M + M0 θ)(tm − tm−1 )[M (tm − tm−1 )] [2(c0 + α − M0 tm )] θ ≤ (M + M0 θ)tm + (M + M0 θ)(tm − tm−1 )( ) 2 = (M + M0 θ)tm + M (tm − tm−1 ) ≤ (M + M0 θ)tm + θ(c0 + α − M0 tm ) = θ(c0 + α) + M tm .

(M + M0 θ)tm+1 = (M + M0 θ)tm +

Extended Kantorovich Theorem for Generalized Equations ...

43

This shows that the inequality (2.6) holds when m + 1 replaces m and implies because of (2.7) that tm+1 ≤ ≤ =

M tm + θ(c0 + α) M + M0 θ     m  M M (c + α) 1 − + θ(c0 + α) 0 M0 M +M0 θ "

c0 + α 1− M0



M + M0 θ m+1 # M . M + M0 θ

Therefore (2.7) also hold when we replace m with m + 1. We conclude, by induction, that (2.6) - (2.7) hold for all m. Then, it follows from (2.8) and (2.2) that  k  k θ θ θ tk+1 − tk ≤ (tk − tk−1 ) ≤ · · · ≤ (t1 − t0 ) = a 2 2 2 and therefore that tm+1 =

m X k=0

=

(tk+1 − tk ) ≤ a

h a 1−

θ 2

1−


m+1 i θ 2

≤

m  k X θ k=0

2

a . 1 − θ2

This shows that the sequence {tm } is monotonically increasing and bounded, and as such it converges to its supremum t∗ . To complete the proof, we let m tend to infinity in (2.7) and see that the inequality (2.3) follows immediately. The following result from [8] will be useful in the sequel. Lemma 2.2. Let g be a maximal monotone operator satisfying condition (1.1) and let A be a bounded linear operator mapping H into H. If there exists c ∈ R such that c > −α and (Ax, x) ≥ ckxk2 ,

∀x ∈ H.

Then, for any b ∈ H, the problem Az + g(z) 3 b

44

Gus I. Argyros

has a unique solution z ∈ H. Next, we use the majorant sequence (2.2) to prove a weak Kantorovich-type existence theorem for problem (1.3). One of the key aspects of this theorem is the use of center-Lipschitz condition that have been used by Argyros [1-7] in other contexts. Theorem 2.3. Let g be a maximal operator satisfying condition (1.1) and suppose that there exist u0 ∈ D and v0 ∈ H such that v0 ∈ g(u0 ),

kf 0 (x) − f 0 (u0 )k ≤ M0 kx − x0 k 2

0

(f (u0 )x, x) ≥ c0 kxk , 0

0

∀x ∈ D0 ,

∀x ∈ D,

kf (x) − f (y)k ≤ M kx − yk ∀x, y ∈ D0 Set h c0 + α  D0 = D ∩ B u0 , M0 a kf (u0 ) + v0 k ≤ b, b where b ≥ 0, M ≥ M0 ≥ 0 and c0 > −α. Let a = c0 +α and suppose that condition (2.1) holds and that B[u0 , t∗ ] ⊆ D, where the t∗ is the limit of the sequence {tm } defined in (2.2). Then equation (1.3) has a unique solution u in B[u0 , r], where 2a q r= , 0a 1 + 1 − 2M c0 +α

and the Newton iterations generated from (1.4) converges to u and satisfy the estimates kum − um−1 k ≤ tm − tm−1 , kum − u0 k ≤ tm , ku − um k ≤ t∗ − tm .

The solution u is also unique in the set B[u0 , t∗ ] ∪ [B(u0 , R) ∩ D] where 0 +α) R = 2(cM − r. 0

Extended Kantorovich Theorem for Generalized Equations ...

45

Remark 2.4. In [8] the following condition was used instead kf 0 (x) − f 0 (y)k ≤ M1 kx − yk ∀x, y ∈ D. But, M ≤ M1 , since D0 ⊆ D leading to a finer analysis. Remark 2.5. If M = M0 , then the Theorem 2.3 becomes Theorem 2.11 of √ 4M0 +M + M 2+8M M0 [12]. However, if M0 < M then < 2M and in this case, 4 condition (2.1) is weaker than the condition 2M a ≤ 1 employed in [12]. We now give two corollaries that specialize Theorem 2.3 to several different situations. Our first corollary of Theorem 2.3 is a weak Kantorovich theorem on the solvability of nonlinear operator equations. Corollary 2.6. Let F : C → H be a continuous function that is Frchet differentiable on D. Suppose that F 0 (u0 ) is invertible for some u0 ∈ D, and ∃M ≥ M ≥ M0 ≥ 0 and a ≥ 0 such that kF 0 (u0 )−1 F (u0 )k ≤ a and kF 0 (u0 )−1 [F 0 (x) − F 0 (y)]k ≤ M kx − yk, 0

kF (u0 )

−1

0

0

[F (x) − F (u0 )]k ≤ M0 kx − x0 k,

∀x, y ∈ D0 ; ∀x ∈ D0 .

Let w∗ be the limit of the sequence w0 = 0, w1 = a, wm+1 := wm + M (wm −wm−1 )2 √ 2a 2(1−M0 wm ) and suppose that B[u0 , w∗] ⊆ D, r = 1+ 1−2M a , and 0

p (4M0 + M + M 2 + 8M M0 )a ≤ 4;

This inequality is strict if M0 = 0. Then, the equation F (u) = 0 has a unique solution in B[u0 , r]. The solution u is also unique in the set B[u0 , w∗] ∪ [B(u0 , R) ∩ D] where R = M20 − r. If we set M = M0 in this theorem, we recover the Hilbert space version of the Kantorovich theorem [14] on the solvability of the equation F (u) = 0. However, if M0 < M , then previous inequalities is weaker than the classical Kantorovich theorem 2M a ≤ 1 and the weaker
Argyros sufficient 3condition (M + M0 )a ≤ 1. For instance, if D = 89 , 10 9 , u0 = 1, F (x) = 6x − 1 and u0 = 1, then it is easy to verify that M = 2, M0 = 1.9 and a = 0.258646. In this case condition mentioned before holds, but the Kantorovich condition and the Argyros condition do not hold.

46

Gus I. Argyros

The following corollary of Theorem 2.3 is a new Kantorovich-type theorem for variational inequalities. Corollary 2.7. Let u0 ∈ D and suppose that kf (u0 )k ≤ b and that conditions from Theorem 2.3 hold, where M ≥ M0 ≥ 0 and c0 > 0. Let a = cb0 and r = q 2a 2M0 a and suppose that B[u0 , s∗ ] ⊆ D where s∗ is the limit of the 1+

1−

c0

sequence s0 = 0, s1 = a, sm+1 := sm + (4M0 + M +

M (sm −sm−1 )2 , 2(c0 −M0 sm )

and that

p M 2 + 8M M0 )a ≤ 4c0 ;

This inequality is strict is M0 = 0. Then, there exists a unique u ∈ B[u0 , r] satisfying the variational inequality (f (u), u − v) ≤ 0,

∀v ∈ C.

(2.9)

The solution u is also unique in the set B[u0 , s∗ ] ∪ [B(u0 , R) ∩ D] where 2c0 R= M − r. 0 Remark 2.8. If set H = Rn with its usual Euclidean scalar product and take C as the positive octant Rn+ in Corollary 2.7 we obtain a new Kantorovich-type existence theorem for the nonlinear complementarity problem f (u) ≥ 0,

u ≥ 0,

(u, f (u)) = 0

(2.10)

in which vector inequalities are interpreted in the component wise sense. Remark 2.9. When we specialize the generalized Newton scheme (1.4) to the variational inequality (2.9) we obtain the Josephy-Newton scheme.   f 0 (um )(um+1 − um ) + f (um ).um+1 − v ≤ 0, ∀c ∈ C, m = 0, 1, . . . (2.11) proposed by Josephy [12] in the late 1970s. It is instructive to compare Corollary 2.7 with the other Kantorovichtype results that have appeared in the Literature for the variational inequality (2.9) and the nonlinear complementarity problem (2.10) approximated with the Josephy-Newton scheme (2.11). Such results have been proved by Josephy [12],

Extended Kantorovich Theorem for Generalized Equations ...

47

Wang [22] and Wang [23] (improving earlier results in Wang and Shen [24]). These results differ from Corollary 2.7 because they use an estimate the form ku1 − u0 k ≤ η as part of their hypotheses, they do not employ the centerLipschitz condition and their convergence criteria are stronger. Josephy’s theorem is much more different from our results because it is based on a concept of regular solution developed by Robinson [17].

R EFERENCES [1] Argyros, I.K., Computational Theory for iterative methods, Studies in Computational Mathematics, 15, Elsevier, New York, USA¡ 2007. [2] Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer Verlag, Berlin, Germany, 2008. [3] Argyros, I.K., On the Newton-Kantorovich hypothesis for solving equations, Journal of Computational and Applied Mathematics, 11, 2004, 103110. [4] Argyros, I.K. and George, S., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications: Volume-III, Nova Science Publisher, New York, 2019. [5] Argyros, I.K. and Hilout, S., Weaker conditions for the convergence of Newton’s method, J. Complexity, 28, 3, 2012, 364 - 387. [6] Argyros, I.K. and Magren, A.A., A Contemporary Study of Iterative Methods, Academic Press, Elsevier, San Diego, CA, USA, 2018. [7] Argyros, I.K. and Regmi, S., Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces, Nova Science Publisher, New York, USA, 2019. [8] Argyros, I.K. and Uko. L., Generalized equations, Variations inequalities and a weak Kantorovich theorem, Appl. Math. Comput., 2005, 131-143. [9] Dennis, J.E., On the Kantorovich hypotheses for Newton’s method, SIAM Journal on Numerical Analysis, 6, 1969, 493-507.

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[10] Gragg, W.B. and Tapia, R.A., Optimal error bounds for the NewtonKantorovich theorem, SIAM Journal on Numerical Analysis, 11, 1974, 10-13. [11] Harker, P.T. and Pang, J.S., Finite dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Mathematical Programming, 48, 1990, 161-220. [12] Josephy, N.H., Newton’s method for generalized equations, Technical Report No. 1965, Mathematics Research Center, University of Wisconsin, Madison, WI, 1979. [13] Kantorovich, L.V., On Newton’s method for functional equations, Doklady Akademii Nauk SSSR, 59, 1948, 1237-1240. [14] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [15] Magren, A.A. and Argyros, I.K., Iterative Methods and Their Dynamics with Applications, CRC Press, New York, USA, 2018. [16] Minty, G.J., Monotone (nonlinear) operators in Hilbert space, Duke Mathematics Journal, 29, 1973, 341-346. [17] Robinson, S.M., Generalized equations, Mathematical programming: the sate of the art, Springer, Berlin, 1982, 346-367. [18] Stampacchia, G., Formes bilineares coercitives sur les ensembles convex, Computes Rendus del L’Academie des Science de Paris, 258, 1964, 44134416. [19] Uko, L.U., Generalized equations and the generalized Newton method, Mathematical Programming, 73, 1996, 251-268. [20] Uko, L.U., On a class of general strongly nonlinear quasivartional inequalities, Rivista di Matematica Pura ed Applicata, 11, 1992, 47-55. [21] Uko, L.U. and Argyros, I.K. Generalized equations, variational inequalities and a weak Kantorovich theorem, Numer. Algor., 52, 2009, 321-333.

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[22] Wang, Z., Extensions of Kantorovich theorems to complementarity problems, ZAMM - Journal of Applied Mathematics and Mechanics, 88, 2008, 179-190. [23] Wang, Z., Semilocal convergence of Newton’s method for finitedimensional variational inequalities and nonlinear complementarity problems, Doctoral thesis, Fakultat fur Mathematik, Universitat Karlsruhe, Germany, 2005. [24] Wang, Z. and Shen, Z., Kantorovich theorem for variational inequalities, Applied Mathematics and Mechanics, 25, 2004, 1291-1297.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 5

E XTENDED THE A PPLICABILITY OF N EWTON ’ S M ETHOD FOR E QUATIONS WITH M ONOTONE O PERATOR Michael I. Argyros∗ Department of Computing and Technology, Cameron University, Lawton, Oklahoma, US

Abstract Newton’s method has been used extensively to solve Hilbert space valued equations with monotone operators. We show that applications can be extended without additional hypotheses.

AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, local convergence, Iterative Methods, algorithm, convergence criteria, Newton method

1.

INTRODUCTION

We are concerned with the problem of finding a solution x∗ for the generalized equation F (x) + M (x) 3 0, (1.1) ∗

Corresponding Author’s Email: [email protected].

52

Michael I. Argyros

where F : D → H is a Frchet differentiable function, H is a Hilbert space, D ⊆ H is an open set M : H ⇒ H is a set-valued and maximal monotone, plays a significant role in classical analysis and its applications for systems of nonlinear equations, abstract inequality systems, and liner and nonlinear complementary problems [1-23]. Let H be a Hilbert space with scalar product < ·, · > and norm k · k, the open and closed balls at x wit radius δ ≥ 0 are denoted, respectively, by U (x, δ) and U (x, δ). We denote by L(H, H) the space consisting of all continuous linear mapping A : H → H and the operator norm of A is defined by kAk : sup{kAk : kxk ≤ 1}. A bounded linear operator Q : H → H is called a positive operator if Q is a self-conjugate and < Qx, x >≥ 0 for each x ∈ H. The domain and the range of Q are, respectively, the set domain of Q := {x ∈ H : Q(x) 6= ∅} and range of Q : {y ∈ H : y ∈ Q(x) for some x ∈ X}. The inverse of Q is setvalued mapping Q−1 : H ⇒ H defined by Q−1 (y) := {x ∈ H : y ∈ Q(x)}. We refer the reader to [20] for concepts and properties of monotonicity. Let Q : H → H be a bounded linear operator. We use the convention that b is b : 1 (Q + Q∗ ) where Q∗ is the conjugate operator of Q. As we can see, Q Q 2 a self-conjugate operator. The next result is of major significance to prove the definition of Newton’s method. Its proof can be found in [20].

2.

L OCAL C ONVERGENCE

We consider different types of Lipschitz majorant functions, so we can first compare them. Definition 2.1. We say that F 0 satisfies the center-majorant Lipschitz condition if there exists a twice continuously differentiable function f0 : [0, R) → R such that for all θ ∈ [0, 1], x ∈ U (x∗ , µ) kF 0 (c x∗ )−1 kkF 0 (x) − F 0 (x∗ + θ(x − x∗ ))k ≤ f00 (kx − x0 k) − f00 (0). (2.1) Suppose that equation f00 (t) = 0

(2.2)

has a minimal positive solution denoted by %0 . Define % = min{µ, %0 , R} and D0 = U (x∗ , %).

(2.3)

Extended the Applicability of Newton’s Method for Equations ...

53

Definition 2.2. We say that F 0 satisfies the restricted-majorant Lipschitz condition if there exists a twice continuously differentiable function f : [0, %) → R such that for all θ ∈ [0, 1], x ∈ U (x∗ , %) kF 0 (c x∗ )−1 kkF 0 (x) − F 0 (x∗ + θ(x − x∗ ))k ≤ f 0 (kx − x∗ k) − f 0 (θkx − x∗ k). (2.4) 0 Definition 2.3. We say that F satisfies the majorant Lipschitz condition, if there exists a twice continuously differentiable function f1 : [0, R) → R such that for all θ ∈ [0, 1], x ∈ U (x∗ , µ) kF 0 (c x∗ )−1 kkF 0 (x) − F 0 (x∗ + θ(x − x∗ ))k ≤ f10 (kx − x∗ k) − f10 (θkx − x∗ k). (2.5) Remark 2.4. In view of (2.1), (2.3) - (2.5), we have f00 (t) ≤ f10 (t)

(2.6)

f 0 (t) ≤ f10 (t)

(2.7)

and

for all t ∈ [0, %). We suppose from now on that f00 (t) ≤ f 0 (t) for all t ∈ [0, %).

(2.8)

Otherwise, our results hold with f replacing f (t), where f is the largest of f0 and f on the interval [0, %). Moreover, we also suppose f0 (t) ≤ f1 (t)

(2.9)

f0 (t) ≤ f (t)

(2.10)

f (t) ≤ f1 (t)

(2.11)

and

for all t ∈ [0, %). Function f1 was used in [20] to obtain their results. But in view of (2.6) - (2.11), function f can replace f1 in the proofs. That is we have obtained a better location U (x∗ , %) for the iterates that U (x∗ , µ) used in [20]. This way we obtain stricter functions f0 and f than f1 . Notice also that f = f (f0 , R, %, µ), whereas f1 = f1 (R, µ) and f0 is needed to define f . It is worth noticing that under (2.5), the following estimate was given in [20]: kF 0 (b x)−1 k ≤

kF 0 (c x∗ )−1 k . |f10 (kx − x∗ k)|

(2.12)

54

Michael I. Argyros

But, we obtain using (2.1) and (2.8) the weaker, actually needed and tighter kF 0 (b x)−1 k ≤

kF 0 (x∗ )−1 k kF 0 (x∗ )−1 k ≤ . |f00 (kx − x∗ k)| |f 0 (kx − x∗ k)|

(2.13)

Next, we present the extended results. Theorem 2.5. Let H be a Hilbert space, D be an open nonempty subset of H, F : D → H be continuous with Frchet derivative F 0 continuous, M : H ⇒ H be a set-valued operator and x∗ ∈ D. Suppose that 0 ∈ F (x∗ ) + M (x∗ ), F 0 (x∗ ) is a positive operator and F 0 (c x∗ )−1 exists. Let R > 0 and µ := sup{t ∈ [0, R) : U (x∗ , t) ⊂ D}. Suppose (2.1) and (2.3) hold (c1 ) f0 (0) = f (0) = 0 and f00 (0) = f 0 (0) = −1; (c2 ) f00 and f 0 are convex and strictly increasing. f (t)

Let v : sup{t ∈ [0, R) : f 0 (t) < 0}, ρ := sup{t ∈ (0, v) : tf 0 (t) − 1 < 1} and r := min{µ, ρ}. Then, the sequences with starting point x0 ∈ U (x∗ , r)/{x∗} and t0 = kx∗ − x0 k, respectively, 0 ∈ F (xn) + F 0 (xn )(xn+1 − xn ) + M (xn+1 ),

tn+1 = |tn −

f (tn ) |, f 0 (tn )

n = 0, 1, . . . ,

are well defined, {tn } is strictly decreasing, is contained in (0, r) and converges to 0, {xn } is contained in U (x∗ , r) and converges to the point x∗ , which is a unique solution of the generalized equation F (x) + M (x) 3 0 in U (x∗ , σ), where σ = min{r, σ} and σ := sup{0 < t < µ : f (t) < 0}. Moreover, the sequence { tn+1 } is strictly decreasing, t2n   tn+1 tn+1 f 00 (t0 ) 2 kx∗ − xn+1 k ≤ kx − x k , ≤ , n = 0, 1, . . . . n ∗ t2n t2n 2|f 0 (t0 )| f (ρ)

If additionally, ρf 0 (ρ) − 1 = 1 and ρ < µ, then r = ρ is the optimal convergence radius. Remark 2.6. If f0 = f = f1 , then Theorem 2.5 reduces to Theorem 4 in [20]. Otherwise it constitutes an improvement with advantages as already stated in the introduction. Notice that the corresponding equation for the radius in f1 (%) −1=1 %f1 (%)

Extended the Applicability of Newton’s Method for Equations ...

55

with solution denoted by %1 . But in view of (2.7), we obtain %1 ≤ %. Estimates (2.6) - (2.8) can be strict. As an example consider functions L0 2 t − t, 2 L f (t) = t2 − t 2

f0 (t) =

and f1 (t) =

L1 2 t −t 2

where H = R, D = U (x∗ , 1), x0 = 0 and R = 1 with F (t) = et − 1 = 0 1

Then, we have L0 = e − 1 < L = e e−1 < L1 = e. Otherwise examples can be found in [1-6].

R EFERENCES [1] Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer Verlag, Berlin, Germany, 2008. [2] Argyros, I.K. and George, S., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications: Volume-III, Nova Science Publisher, New York, 2019. [3] Argyros, I.K. and Hilout, S., Weaker conditions for the convergence of Newton’s method, J. Complexity, 28, 3, 2012, 364 - 387. [4] Argyros, I.K. and Magren, A.A., A Contemporary Study of Iterative Methods, Academic Press, Elsevier, San Diego, CA, USA, 2018. [5] Argyros, I.K. and Regmi, S., Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces, Nova Science Publisher, New York, USA, 2019.

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[6] Blum F., Cucker, F., Shub, M. and Smale, S., Complexity and real computation, Springer-Verlag, New York, 1998. [7] Chang, D.C., Wang, J. and Yao, J.C., Newtons method for variational inequality problems: Smales point estimate theory under the γ-condition, Applicable Analysis, 94, 1, 2015, 44-55. [8] Daniel, J.W., Newton’s method for nonlinear inequalities, Numer. Math., 21, 1973, 381-387. [9] Dokov, S.P. and Dontchev, A.L., Robinson’s strong regularity implies robust local convergence of Newton’s method, Appl. Optim., Kluwer Acad. Publ., Dordrechet, 15, 1998, 116-129. [10] Dontchev, A.L. and Rockafellar, R.T., Implicit functions and solution mappings, Springer Monographs in Mathematics, Springer, Dordrecht, 2009. [11] Dontchev, A.L. and Rockafellar, R.T., Newton’s method for generalized equations: a sequential implicit function theorem, Math. Program., 123, 1, Ser. B, 2010, 139-159. [12] Facchinei, F., and Pang, J.S., Finite-dimensional variational inequalities and complementary problems, Springer-Verlag, New York, 2003. [13] Ferreira, O.P., Gonalves, M.L.N. and Oliveira, P.R., Local convergence analysis of the Gauss-Newton method under a majorant condition, J. Complexity, 27, 1, 2011, 111-125. [14] Gonalves, M.L.N., Inexact Gauss-Newton like methods for injectiveoverdetermined systems of equations under a majorant condition, Numerical Algorithms, 72, 2, 2016, 377-392. [15] Josephy, M., Newton’s Method for Generalized Equations and the PIES Energy Model, University of Wisconsin-Madison, 1979. [16] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [17] Magren, A.A. and Argyros, I.K., Iterative Methods and Their Dynamics with Applications, CRC Press, New York, USA, 2018.

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[18] Robinson, S.M., Extension of Newton’s method to nonlinear functions with values in a cone, Numer. Math., 19, 1972, 341-347. [19] Silva, G.N., Kantorovichs theorem on Newtons method for solving generalized equations under the majorant conditions, Applied Mathematics and Computation, 286, 2016, 178-188. [20] Silva, G.N., Local convergence of Newton’s method for solving generalized equations with monotone operator, Applicable Analysis, 97, 7, 2018, 1094-1105. [21] Traub, J.F. and Woniakowski, H., Convergence and complexity of Newton iteration for operator equations, J. Assoc. Comput. Mach., 26, 2, 1979, 250-258. [22] Uko, L.U. and Argyros, I.K., Generalized equations, variational inequalities and a weak Kantorovich theorem, Numerical Algorithms, 52, 3, 2009, 321-333. [23] Zabrejko, P.P. and Nguen, D.F., The majorant method in the theory of Newton-Kantorovich approximations and the Ptk error estimates, Numer. Funct. Anal. Optim., 9, 5-6, 1987, 671-684.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 6

I MPROVED L OCAL C ONVERGENCE FOR A P ROXIMAL G AUSS -N EWTON S OLVER Samundra Regmi∗ Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma, US

Abstract Proximal Gauss-Newton solvers perform well on Hilbert spaces. But the convergence domain needs to be extended without additional hypotheses. We obtain this goal by utilizing more precise estimates.

AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, local convergence, Iterative Methods, algorithm, convergence criteria, Gauss-Newton solver

1.

INTRODUCTION

We consider finding a solution x∗ the optimization problem Ψ(x) := ∗

1 min kF (x) − yk2 + I(x), 2 x∈H1

Corresponding Author’s Email: [email protected].

(1.1)

60

Samundra Regmi

where H1 , H2 are Hilbert spaces, y ∈ H2 , F : H1 → H2 is Gteaux differentiable operator and I : H1 → R ∪ {∞} is a lower convex and lower semicontinuous functional. We use the proximal Gauss-Newton solver defined for all n = 0, 1, 2, . . . by H(xn)

xn+1 = proxj

(xn − F 0 (xn )+ F (xn ))

(1.2)

to generate a sequence approximating x∗ under certain conditions. We let y = 0 without any loss of generality, since otherwise our results can be obtained by replacing F by F − y. We are motivated by optimization considerations and the work in [47]. In particular, we present a finder local convergence analysis of solver (1.2) without additional conditions. As in [47], we use the conditions ( F : D ⊆ H1 → H2 is a differentiable Gtaux differentiable operator (C) : I : H1 → R ∪ {∞} is proper, lower semicontinious and convex functional. From now on we assume that (C) conditions hold. For brevity, we refer the reader to [1-56] for some standard concepts introduced in this chapter.

2.

L OCAL C ONVERGENCE

From now on F : D → H2 is a Gteaux differentiable operator, and for R > 0, L0 , L1, L : [0, R) → R are positive, continuous and increasing functions. We need the following Lipschitz definitions. Definition 2.1. Operator F 0 satisfies the center Lipschitz condition with L0 average at x∗ on D, if for all x ∈ D kF (x∗ ) − F (x) − F 0 (x)(x∗ − x)k ≤

∗ kx−x Z k

(2kx − x∗ k − u)L0 (u)du. (2.1)

0

Suppose that equation Zt 0

(2t − u)L0 (u)du = 1

(2.2)

Improved Local Convergence for a Proximal Gauss-Newton Solver

61

has a minimal positive solution δ. Define D0 = D ∩ U (x∗ , δ).

(2.3)

Definition 2.2. Operator F 0 satisfies the restricted radius Lipschitz condition with L average at x∗ on D0 , if for all x ∈ D0 . kF (x∗ ) − F (x) − F 0 (x)(x∗ − x)k ≤

∗ kx−x Z k

L(u)udu

(2.4)

0

Definition 2.3. Operator F 0 satisfies the radius Lipschitz condition with L1 average at x∗ on D, if for all x ∈ D kF (x∗ ) − F (x) − F 0 (x)(x∗ − x)k ≤

∗ kx−x Z k

L1 (u)udu

(2.5)

0

Remark 2.4. By (2.3), we have D0 ⊆ D

(2.6)

L0 (u) ≤ L1 (u)

(2.7)

L(u) ≤ L(u)

(2.8)

so and for all u ∈ [0, δ). Hence, the results in [47] using L1 can be rewritten using L instead. But this modification leads to a larger radius of convergence and tighter error estimates on kx∗ − xn k, without additional conditions, since L is a special case of L1 . Due to the above, we skip the proofs. Next, we present the main local convergence result for solver (1.2). Theorem 2.5. Let V ⊆ D be an open star-shaped set with respect to x∗ , where x∗ ∈ dom(j) ∩ V is a local minimized of Ψ0 . Moreover assume that 1. F 0 (x∗ ) is injective with closed range; 2. F 0 : D ⊆ H1 → L(H1 , H2 ) is restricted radius Lipschitz continuous of center x∗ with L average on v;

62

Samundra Regmi √

3. [(1 + 2)k + 1]αβ 2 L0 (0) < 1, where α = kF (x∗ )k, β = kF 0 (x∗ )+ k, k = kF 0 (x∗ )+ kkF 0 (x∗ )k, the conditioning number of F 0 (x∗ ). Define R and p : [0, R) → R+ by setting R = sup{r ∈ (0, R) : γ0 (r)r < β1 } and for λ ∈ [0, 1],  r  1 R uλ L(u)du, if r ∈ (0, R) r 1+λ γλ (r) = 0  L(0) , if r = 0 1+λ ( √ β βγ0 (r)γ1(r)r 2 + kγ1 (r)r (1 + 2)αβ 2 γ0 (r)2 r + p(r) = 1 − βγ0 (r)r (1 − βγ0 (r)r) 1 − βγ0 (r)r ) √ [(1 + 2)k + 1]αβγ0 (r) + . 1 − βγ0 (r)r The function p is continuous and strictly increasing. If we define r = sup{r ∈ (0, R) : p(r) < 1}, and we fix r ∈ R, with 0 < r ≤ r, such that U (x∗ , r) ⊆ D, we get that the sequence x0 ∈ U (x∗ , r), H(xn)

xn+1 = proxI

(xn − F 0 (xn )+ F (xn ))

with H(xn) := F 0 (xn )∗ F 0 (xn ), is well-defined, i.e. xn ∈ U (x∗ , r) and F 0 (xn ) is injective with closed range and it holds kxn − x∗ k ≤ pn0 kx0 − x∗ k where p0 := p(kx0 − x∗ k) < 1. Moreover, the following item is true kxn+1 − x∗ k ≤ α2 kxn − x∗ k2 + α1 kxn − x∗ k, where

√ [(1 + 2)k + 1]αβ 2 γ0 (sx0 ) ; α1 = (1 − βγ0 (sx0 )sx0 )2 √ kβγ1 (sx0 ) + (1 + 2)αβ 3 γ0 (sx0 )2 + β 2 γ0 (sx0 )γ1 (sx0 )sx0 α2 = (1 − βγ0 (sx0 )sx0 )2

and sx0 = kx0 − x∗ k.

Improved Local Convergence for a Proximal Gauss-Newton Solver

63

Remark 2.6. If L0 = L = L1 , then our results reduce to the corresponding ones in [47, Theorem 1]. Otherwise, in view of (2.6) - (2.8) our results constitute an improvement with advantages already stated in the introduction. Examples where estimates (2.6) - (2.8) are strict can be found in [1-9]. Clearly, the rest of the specializations Theorem 1 in [47] are immediately extended too. We leave the details to the motivated reader.

R EFERENCES [1] Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer Verlag, Berlin, Germany, 2008. [2] Argyros, I.K., On the semilocal convergence of the Gauss-Newton method, Adv. Nonlinear Var. Inequal., 8, 2, 93-99, 2005. [3] Argyros, I.K. and George, S., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications: Volume-III, Nova Science Publisher, New York, 2019. [4] Argyros, I.K. and Hilout. S., Extending the applicability of the GaussNewton method under average Lipschitz-type conditions, Numer. Algorithms, 58, 1, 2011, 23-52. [5] Argyros, I.K. and Hilout, S., On the Gauss-Newton method, J. Appl. Math. Comput., 2010, 1-14. [6] Argyros, I.K. and Hilout, S., On the local convergence of the GaussNewton method, Punjab Univ. J. Math., 41, 2009, 23-33. [7] Argyros, I.K. and Hilout, S., Weaker conditions for the convergence of Newton’s method, J. Complexity, 28, 3, 2012, 364 - 387. [8] Argyros, I.K. and Magren, A.A., A Contemporary Study of Iterative Methods, Academic Press, Elsevier, San Diego, CA, USA, 2018. [9] Argyros, I.K. and Regmi, S., Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces, Nova Science Publisher, New York, USA, 2019.

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[10] Bachmayr, M. and Burger, M., Iterative total variation schemes for nonlinear inverse problems, Inverse Probl., 25, 10, 2009, 105004. [11] Bakushinsky, A.B. and Kokurin, M.Y., Iterative Methods for Approximate Solution of Inverse Problems, Mathematics and Its Applications, Springer, Dordrecht, 577, 2004. [12] Bellavia, S., Macconi, M. and Morini, B., STRSCNE: a scaled trust-region solver for constrained nonlinear equations, Comput. Optim. Appl., 28, 1, 2004, 31-50. [13] Ben, A.I., A modified Newton-Raphson method for the solution of systems of equations, Isr. J. Math., 3, 1965, 94-98. [14] Blaschke, B., Neubauer, A. and Scherzer, O., On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numer. Anal., 17, 3, 1997, 421-436. [15] Borwein, J.M. and Lewis, A.S., Convex Analysis and Nonlinear Optimization, Advanced Books in Mathematics, Canadian Mathematical Society, Ottawa, 2000. [16] Burke, J.V. and Ferris, M.C., A Gauss-Newton method for convex composite optimization, Math. Program. Ser. A., 71, 2, 1995, 179-194. [17] Combettes, P.L. and Wajs, V.R., Signal recovery by proximal forwardbackward splitting, Multiscale Mode. Simul., 4, 4, 2005, 1168-1200. [18] Dennis, J.E. and Schnabel, R.B., Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, 16, SIAM, Philadelphia, 1996. [19] Engl, H.W., Hanke, M. and Neubauer, A., Regularization of Inverse Problems, Mathematics and Its Applications, 375, 1996. [20] Ferreira, O.P. and Svaiter, B.F., Kantorovich’s majorants principle for Newton’s method, Comput. Optim. Appl., 42, 2, 2009, 213-229. [21] Floudas, C.A. and Pardalos, P.M., A Collection of Test Problems for Constrained Global Optimization Algorithms, Lecture Notes in Computer Science, 455, Springer, Berlin, 1990.

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[22] Groetsch, C.W., Generalized Inverses of Linear Operators: Representation and Approximation, Monographs and Textbooks in Pure and Applied Mathematics, 37, Marcel Dekker, New York, 1977. [23] Hussler, W.M., A Kantorovich-type convergence analysis for the GaussNewton-method, Numer. Math., 48, 1, 1986, 119-125. [24] Hiriart, J.B. and Lemarchal, C., Convex Analysis and Minimization Algorithms. II., Fundamental Principles of Mathematical Sciences, 306, Springer, Berlin, 1993. [25] Jin, Q., A convergence analysis of the iteratively regularized GaussNewton method under the Lipschitz condition, Inverse Probl., 24, 4, 2008, 045002. [26] Kaltenbacher, B. and Hofmann, B., Convergence rates for the iteratively regularized Gauss-Newton method in Banach spaces, Inverse Probl., 26, 3, 2010, 035007. [27] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [28] Kanzow, C., An active set-type Newton method for constrained nonlinear systems, Appl. Optim., 50, 2001, 179-200. [29] Kowalik, J. and Osborne, M.R., Methods for Unconstrained Optimization Problems, Elsevier, New York, 1969. [30] Kozakevich, D.N., Martnez, J.M. and Santos, S.A., Solving nonlinear systems of equations with simple constraints, Comput. Appl. Math., 16, 3, 1997, 215-235. [31] Langer, S., Investigation of preconditioning techniques for the iteratively regularized Gauss-Newton method for exponentially ill-posed problems, SIAM J. Sci. Comput., 32, 5, 2010, 2543-2559. [32] Lewis, A. and Wright, S.J., A proximal method for composite optimization, 2008, 0812.0423v1 [math.OC]. [33] Li, C. and Ng, K.F., Majorizing functions and convergence of the GaussNewton method for convex composite optimization, SIAM J. Optim., 18, 2, 2007, 613-642.

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[34] Li, C. and Wang, X., Convergence of Newton’s method and uniqueness of the solution of equations in Banach Spaces II, Acta Math. Sin. Engl. Ser., 19, 2, 2003, 405-412. [35] Li, C. and Wang, X., On convergence of the Gauss-Newton method for convex composite optimization, Math. Program., Ser. A, 91, 2, 2002, 349356. [36] Li, C., Hu, N. and Wang, J., Convergence behavior of Gauss-Newton’s method and extensions of the Smale point estimate theory, J. Complexity, 26, 3, 2010, 268-295. [37] Li, C., Zhang, W. and Jin, X., Convergence and uniqueness properties of Gauss-Newton’s method, Comput. Math. Appl., 47, 6-7, 2004, 1057-1067. [38] Magren, A.A. and Argyros, I.K., Iterative Methods and Their Dynamics with Applications, CRC Press, New York, USA, 2018. [39] Moreau, J.J., Fonctions convexes duales et points proximaux dans un espace hilbertein, C.R. Acad. Sci. Paris, 255, 1962, 2897-2899. [40] Moreau, J.J., Proprits des applications prox, C.R. Acad. Sci. Paris, 256, 1963, 1069-1071. [41] Moreau, J.J., Proximit et dualit dans un espace hilbertien, Bull. Soc. Math. Fr., 93, 1965, 273-299. [42] Mosci, S., Rosasco, L., Santoro, M., Verri, A. and Villa, S., Solving structured sparsity regularization with proximal methods, Machine Learning and Knowledge Discovery in Databases, Lectures Notes in Computer Science, Springer, Berlin, 6322, 2010, 418-433. [43] Osborne, M.R., Some aspects of non-linear least squares calculations, Numerical Methods for Non-Linear Optimization, Conf. Univ. Dundee, Dundee, 1971, 171-189. [44] Polyak, B.T., Introduction on Optimization, Translations Series in Mathematics and Engineering, Optimization Software Inc. Publications Division, New York, 1987.

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[45] Ramlau, R. and Teschke, G., A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints, Numer. Math., 104, 2, 2006, 177-203. [46] Ramlau, R. and Teschke, G., Tikhonov replacement functionals for iteratively solving nonlinear operator equations, Inverse Probl., 21, 5, 2005, 1571-1592. [47] Salzo, S. and Villa, S., Convergence analysis of a proximal Gauss-Newton method, Comput. Optim. Appl., 53, 2012, 557-589. [48] Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M. and Lenzen, F., Variational Methods in Imaging, Applied Mathematical Sciences, Springer, New York, 167, 2009. [49] Shacham, M., Brauner, N. and Cultlib, M., A web-based library for testing performance of numerical software for solving nonlinear algebraic equations, Comput. Chem. Eng., 26, 2002, 547-554. [50] Stewart, G.W., On the continuity of the generalized inverse, SIAM J. Appl. Math., 17, 1969, 547-554. [51] Ulbrich, M., Nonmonotone trust-region methods for bound-constrained semismooth equations with applications to nonlinear mixed complementarity problems, SIAM J. Optim., 11, 4, 2001, 889-917. [52] Wang, X., Convergence of Newton’s method and inverse function theorem in Banach space, Math. Comput., 68, 225, 1999, 169-186. [53] Wang, X., Convergence of Newton’s method and uniqueness of the solution of equations in Banach space, IMA J. Numer. Anal., 20, 1, 2000, 123-134. [54] Wedin, P., Perturbation theory for pseudo-inverses, BIT Numer. Math., 13, 1973, 217-232. [55] Womersley, R.S., Local properties of algorithms for minimizing nonsmooth composite functions, Math. Program. Ser. A., 31, 1, 1985, 69-89. [56] Xu, C. and Li, C., Convergence criterion of Newton’s method for singular systems with constant rank derivatives, J. Math. Anal. Appl., 345, 2, 2008, 689-701.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 7

I MPROVED E RROR E STIMATES FOR S OME N EWTON -T YPE M ETHODS Samundra Regmi∗ Department of Mathematical Sciences, Cameron University, Lawton, Oklahoma, US

Abstract Tighter Error estimates are very important in a Hilbert space setting, since if obtained, then fewer iterates are needed to obtained a predetermined error tolerance. We achieve this objective without adding hypotheses. Hence, we extend the applications of Newton-type methods on Hilbert space.

AMS Subject Classifications: 45G10, 65H05, 65J15, 49M15 Keywords: Hilbert Space, local convergence, Iterative Methods, algorithm, convergence criteria, Newton method

1.

INTRODUCTION

To solve equation F (x) = 0, ∗

Corresponding Author’s Email: [email protected].

(1.1)

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Samundra Regmi

we introduce: Newton’s Method: xn+1 = xn − F 0 (xn )−1 F (xn )

(1.2)

xn+1 = xn − F [xn−1 , xn]−1 F (xn )

(1.3)

Secant Method:

and Ostrowski-Traub Method: xn,1 = xn − F 0 (xn )−1 F (xn ) xn,1 − xn F (xn,1 ) xn+1 = xn,1 − 2F (xn,1 ) − F (xn )

(1.4)

to generate sequence converging to a solution x∗ of equation (1.1). Here F : Ω ⊂ B1 → B2 , where B1 = B2 = R or B1 = B2 = C, Ω is open and convex, and [·, ·] stands for [·, ·; F ] : Ω × Ω → L(B1 , B2 ) which is a divided difference of order one [13]. Our new technique is based on the condition |F [x−1, x0 ]−1 (F [x, y] − F [x−1, x0 ])| ≤ L1 |x − x−1 | + L2 |y − x0 | for all x, y ∈ Ω (1.5)

By this condition and the Banach lemma on invertible operators [12], we have |F [x−1 , x0 ]−1 (F [x, y] − F [x−1 , x0])|

(1.6)

≤ L1 (|x − x0 | + |x0 − x−1 |) + L2 |y − x0 |

(1.7)

< (L1 + L2 )r0 + L1 |x0 − x−1 | < 1

(1.8)

provided that x, y ∈ U (x0 , r0 ),

r=

1 − L1 α L1 + L2

and L1 α < 1.

(1.9)

Ω0 = Ω ∩ U (x0 , r0)

(1.10)

Set Consider also condition

Improved Error Estimates for Some Newton-Type Methods

71

|F [x−1 , x0 ]−1 [F 00 (x) − F 00 (x−1 )]| ≤ K|x − x−1 | for all x ∈ Ω0 .

(1.11)

The corresponding condition in [10] is: |F [x−1 , x0 ]−1 [F 00 (x) − F 00 (x−1 )]| ≤ K1 |x − x−1 | for all x ∈ Ω.

(1.12)

It follows by (1.8) - (1.10) that K ≤ K1 .

(1.13)

Hence, K can replace K1 in all the results in [10] leading to the improvements: weaker convergence criteria, tighter error bounds on |tn − x∗ |, |tn+1 − tn | and more precise information on the location of the solution x∗ . These improvements are obtained under the same computation effort since the computation of K1 requires the computation of K as a special case.

2.

SEMI -L OCAL C ONVERGENCE

Theorem 2.1. Given x0 , x−1 ∈ Ω ⊆ B1 , a convex subset of B1 and F (x) has first- and second-order derivatives on Ω. |x0 − x−1 | ≤ τ, |F [x−1 , x0 ]−1 F (x−1 )| ≤ η,

|F [x−1 , x0]−1 F 00 (x−1 )| ≤ γ,

|F [x−1 , x0 ]−1 [F 00 (x) − F 00 (x−1 )]| ≤ K|x − x−1 | ∀x, y ∈ Ω0 . If U (x−1 , t∗ ) ⊂ Ω, p 2[p(τ )]2 γ + 2 γ 2 + 2p(τ )K p η≤ , 3 (γ + γ 2 + 2p(τ )K)2

where p(τ ) = 1 + 12 γτ + 16 Kτ 2 .

(i) Then, the sequence {xn } starting from x−1 , x0 defined by (1.2) converges to x∗ , the unique solution of F (x) in U (x0 , t∗ ) ∪ U (x0 , t∗∗ ), |x∗ − xn | ≤

t∗∗ − t∗

1

G θGn θ−1n−1 Gn Gn−1 0 − θ0 θ−1

n = 0, 1, . . .

72

Samundra Regmi and there exists a positive constant 0 < q0 < 1, such that Pk−1

|F (xn+k )| ≤ q0

j=0

G(n+j)

where θ0 = t∗ − tτ∗∗ − τ, θ−1 = of the polynomial defined by

|F (xn )|,

t∗ t∗∗ , 0

k = 0, 1, . . .,

< t∗ ≤ t∗∗ are two positive zeros

  K 3 γ 2 1 1 2 φ1 (t) = t + t − 1 + γt + Kτ t + η, 6 2 2 6 and {Gn } is a Fibonacci sequence with G−1 = 0. (ii) Let {tn } be defined by tn+1 = tn − φ1 [tn , tn−1 ]−1 F (tn ), Then, for all n = 0, 1, 2, . . . F (xn+1 ) F (x ) ≤ n F (xn+1 ) F (xn ) ≤

t−1 = 0, n = 0, 1, . . ..

φ(tn+1 ) , φ(tn ) t∗ − tn+1 . t∗ − tn

Theorem 2.2. Given x0 ∈ Ω ⊂ B1 , a convex subset of B1 , and F (x) has firstand second-order derivatives on Ω, |F 0 (x0 )−1 F (x0 )| ≤ η,

|F 0 (x0 )−1 F 00 (x0 )| ≤ γ,

|F 0 (x0 )−1 [F 00 (x) − F 00 (x0 )]| ≤ K|x − x0 |, If U (x∗ , t∗ ) ⊂ Ω,

∀x, y ∈ Ω0 .

p 2(γ + 2 γ 2 + 2K) p η≤ 3(γ + γ 2 + 2K)2

(2.1)

(i) then, the sequence {xn } starting from x0 defined by (1.3) converges to x∗ , the unique solution of F (x) in U (x0 , t∗ ) ∪ U (x0 , t∗∗ ), |x∗ − xn | ≤

t∗∗ − t∗ 4n n θ 1 − θ04 0

n = 0, 1, 2, . . .

Improved Error Estimates for Some Newton-Type Methods

73

and |F (xn+1 )| ≤ θ04 where θ0 = mial

t∗ t∗∗ ,

n


4k3−1

|F (xn )|,

k = 0, 1, . . .,

and 0 < t∗ ≤ t∗∗ are two positive zeros of the polynoφ2 (t) =

K 3 1 2 t + γt − t + η. 6 2

(ii) Let {tn } with t0 = 0 be defined by tn,1 = tn − φ02 (tn )−1 φ2 (tn ) n = 0, 1, 2, . . . , tn,1 − tn φ2 (tn,1 ) n = 0, 1, 2 . . ., tn+1 = tn,1 − 2φ2 (tn,1 ) − φ2 (tn ) the for all n = 0, 1, 2, . . . F (xn+1 ) F (xn ) ≤ F (xn+1 ) F (xn ) ≤

φ(tn+1 ) , φ(tn ) t∗ − tn+1 . t∗ − tn

Remark 2.3. In the case τ = 0, we see that we can use the same condition to guarantee the convergence of Eqs. (1.3) and (1.4) as we do for Newton method Eq. (1.2) in Ref. [10].

3.

E RROR B OUNDS OF N EWTON -L IKE METHOD FOR P OLYNOMIALS

Lemma 3.1. If

p 2p2 γ + 2 γ 2 + 2pK p q≤ · , 3 (γ + γ 2 + 2pK)2

(3.1)

where K, τ, p, q are positive number, then the polynomial defined by φ(t) =

K 3 1 2 t + γt − pt + q 6 2

(3.2)

has two positive zeros satisfying 0 ≤ ζ ∗ ≤ ζ ∗∗ . Applying method (1.3) to φ1 (t), we have

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Samundra Regmi

Theorem 3.2. {tn } converges to t∗ increasingly, 0 ≤ t∗ − tn ≤

t∗∗ − t∗

1

G θGn θ−1n−1 , Gn Gn−1 0 − θ0 θ−1

n = −1, 0, 1, . . .,

and there exists a positive constant 0 < q0 < 1, such that t∗ − tn+1 φ1 (tn+1 ) ≤ ∗ ≤ q0Gn , φ1 (tn ) t − tn

n = −1, 0, 1, . . . ,

where t∗ , t∗∗ , θ0 , θ−1 and {Gn } are defined in Theorem 2.1. Applying method (1.4) to φ2 (t), we have the following theorem. Theorem 3.3. {tn } converges to t∗ increasingly, t∗∗ − t∗ 4n , n = 0, 1, . . ., n θ 1 − θ04 0 φ2 (tn+1 ) t∗ − tn+1 n ≤ ∗ ≤ θ04 , n = −1, 0, 1, . . ., φ2 (tn ) t − tn

0 ≤ t∗ − tn ≤

(3.3) (3.4)

where t∗ , t∗∗ , and θ0 are defined in Theorem 2.2.

R EFERENCES [1] Argyros, I.K., Convergence and Applications of Newton-type Iterations, Springer Verlag, Berlin, Germany, 2008. [2] Argyros, I.K. and George, S., Mathematical Modeling For The Solution Of Equations And Systems Of Equations With Applications: Volume-III, Nova Science Publisher, New York, 2019. [3] Argyros, I.K. and Hilout, S., Weaker conditions for the convergence of Newton’s method, J. Complexity, 28, 3, 2012, 364 - 387. [4] Argyros, I.K. and Magren, A.A., A Contemporary Study of Iterative Methods, Academic Press, Elsevier, San Diego, CA, USA, 2018.

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75

[5] Argyros, I.K. and Regmi, S., Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces, Nova Science Publisher, New York, USA, 2019. [6] Ezquerro, J.A., Guiterrez, J.M. and Hernandez, M.A., A construction procedure of iteratives methods with cubical convergence, Appl. Math. Comput., 85, 1997, 181-198. [7] Gragg, W. and Tapia, R.A., Optimal error bounds for the NewtonKantorovich theorem, SIAM. J. Numer. Anal., 11, 1974, 10-13. [8] Guiterrez, J.M., A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math., 79, 1997, 131-145. [9] Huang, Z., A note on the Kantorovich theorem for the Newton iteration, J. Comput. Appl. Math., 47, 1993, 211-217. [10] Huang, Z., On the error estimates of several Newton-like methods, Appl. Math. Comput., 106, 1999, 1-16. [11] Kantorovich, L.V., On Newton’s method, Trudy, Mat. Inst. Steklov, 28, 1949, 104-144. [12] Kantorovich, L.V. and Akilov, G.P., Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964. [13] Magren, A.A. and Argyros, I.K., Iterative Methods and Their Dynamics with Applications, CRC Press, New York, USA, 2018. [14] Mysovskii, I., On the covnergence of L.V. Kantorovich’s method of solution of functional equations with its applications, Dokl. Akad. Nauk. SSSR, 70, 1950, 148, 565-568. [15] Ortega, J. and Rheinboldt, W., Iterative solution of nonlinear equations in several variables, Academic Press, New York, 1970. [16] Ostrowski, A.M., Solution of equations in Euclidean and Banach spaces, Academic Press, New York, 1960. [17] Rall, L.B., A Note on the convergence of Newton’s method, SIAM J. Numer. Anal., 11, 1974, 34-36.

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[18] Ralston, A. and Wilf, H.S., Mathematical Methods for Digital Computers (Vol II), John Wiley & and Sons, Inc. New York-London-Sydney, 1967. [19] Wang, X., On convergence of iterative process, Kexue Tongbao, 20, 1975, 558-559. [20] Wang, X., On error estimates for some numerical root-finding methods, Acta. Math. Sinica, 22, 1979, 638-642. [21] Yamamoto, T., A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions, Numer. Math., 49, 1986, 203220. [22] Yamamoto, T., On the method of Tangent hyperbolas in Banach Spaces, J. Comput. Appl. Math., 21, 1988, 75-86.

In: Hilbert Spaces and Its Applications ISBN: 978-1-53618-983-4 c 2021 Nova Science Publishers, Inc. Editors: M. Argyros et al.

Chapter 8

T WO N ON C LASSICAL Q UANTUM L OGICS OF P ROJECTIONS IN H ILBERT S PACE AND T HEIR M EASURES 1

Marjan Matvejchuk1,∗ and Elena Vladova2† Kazan National Research Technical University, Kazan, Russia 2 Moscow State Technical University of Civil Aviation, Moscow, Russia

Abstract Many scientific works are devoted to the algebraic axiomatics of quantum logics. Classical quantum logics consist of all orthogonal projections (= idempotents) of a complex Hilbert space. The following question remains open. Are there logics of projections in Hilbert space other than orthogonal projections, which allow one to construct the theory of quantum mechanics as productive as the logic of orthogonal projections? In this chapter, we study linear operators on complex Hilbert spaces which are real-orthogonal projections. It is a generalization of such standard (complex) orthogonal projections for which only the real part of scalar product vanishes. We compare some partial order properties of the orthogonal and of the real-orthogonal projections. We prove that the set of all real-orthogonal projections in complex space is a quantum pseudo logic. ∗ †

Corresponding Author’s Email: [email protected]. Author’s Email: evv−[email protected].

78

Marjan Matvejchuk and Elena Vladova In the second part of our work, we study quantum logics of projections in a Hilbert space with an indefinite metric (in the Kerin space). Note that spaces with indefinite metrics are fruitfully used in quantum field theory. It turns out that the main class of measures on the logics of projections in Kerin’s spaces are indefinite measures.

AMS Subject Classifications: 15A54, 46L50, 46B09, 81P10, 28A60 Keywords: Hilbert space, real-orthogonality, idempotent, projection, partial order, logic

1.

INTRODUCTION

In the book [2] Russian transl: M. Nauka, (1984). Chapter XII (see Problem 110, page 371, and Problem 88, page 547, [in Russian]) the problem of describing the measures on quantum logics of projections (or the problem of construction of probability theory to quantum mechanics) has been posed (see also [3]). The Birkhoff problem sounds like: To create an algebra of probability for quantum mechanics . . . without turning to Hilbert space or W ∗ -algebra. (see also [6]). Many papers are devoted to quantum logic. Quantum logic [15] (or orthomodular partial ordered set =OPS ) is a set L endowed with a partial order ≤ and unary operation ⊥ such that the following conditions are satisfied (the symbols ∨, ∧ denote the lattice-theoretic operations induced by ≤): (i) L possesses the least and the greatest elements, 0 and I, and 0 6= I; (ii) a ≤ b implies b⊥ ≤ a⊥ for any a, b ∈ L; (iii) (a⊥ )⊥ = a for any a ∈ L; (iv) if {ai }i∈X is a finite subset of L such that ai ≤ a⊥ j for i 6= j, then there exists a supremum ∨i∈X ai in L; (v) if a, b ∈ L and a ≤ b, then b = a ∨ (b ∧ a⊥ ). Sometimes axioms (iv), (v) are replaced by the following: (iv0 ) if a ≤ b⊥ then there exist a ∨ b; (v0 ) if a, b ∈ L and a ≤ b, then there exist c ≤ a⊥ such that b = a ∨ c.

Two Non Classical Quantum Logics of Projections in Hilbert Space ... 79 Algebraically, quantum logics are called orthomodular partially ordered sets (or, in short, orthomodular posets). Logic L does not have to be neither distributive nor a lattice. Two elements a, b ∈ L are called orthogonal if a ≤ b⊥ . We will denote the orthogonality of a, b by the symbol a ⊥ b. An element a, a 6= 0, is said to be atom, if b ≤ a, b 6= 0 implies b = a. An important interpretation of quantum logic is the set Π of all orthogonal (=self-adjoint) projections (=idempotents) on a Hilbert space H. We can compare some partial order properties of orthogonal and real-orthogonal (=Rorthogonal) projections. Projections are widely studied at the present time. All known logics consisting of projections (not orthogonal in general) possess the following property: for any  ≥ 1 there exists a projection A with the norm kAk ≥ . In the chapter, we propose a variant of logic with the property: for any  > 0 there exists an R-orthogonal projection A with the norm kA| ≤ . We prove that for the set of all R-orthogonal projections on complex Hilbert space the conditions (i)-(iii) and the condition (iv00 ) if a family {ai } is mutually orthogonal then supi {ai } exists (and = P i ai ).

are fulfilled. Note also the article [14]. It examines R-orthogonal projections in the real Euclidean space. Some definitions and properties. Let us denote the real part (the imaginary part) of complex number λ by Re(λ) (by Im(λ), respectively). Let H (Rn , Cn ) denote the complex Hilbert space (real, complex Euclidean space of dimension n, respectively) with the inner product (x, y), x, y ∈ H ((x, y) = x1 y1 + x2 y2 + · · · + xn yn if x = (x1 , xp 2, . . . , xn ), y = (y1 , y2 , . . . , yn ∈ Cn ), respectively) and with the norm kxk := (x, x). Let B(H) be the set of all bounded linear operator on H with the norm kAk := sup{kAxk : kxk = 1}. An operator V ∈ B(H) is said to be a partial isometry with an initial subspace H1 ⊆ H and with a final one H2 ⊆ H if (V x, V y) = (x, y) for all x, y ∈ H1 , V x = 0 for all x ⊥ H1 , and V H1 = H2 . Two vectors x, y are called orthogonal if (x, y) = 0 for all x, y. Definition 1.1. Two vectors x, y are said to be real-orthogonal (or shortly R-orthogonal) if Re(x, y) = 0.

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Marjan Matvejchuk and Elena Vladova

Note that this property has been called semi-orthogonal in [5]. Dominic Widdows [13] proposed to use the term real-orthogonal (or R-orthogonal in short) instead, to avoid any confusion with the definition of semi-orthogonal matrices1 . Let A ∈ B(H). Then, A∗ is the bounded operator with (Ax, y) = (x, A∗y) ∀x, y. By H(A) we denote the Hermitian part of A, i.e. H(A) = 21 (A + A∗ ). It is well known that A is an orthogonal projection, i.e. an Hermitian idempotent, if and only if (I −A)x and Ax are orthogonal for all x. Similarly [5], an operator A is called R-orthogonal projection (= real-orthogonal projection) if the vectors (I − A)x, Ax are R-orthogonal for all x ∈ H. This is equivalent to the fact that (I − A∗ )A is skew-Hermitian, which is satisfied if and only if A∗ A equals the Hermitian part of A, i.e. A∗ A =

1 (A + A∗ ) = H(A). 2

(1.1)

An operator A ∈ B(H) satisfying (1.1) is said to be R- orthogonal projection, also.

2.

R EAL -O RTHOGONAL P ROJECTIONS

For example, A = (1/2)(I + V ) is R-orthogonal projection in H, dim H = ∞. Here V is a partial isometry with the initial subspace H and the final one H0 (⊂ H). But A∗ = (1/2)(I + V ∗ ) is not R-orthogonal projection. Let A be R-orthogonal projection. It is clear that, kAk ≤ 1. An operator A on H is R-orthogonal projection if and only if kAxk2 = Re(Ax, x) (= (H(A)x, x)) for all x. Note that any orthogonal (=self adjoint) projection is R-orthogonal projection, also. Example 2.1. Any one-dimensional R-orthogonal projection on Rn is an orthogonal projection. Proof. Any one-dimensional operator P has the form P = (·, x)y, x, y ∈ Rn . Without loss of generality we may assume that kxk = kyk. It is clear that P + P ∗ = (·, x)y + (·, y)x = (1/2)[(·, x + y)(x + y) − (·, x − y)(x − y)] and 1

see https://en.wikipedia.org/wiki/Semiorthogonal matrix

Two Non Classical Quantum Logics of Projections in Hilbert Space ... 81 (x + y, x − y) = 0. Let, in addition, P be R-orthogonal projection. Due to (1.1), (0 0, [x, x] ≥ 0, [x, x] < 0, [x, x] ≤ 0 respectively). Let us denote by β 0 (β ++ , β + , β −− , β − ) the set of all J-neutral (J-positive, Jnonnegative, J-negative, J-non positive, respectively) vectors. Let β := β ++ ∪ β −− . The vectors from β are said to be definite vectors. Put Γ+ ≡ {f ∈ H : [f, f ] = 1} and Γ− ≡ {f ∈ H : [f, f ] = −1}. It is clear that JΓ± = Γ± . The set Γ := Γ+ ∪ Γ− is an indefinite analogy of the unit sphere S of H. An operator A ∈ B(H) is J-positive (J-negative) if [Ax, x] ≥ 0 ([Ax, x] ≤ 0), for all x ∈ H. Note that A ∈ B(H) is J-positive (negative) if and only if JA ≥ 0 (JA ≤ 0, respectively). Let A ∈ B(H). The operator A# := JA∗ J (≡ [Ax, y] = [x, A#y], for all x, y ∈ H) is said to be J-adjoint to A. Write 1 (A − A∗ ), for all A ∈ B(H). Set L1 := {A ∈ 0 and find a vector p ∈ Γ+ such that f (p) ≤ . To the vector p there corresponds a ˙ 1 . Let J1 be the canonical symmetry canonical decomposition H = lin{p}[+]H corresponding to this decomposition. J1 generates a new Hilbert space structure in H topologically equivalent to the original one. Here the point p becomes the pole of the unit hyperboloid Γ+ . We denote by σ the rotation about the pole p by the angle π/2. We set ν(x) ≡ f (x) + f (σx), x ∈ Γ. It is clear that ν is an indefinite J-frame function. For every point y on the equator (= Γ ∩ H1 ) the

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points y, σy, p form a J-frame and therefore (0 ≥) ν(y) = f (y) + f (σy) = w − f (p) (≥ w − ). Thus ν is constant on the equator Γ ∩ H1 . We remark that ν(x) ≥ 0 ≥ ν(p) − 2 for any x ∈ Γ+ and ν(g) = ν(y) + 2 for any g ∈ Γ− . We denote by S1 the unit sphere in H with respect to the Hilbert norm kxk1 = [J1 x, x], x ∈ H, and by E1 (g/kgk1) the unique great circle in S1 passing through g/kgk1 ∈ S1 and meeting the equator Γ ∩ H1 in the point x in such way that [x, g] = 0. N1 denotes the open upper hemisphere of S1 . Next, let t, s ∈ Γ− be points such that [t, s] = 0 and t/ktk1 , s/ksk1 ∈ E1 (g/kgk1). Then, for a point q lying on the equator Γ ∩ H1 with [q, g] = 0, we have ν(q) + ν(g) = ν(t) + ν(s). Hence, ν(g) = ν(t) + ν(s) − ν(q) ≤ ν(t) + ν(s) − (ν(s) − 2), i.e., ν(g) − 2 ≤ ν(t). Let N − ≡ {x ∈ Γ− : kxk1 6= 1} and β ≡ sup{ν(x) : x ∈ N − }. Choose z ∈ N − in such way that ν(z) ≥ β − , and denote by U the set of points x ∈ N − such that for some y ∈ N − we have a) y/kyk1 ∈ E1 (z/kzk1 ) and b) x/kxk1 ∈ E1 (y/kyk1). Then, for any x ∈ U we shall have ν(x) ≥ ν(y) − 2,

ν(y) ≥ ν(z) − 2,

β ≥ ν(x) ≥ ν(z) − 4 ≥ β − 5.

Therefore osc(ν, U ) ≤ 5. It follows from Lemma 1.5 that U has nonempty interior U 0 . Suppose the vector y belongs to U 0 . We find a vector z ∈ Γ+ such that [x, y] = 0. To this vector there correspond a canonical decomposition H = ˙ 2 and a canonical symmetry J2 defining a Hilbert space structure lin{z}[+]H in H with inner product (t, s)2 = [J2 t, s]. Here the point y lies on the equator Γ ∩ H2 . Using Lemma 9.4 we find that each point g ∈ Γ has a neighborhood V in the topology of the norm k · k2 for which osc(ν, V ) ≤ 40. Since, the norms k · k and k · k2 are equivalent, the point p has a neighborhood W (in the topology of the norm k · k ) for which osc(ν, W ) ≤ 40. But 0 ≤ ν(p) = 2f (p) ≤ 2. Hence, sup{ν(x) : x ∈ W } ≤ 42. Since, 0 ≤ f (x) ≤ ν(x) when x ∈ Γ+ , we have sup{f (x) : x ∈ W } ≤ 42. Again using Lemma 9.4 we conclude that each point of Γ has a neighborhood U in which osc(f, U ) ≤ 24 · 42 · . By the arbitrariness of  > 0 the function f is continuous. By Lemma 9.2 the function f is regular up to a semi-trace. Lemma 9.7. Let f (ψ) = tr(Apψ ) + c, c ∈ R, for ψ ∈ Γ+ and f (ψ) = tr(Apψ ) for ψ ∈ Γ− be an indefinite J-frame function in the two-dimensional indefinite

Two Non Classical Quantum Logics of Projections in Hilbert Space ... 115 space H. Then, A is a J-self-adjoint operator uniquely determined by the values of f on Γ− . Proof. Let e+ ∈ S ∩ H + , e− ∈ S ∩ H − and let A be represented by the matrix (aij ) in the basis e+ , e− . Moreover, f (e+ ) = a11 + c ≥ 0, f (e− ) = a22 ≤ 0. Hence, aii ∈ R. Let ψ ≡ αe+ + βe− ∈ Γ− . Then, |β|2 − |α|2 = 1 and f (ψ) = (Aψ, −ψ) = a22 |β|2 − a11 |α|2 + a21 αβ − a12 βα (9.6) p p Setting α = t/(1 − 2t), β = (1 − t)/(1 − 2t), where t ∈ [0, 1/2), we verify that =a21 = =a12 . Similarly we find that a2 ≥ 0 and e1 , e2 ∈ (H + ∪ H − ) ∩ S, (e1 , e2 ) = 0. Any vector ϕ[⊥] ∈ S such that [ϕ[⊥], ϕ] = 0 has the form eiα (a2 e1 + a1 e2 ). Let us estimate F (ϕ[⊥] ). It follows from the regularity of F on linR {e1 , e2 } that |F (ϕ[⊥]) − m|(ϕ[⊥], ϕ)|2| ≤ δ(),

(9.9)

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where δ() →→0 0. Suppose the vector g belongs to S. There exists a number eiθ , θ ∈ R, such that eiθ ∈ H ≡ linR {ϕ, eiβ (a2 )e1 + a1 e2 )}. We set q
ϕ[⊥] ≡ eiβ (a2 e1 +a1 e2 ), ϕ⊥ ≡ 1/ 1 −