Extending the Kantorovich’s theorem on Newton’s method for solving strongly regular generalized equation View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

2018-04-27

AUTHORS

I. K. Argyros, G. N. Silva

ABSTRACT

The aim of this paper, is to extend the applicability of Newton’s method for solving a generalized equation of the type f(x)+F(x)∋0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)+F(x)\ni 0$$\end{document} in Banach spaces, where f is a Fréchet differentiable function and F is a set-valued mapping. The novelty of the paper is the introduction of a restricted convergence domain. Using the idea of a weaker majorant, the convergence of the method, the optimal convergence radius, and results of the convergence rate are established. That is we find a more precise location where the Newton iterates lie than in earlier studies. Consequently, the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the semilocal convergence of the Newton iteration for solving f(x)+F(x)∋0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)+F(x)\ni 0$$\end{document}. The strong regularity concept plays an important role in our analysis. We finally present numerical examples, where we can solve equations in cases not possible before without using additional hypotheses. More... »

PAGES

213-226

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s11590-018-1266-6

DOI

http://dx.doi.org/10.1007/s11590-018-1266-6

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1103668458


Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

[
  {
    "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
    "about": [
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Mathematical Sciences", 
        "type": "DefinedTerm"
      }, 
      {
        "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0103", 
        "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
        "name": "Numerical and Computational Mathematics", 
        "type": "DefinedTerm"
      }
    ], 
    "author": [
      {
        "affiliation": {
          "alternateName": "Department of Mathematics Sciences, Cameron University, 73505, Lawton, OK, USA", 
          "id": "http://www.grid.ac/institutes/grid.253592.a", 
          "name": [
            "Department of Mathematics Sciences, Cameron University, 73505, Lawton, OK, USA"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Argyros", 
        "givenName": "I. K.", 
        "id": "sg:person.015707547201.06", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015707547201.06"
        ], 
        "type": "Person"
      }, 
      {
        "affiliation": {
          "alternateName": "Universidade Federal do Oeste da Bahia, CEP 47808-021, Barreiras, BA, Brazil", 
          "id": "http://www.grid.ac/institutes/grid.472638.c", 
          "name": [
            "Universidade Federal do Oeste da Bahia, CEP 47808-021, Barreiras, BA, Brazil"
          ], 
          "type": "Organization"
        }, 
        "familyName": "Silva", 
        "givenName": "G. N.", 
        "id": "sg:person.016672542147.21", 
        "sameAs": [
          "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016672542147.21"
        ], 
        "type": "Person"
      }
    ], 
    "citation": [
      {
        "id": "sg:pub.10.1007/s10589-007-9082-4", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1053719030", 
          "https://doi.org/10.1007/s10589-007-9082-4"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/s10107-009-0322-5", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1007546834", 
          "https://doi.org/10.1007/s10107-009-0322-5"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/978-0-387-87821-8", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1022362018", 
          "https://doi.org/10.1007/978-0-387-87821-8"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/bf01404880", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1048822194", 
          "https://doi.org/10.1007/bf01404880"
        ], 
        "type": "CreativeWork"
      }, 
      {
        "id": "sg:pub.10.1007/978-1-4612-0701-6", 
        "sameAs": [
          "https://app.dimensions.ai/details/publication/pub.1020016656", 
          "https://doi.org/10.1007/978-1-4612-0701-6"
        ], 
        "type": "CreativeWork"
      }
    ], 
    "datePublished": "2018-04-27", 
    "datePublishedReg": "2018-04-27", 
    "description": "The aim of this paper, is to extend the applicability of Newton\u2019s method for solving a generalized equation of the type f(x)+F(x)\u220b0\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f(x)+F(x)\\ni 0$$\\end{document} in Banach spaces, where f is a Fr\u00e9chet differentiable function and F is a set-valued mapping. The novelty of the paper is the introduction of a restricted convergence domain. Using the idea of a weaker majorant, the convergence of the method, the optimal convergence radius, and results of the convergence rate are established. That is we find a more precise location where the Newton iterates lie than in earlier studies. Consequently, the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the semilocal convergence of the Newton iteration for solving f(x)+F(x)\u220b0\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$$f(x)+F(x)\\ni 0$$\\end{document}. The strong regularity concept plays an important role in our analysis. We finally present numerical examples, where we can solve equations in cases not possible before without using additional hypotheses.", 
    "genre": "article", 
    "id": "sg:pub.10.1007/s11590-018-1266-6", 
    "inLanguage": "en", 
    "isAccessibleForFree": false, 
    "isPartOf": [
      {
        "id": "sg:journal.1052645", 
        "issn": [
          "1862-4472", 
          "1862-4480"
        ], 
        "name": "Optimization Letters", 
        "publisher": "Springer Nature", 
        "type": "Periodical"
      }, 
      {
        "issueNumber": "1", 
        "type": "PublicationIssue"
      }, 
      {
        "type": "PublicationVolume", 
        "volumeNumber": "13"
      }
    ], 
    "keywords": [
      "Newton method", 
      "generalized equation", 
      "set-valued mappings", 
      "optimal convergence radius", 
      "restricted convergence domain", 
      "Fr\u00e9chet differentiable function", 
      "same computational cost", 
      "Newton iterates", 
      "semilocal convergence", 
      "Newton iteration", 
      "Kantorovich theorem", 
      "Lipschitz constants", 
      "Banach spaces", 
      "differentiable functions", 
      "convergence rate", 
      "numerical examples", 
      "convergence domain", 
      "convergence radius", 
      "regularity concepts", 
      "computational cost", 
      "additional hypotheses", 
      "equations", 
      "theorem", 
      "convergence", 
      "iterates", 
      "majorant", 
      "iteration", 
      "space", 
      "applicability", 
      "novelty", 
      "idea", 
      "function", 
      "mapping", 
      "one", 
      "cost", 
      "domain", 
      "concept", 
      "way", 
      "cases", 
      "results", 
      "radius", 
      "introduction", 
      "analysis", 
      "constants", 
      "types", 
      "important role", 
      "location", 
      "earlier studies", 
      "hypothesis", 
      "rate", 
      "aim", 
      "precise location", 
      "study", 
      "role", 
      "method", 
      "paper", 
      "example", 
      "weaker majorant", 
      "strong regularity concept", 
      "regular generalized equation"
    ], 
    "name": "Extending the Kantorovich\u2019s theorem on Newton\u2019s method for solving strongly regular generalized equation", 
    "pagination": "213-226", 
    "productId": [
      {
        "name": "dimensions_id", 
        "type": "PropertyValue", 
        "value": [
          "pub.1103668458"
        ]
      }, 
      {
        "name": "doi", 
        "type": "PropertyValue", 
        "value": [
          "10.1007/s11590-018-1266-6"
        ]
      }
    ], 
    "sameAs": [
      "https://doi.org/10.1007/s11590-018-1266-6", 
      "https://app.dimensions.ai/details/publication/pub.1103668458"
    ], 
    "sdDataset": "articles", 
    "sdDatePublished": "2022-01-01T18:51", 
    "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
    "sdPublisher": {
      "name": "Springer Nature - SN SciGraph project", 
      "type": "Organization"
    }, 
    "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/article/article_791.jsonl", 
    "type": "ScholarlyArticle", 
    "url": "https://doi.org/10.1007/s11590-018-1266-6"
  }
]
 

Download the RDF metadata as:  json-ld nt turtle xml License info

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s11590-018-1266-6'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s11590-018-1266-6'

Turtle is a human-readable linked data format.

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s11590-018-1266-6'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s11590-018-1266-6'


 

This table displays all metadata directly associated to this object as RDF triples.

148 TRIPLES      22 PREDICATES      90 URIs      77 LITERALS      6 BLANK NODES

Subject Predicate Object
1 sg:pub.10.1007/s11590-018-1266-6 schema:about anzsrc-for:01
2 anzsrc-for:0103
3 schema:author N94291f0e29714642934015026ae771af
4 schema:citation sg:pub.10.1007/978-0-387-87821-8
5 sg:pub.10.1007/978-1-4612-0701-6
6 sg:pub.10.1007/bf01404880
7 sg:pub.10.1007/s10107-009-0322-5
8 sg:pub.10.1007/s10589-007-9082-4
9 schema:datePublished 2018-04-27
10 schema:datePublishedReg 2018-04-27
11 schema:description The aim of this paper, is to extend the applicability of Newton’s method for solving a generalized equation of the type f(x)+F(x)∋0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)+F(x)\ni 0$$\end{document} in Banach spaces, where f is a Fréchet differentiable function and F is a set-valued mapping. The novelty of the paper is the introduction of a restricted convergence domain. Using the idea of a weaker majorant, the convergence of the method, the optimal convergence radius, and results of the convergence rate are established. That is we find a more precise location where the Newton iterates lie than in earlier studies. Consequently, the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the semilocal convergence of the Newton iteration for solving f(x)+F(x)∋0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x)+F(x)\ni 0$$\end{document}. The strong regularity concept plays an important role in our analysis. We finally present numerical examples, where we can solve equations in cases not possible before without using additional hypotheses.
12 schema:genre article
13 schema:inLanguage en
14 schema:isAccessibleForFree false
15 schema:isPartOf N0472b36653e348a0afa73e7199177417
16 N72c84a23e39f4a4cbfb4bc8941a960a7
17 sg:journal.1052645
18 schema:keywords Banach spaces
19 Fréchet differentiable function
20 Kantorovich theorem
21 Lipschitz constants
22 Newton iterates
23 Newton iteration
24 Newton method
25 additional hypotheses
26 aim
27 analysis
28 applicability
29 cases
30 computational cost
31 concept
32 constants
33 convergence
34 convergence domain
35 convergence radius
36 convergence rate
37 cost
38 differentiable functions
39 domain
40 earlier studies
41 equations
42 example
43 function
44 generalized equation
45 hypothesis
46 idea
47 important role
48 introduction
49 iterates
50 iteration
51 location
52 majorant
53 mapping
54 method
55 novelty
56 numerical examples
57 one
58 optimal convergence radius
59 paper
60 precise location
61 radius
62 rate
63 regular generalized equation
64 regularity concepts
65 restricted convergence domain
66 results
67 role
68 same computational cost
69 semilocal convergence
70 set-valued mappings
71 space
72 strong regularity concept
73 study
74 theorem
75 types
76 way
77 weaker majorant
78 schema:name Extending the Kantorovich’s theorem on Newton’s method for solving strongly regular generalized equation
79 schema:pagination 213-226
80 schema:productId Nbd460f70847a4589ba8dde74e226c055
81 Neae05426e91c409294f51a17c029f574
82 schema:sameAs https://app.dimensions.ai/details/publication/pub.1103668458
83 https://doi.org/10.1007/s11590-018-1266-6
84 schema:sdDatePublished 2022-01-01T18:51
85 schema:sdLicense https://scigraph.springernature.com/explorer/license/
86 schema:sdPublisher Na7ef62cec71d478b9d33cbf7ab39ad28
87 schema:url https://doi.org/10.1007/s11590-018-1266-6
88 sgo:license sg:explorer/license/
89 sgo:sdDataset articles
90 rdf:type schema:ScholarlyArticle
91 N0472b36653e348a0afa73e7199177417 schema:volumeNumber 13
92 rdf:type schema:PublicationVolume
93 N72c84a23e39f4a4cbfb4bc8941a960a7 schema:issueNumber 1
94 rdf:type schema:PublicationIssue
95 N94291f0e29714642934015026ae771af rdf:first sg:person.015707547201.06
96 rdf:rest Nf6f24af292be41968904f1f6b22917d8
97 Na7ef62cec71d478b9d33cbf7ab39ad28 schema:name Springer Nature - SN SciGraph project
98 rdf:type schema:Organization
99 Nbd460f70847a4589ba8dde74e226c055 schema:name doi
100 schema:value 10.1007/s11590-018-1266-6
101 rdf:type schema:PropertyValue
102 Neae05426e91c409294f51a17c029f574 schema:name dimensions_id
103 schema:value pub.1103668458
104 rdf:type schema:PropertyValue
105 Nf6f24af292be41968904f1f6b22917d8 rdf:first sg:person.016672542147.21
106 rdf:rest rdf:nil
107 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
108 schema:name Mathematical Sciences
109 rdf:type schema:DefinedTerm
110 anzsrc-for:0103 schema:inDefinedTermSet anzsrc-for:
111 schema:name Numerical and Computational Mathematics
112 rdf:type schema:DefinedTerm
113 sg:journal.1052645 schema:issn 1862-4472
114 1862-4480
115 schema:name Optimization Letters
116 schema:publisher Springer Nature
117 rdf:type schema:Periodical
118 sg:person.015707547201.06 schema:affiliation grid-institutes:grid.253592.a
119 schema:familyName Argyros
120 schema:givenName I. K.
121 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.015707547201.06
122 rdf:type schema:Person
123 sg:person.016672542147.21 schema:affiliation grid-institutes:grid.472638.c
124 schema:familyName Silva
125 schema:givenName G. N.
126 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016672542147.21
127 rdf:type schema:Person
128 sg:pub.10.1007/978-0-387-87821-8 schema:sameAs https://app.dimensions.ai/details/publication/pub.1022362018
129 https://doi.org/10.1007/978-0-387-87821-8
130 rdf:type schema:CreativeWork
131 sg:pub.10.1007/978-1-4612-0701-6 schema:sameAs https://app.dimensions.ai/details/publication/pub.1020016656
132 https://doi.org/10.1007/978-1-4612-0701-6
133 rdf:type schema:CreativeWork
134 sg:pub.10.1007/bf01404880 schema:sameAs https://app.dimensions.ai/details/publication/pub.1048822194
135 https://doi.org/10.1007/bf01404880
136 rdf:type schema:CreativeWork
137 sg:pub.10.1007/s10107-009-0322-5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1007546834
138 https://doi.org/10.1007/s10107-009-0322-5
139 rdf:type schema:CreativeWork
140 sg:pub.10.1007/s10589-007-9082-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1053719030
141 https://doi.org/10.1007/s10589-007-9082-4
142 rdf:type schema:CreativeWork
143 grid-institutes:grid.253592.a schema:alternateName Department of Mathematics Sciences, Cameron University, 73505, Lawton, OK, USA
144 schema:name Department of Mathematics Sciences, Cameron University, 73505, Lawton, OK, USA
145 rdf:type schema:Organization
146 grid-institutes:grid.472638.c schema:alternateName Universidade Federal do Oeste da Bahia, CEP 47808-021, Barreiras, BA, Brazil
147 schema:name Universidade Federal do Oeste da Bahia, CEP 47808-021, Barreiras, BA, Brazil
148 rdf:type schema:Organization
 




Preview window. Press ESC to close (or click here)


...