Improving the rate of convergence of some Newton-like methods for the solution of nonlinear equations containing a nondifferentiable term

9 years ago

Emil Cătinaş, Ion Păvăloiu

(original), html, Numerical Analysis, paper, PDF, works

Abstract

Authors

Ioannis K. Argyros
(Cameron University, USA)

Emil Cătinaş

Emil Cătinaş

Ion Păvăloiu

Ion Păvăloiu

Keywords

chord/secant method; semilocal convergence; r-convergence order.

Cite this paper as:

I. Argyros, E. Cătinaş, I. Păvăloiu, Improving the rate of convergence of some Newton-like methods for the solution of nonlinear equations containing a nondifferentiable term, Rev. Anal. Numér. Théor. Approx., 27 (1998) no. 2, pp. 191-202.

PDF

Scanned paper: on the journal website.

Latex-pdf version of the paper.

About this paper

Print ISSN

1222-9024

Online ISSN

2457-8126

MR

ZBL

Google Scholar citations

[1] K. Argyros, On the solution of equations with nondifferentiable operators and the Ptak error estimates, BIT, 30 (1990), pp. 752-754.

[2] I. K. Argyros, On the solution of nonlinear equations with a nondifferentiable term, Rev. Anal. Numer. Theor. Approx., 22 (1993) 2, pp. 125-135.

[3] I. K. Argyros, On some iterative methods for solving nonlinear equations with a nondifferentiable term of order between 1.618… and 1.839. (submitted to this joumal).

[4] I. K. Arglros, and F. Szidarovszky, The Theory and Application of lteration Method, CRC Press, Inc., Boca Raton, Florida, 1993.

[5] E. Cătinaș, On some iterative methods for solving nonlinear equations, Rev. Anal. Numer. Theor. Approx., 23, I (1994), pp. 47-53.

[6] G. Goldner, and M. Balazs, Remarks on divided differences and method of chords, Rev. Anal. Numer. Theor. Approx., 3, I (1974), pp. 19-30.

[7] L. V. Kantorovich, The method of successive approximation for functional equations, Acta Math. 71 (1939), pp. 63-97.

[8] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.

[9] I. Păvăloiu, Sur une generalisation de la methode de Steffensen, Rev. Anal. Numer. Theor. Approx., 21, 1 (1992), pp. 59-65.

[10] I. Păvăloiu, A convergence theorem concerning the chord methods, Rev. Anal. Numer. Theor. Approx., 22, 1 (1993), pp. 83-85.

[11] F. A, Potra, On an iterative algorithm of order 1.839 . . . for solving nonlinear equations, Numer. Funct. Anal. Optimiz. 7, I (1984-1985), pp. 75-106.

[12] T. Yamamoto and X. Chen, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optimiz. 10, 1 and 2 (1989), pp.37-48.

Paper (preprint) in HTML form

/*====================================================================== Core CSS for LaTeXML documents converted to (X)HTML */ /* Generic Page layout */ .ltx_page_header, .ltx_page_footer { font-size:0.8em; } .ltx_page_header *[rel~="prev"], .ltx_page_footer *[rel~="prev"] { float:left; } .ltx_page_header *[rel~="up"], .ltx_page_footer *[rel~="up"] { display:block; text-align:center; } .ltx_page_header *[rel~="next"], .ltx_page_footer *[rel~="next"] { float:right; } /* What was I trying for here; need more selective rule! .ltx_page_header .ltx_ref, .ltx_page_footer .ltx_ref { margin:0 1em; } */ .ltx_page_header li { padding:0.1em 0.2em 0.1em 1em;} /* Main content */ .ltx_page_content { clear:both; } .ltx_page_header { border-bottom:1px solid; margin-bottom:5px; } .ltx_page_footer { clear:both; border-top:1px solid; margin-top:5px; } .ltx_page_header:after, .ltx_page_footer:after, .ltx_page_content:after { content:"."; display:block; height:0; clear:both; visibility:hidden; } .ltx_page_footer:before { content:"."; display:block; height:0; clear:both; visibility:hidden; } .ltx_page_logo { font-size:80%; margin-top: 5px; clear:both; float:right; } .ltx_page_logo a { font-variant: small-caps; } .ltx_page_logo img { vertical-align:-3px; } /* if shown */ .ltx_page_navbar li { white-space:nowrap; display:block; overflow:hidden; } /* If ref got turned into span, it's "this section"*/ .ltx_page_navbar li span.ltx_ref { white-space:normal; overflow:visible; } /* Ought to be easily removable/overridable? */ .ltx_pagination.ltx_role_newpage { height:2em; } /*====================================================================== Document Structure; Titles & Frontmatter */ /* undo bold here to remove the browser's native h# styling, at let all other styles override it (with more specific rules)*/ .ltx_title { font-size:100%; font-weight:normal; } /* Hack to simulate run-in! put class="ltx_runin" on a title or tag for it to run-into the following text. */ .ltx_runin { display:inline; } .ltx_runin:after { content:" "; } .ltx_runin + .ltx_para, .ltx_runin + .ltx_para p, .ltx_runin + p { display:inline; } .ltx_outdent { margin-left: -2em; } /* .ltx_chapter_title, etc should be in ltx-article.css etc. */ .ltx_page_main { margin:0px; padding:1em 3em 1em 2em; } .ltx_tocentry { list-style-type:none; } /* support for common author block layouts.*/ /* add class ltx_authors_1line to get authors in single line with pop-up affiliation, etc. */ .ltx_authors_1line .ltx_creator, .ltx_authors_1line .ltx_author_before, .ltx_authors_1line .ltx_author_after { display:inline;} .ltx_authors_1line .ltx_author_notes { display:inline-block; } .ltx_authors_1line .ltx_author_notes:before { content:"*"; color:blue;} .ltx_authors_1line .ltx_author_notes span { display:none; } .ltx_authors_1line .ltx_author_notes:hover span { display:block; position:absolute; z-index:10; background:white; text-align:left; border: 1px solid black; border-radius: 0 5px 5px 5px; box-shadow: 5px 5px 10px gray; } /* add class=ltx_authors_multiline to get authors & affliations on separate lines*/ .ltx_authors_multiline .ltx_creator, .ltx_authors_multiline .ltx_author_before, .ltx_authors_multiline .ltx_author_after, .ltx_authors_multiline .ltx_author_notes, .ltx_authors_multiline .ltx_author_notes .ltx_contact { display:block; } /*====================================================================== Para level */ .ltx_float { margin: 1ex 3em 1ex 3em; } td.ltx_subfigure, td.ltx_subtable, td.ltx_subfloat { width:50%; } /* theorems, figure, tables, floats captions.. */ /*====================================================================== Blocks, Lists, Floats */ .ltx_p, .ltx_quote, .ltx_block, .ltx_para { display: block; } /* alignment within blocks */ .ltx_align_left { text-align:left; } .ltx_align_right { text-align:right; } .ltx_align_center { text-align:center; } .ltx_align_justify { text-align:justify; } .ltx_align_top { vertical-align:top; } .ltx_align_bottom { vertical-align:bottom; } .ltx_align_middle { vertical-align:middle; } .ltx_align_baseline { vertical-align:baseline; } .ltx_align_floatleft { float:left; } .ltx_align_floatright { float:right; } .ltx_td.ltx_align_left, .ltx_th.ltx_align_left, .ltx_td.ltx_align_right, .ltx_th.ltx_align_right, .ltx_td.ltx_align_center, .ltx_th.ltx_align_center { white-space:nowrap; } .ltx_td.ltx_align_left.ltx_wrap, .ltx_th.ltx_align_left.ltx_wrap, .ltx_td.ltx_align_right.ltx_wrap, .ltx_th.ltx_align_right.ltx_wrap, .ltx_td.ltx_align_center.ltx_wrap, .ltx_th.ltx_align_center.ltx_wrap, .ltx_td.ltx_align_justify, .ltx_th.ltx_align_justify { white-space:normal; } .ltx_tabular .ltx_tabular { width:100%; } .ltx_inline-block { display:inline-block; } /* equations in non-aligned mode (not normally used) */ .ltx_eqn_div { display:block; width:95%; text-align:center; } /* equations in aligned mode (aligning tags, etc as well as equations) */ .ltx_eqn_table { display:table; width:100%; border-collapse:collapse; } .ltx_eqn_row { display:table-row; } .ltx_eqn_cell { display:table-cell; width:auto; } /* Padding between column pairs in ams align */ table.ltx_eqn_align tr.ltx_equation td.ltx_align_left + td.ltx_align_right, table.ltx_eqn_align tr.ltx_equation td.ltx_align_left + td.ltx_align_center, table.ltx_eqn_align tr.ltx_equation td.ltx_align_center + td.ltx_align_right, table.ltx_eqn_align tr.ltx_equation td.ltx_align_center + td.ltx_align_center { padding-left:3em; } table.ltx_eqn_eqnarray tr.ltx_eqn_lefteqn + tr td.ltx_align_right { min-width:2em; } .ltx_eqn_eqno { max-width:0em; overflow:visible; white-space: nowrap; } .ltx_eqn_eqno.ltx_align_right .ltx_tag { float:right; } .ltx_eqn_center_padleft, .ltx_eqn_center_padright { width:50%; min-width:2em;} .ltx_eqn_left_padleft, .ltx_eqn_right_padright { min-width:2em; } .ltx_eqn_left_padright, .ltx_eqn_right_padleft { width:100%; } /* Various lists */ .ltx_itemize, .ltx_enumerate, .ltx_description { display:block; } .ltx_itemize .ltx_item, .ltx_enumerate .ltx_item { display: list-item; } /* Position the tag to look like a normal item bullet. */ li.ltx_item > .ltx_tag { display:inline-block; margin-left:-1.5em; min-width:1.5em; text-align:right; } .ltx_item .ltx_tag + .ltx_para, .ltx_item .ltx_tag + .ltx_para .ltx_p { display:inline; } /* NOTE: Need to try harder to get runin appearance? */ dl.ltx_description dt { margin-right:0.5em; float:left; font-weight:bold; font-size:95%; } dl.ltx_description dd { margin-left:5em; } dl.ltx_description dl.ltx_description dd { margin-left:3em; } /* Theorems */ .ltx_theorem {margin:1em 0em 1em 0em; } .ltx_title_theorem { font-size:100%; } /* Bibliographies */ .ltx_bibliography dt { margin-right:0.5em; float:left; } .ltx_bibliography dd { margin-left:3em; } /*.ltx_biblist { list-style-type:none; }*/ .ltx_bibitem { list-style-type:none; } .ltx_bibitem .ltx_tag { font-weight:bold; margin-left:-2em; width:3em; } /*.bibitem-tag + div { display:inline; }*/ .ltx_bib_title { font-style:italic; } .ltx_bib_article .bib-title { font-style:normal !important; } .ltx_bib_journal { font-style:italic; } .ltx_bib_volume { font-weight:bold; } /* Indices */ .ltx_indexlist li { list-style-type:none; } .ltx_indexlist { margin-left:1em; padding-left:1em;} /* Listings */ .ltx_listing { display:block; margin: 1ex 3em 1ex 0em; overflow-x:auto; text-align: left; } .ltx_float .ltx_listing { margin: 0; } .ltx_listingline { white-space:nowrap; min-height:1em; } .ltx_lst_numbers_left .ltx_listingline .ltx_tag { background-color:transparent; margin-left:-3em; width:2.5em; position:absolute; text-align:right; } .ltx_lst_numbers_right .ltx_listingline .ltx_tag { background-color:transparent; width:2.5em; position:absolute; right:3em; text-align:right; } /* position:absolute; left:0em; max-width:0em; text-align:right; } */ .ltx_parbox {text-indent:0em; } /* NOTE that it is CRITICAL to put position:relative outside & absolute inside!! I wish I understood why! Outer box establishes resulting size, neutralizes any outer positioning, etc; inner establishes position of stuff to be rotated */ .ltx_transformed_outer { position:relative; bottom:0pt;left:0pt; overflow:visible; } .ltx_transformed_inner { display:block; position:absolute;bottom:0pt;left:0pt; } .ltx_transformed_inner > .ltx_p {text-indent:0em; margin:0; padding:0; } /* If simulating a table (html5), try to get rowspan to work...sorta? */ span.ltx_rowspan { position:absolute; top:0; bottom:0; } /* by default, p doesn't indent */ .ltx_p { text-indent:0em; white-space:normal; } /* explicit control of indentation (on ltx_para) */ .ltx_indent > .ltx_p:first-child { text-indent:2em!important; } .ltx_noindent > .ltx_p:first-child { text-indent:0em!important; } /*====================================================================== Columns */ .ltx_page_column1 { width:44%; float:left; } /* IE uses % of wrong container*/ .ltx_page_column2 { width:44%; float:right; } .ltx_page_columns > .ltx_page_column1 { width:48%; float:left; } .ltx_page_columns > .ltx_page_column2 { width:48%; float:right; } .ltx_page_columns:after { content:"."; display:block; height:0; clear:both; visibility:hidden; } /*====================================================================== Borders and such */ .ltx_tabular { display:inline-table; border-collapse:collapse; } .ltx_tabular.ltx_centering { display:table; } .ltx_thead, .ltx_tfoot, .ltx_tbody { display:table-row-group; } .ltx_tr { display:table-row; } .ltx_td, .ltx_th { display:table-cell; } .ltx_framed { border:1px solid black;} .ltx_tabular .ltx_td, .ltx_tabular .ltx_th { padding:0.1em 0.5em; } /* regular lines */ .ltx_border_t { border-top:1px solid black; } .ltx_border_r { border-right:1px solid black; } .ltx_border_b { border-bottom:1px solid black; } .ltx_border_l { border-left:1px solid black; } /* double lines */ .ltx_border_tt { border-top:3px double black; } .ltx_border_rr { border-right:3px double black; } .ltx_border_bb { border-bottom:3px double black; } .ltx_border_ll { border-left:3px double black; } /* Light lines */ .ltx_border_T { border-top:1px solid gray; } .ltx_border_R { border-right:1px solid gray; } .ltx_border_B { border-bottom:1px solid gray; } .ltx_border_L { border-left:1px solid gray; } /* Framing */ .ltx_framed_rectangle { border-style:solid; border-width:1px; } .ltx_framed_top { border-top-style:solid; border-top-width:1px; } .ltx_framed_left { border-left-style:solid; border-left-width:1px; } .ltx_framed_right { border-right-style:solid; border-right-width:1px; } .ltx_framed_bottom, .ltx_framed_underline { border-bottom-style:solid; border-bottom-width:1px; } .ltx_framed_topbottom { border-top-style:solid; border-top-width:1px; border-bottom-style:solid; border-bottom-width:1px; } .ltx_framed_leftright { border-left-style:solid; border-left-width:1px; border-right-style:solid; border-right-width:1px; } /*====================================================================== Misc */ /* .ltx_verbatim*/ .ltx_verbatim { text-align:left; } /*====================================================================== Meta stuff, footnotes */ .ltx_note_content { display:none; } /*right:5%; */ .ltx_note_content { max-width: 70%; font-size:90%; left:15%; text-align:left; background-color: white; padding: 0.5em 1em 0.5em 1.5em; border: 1px solid black; border-radius: 0 5px 5px 5px; box-shadow: 5px 5px 10px gray; } .ltx_note_mark { color:blue; } .ltx_note_type { font-weight: bold; } .ltx_note { display:inline-block; text-indent:0; } /* So we establish containing block */ .ltx_note_content .ltx_note_mark { position:absolute; left:0.2em; top:-0.1em; } .ltx_note:hover .ltx_note_content, .ltx_note .ltx_note_content:hover { display:block; position:absolute; z-index:10; } .ltx_ERROR { color:red; } .ltx_rdf { display:none; } .ltx_missing { color:red;} .ltx_nounicode { color:red; } /*====================================================================== SVG (pgf/tikz ?) basics */ /* Stuff appearing in svg:foreignObject */ .ltx_svg_fog foreignObject { margin:0; padding:0; overflow:visible; } .ltx_svg_fog foreignObject > p { margin:0; padding:0; display:block; } /*.ltx_svg_fog foreignObject > p { margin:0; padding:0; display:block; white-space:nowrap; }*/ /*====================================================================== Low-level Basics */ /* Note that LaTeX(ML)'s font model doesn't map quite exactly to CSS's */ /* Font Families => font-family */ .ltx_font_serif { font-family: serif; } .ltx_font_sansserif { font-family: sans-serif; } .ltx_font_typewriter { font-family: monospace; } /* dingbats should be converted to unicode? */ /* Math font families handled within math: script, symbol, fraktur, blackboard ? */ /* Font Series => font-weight */ .ltx_font_bold { font-weight: bold; } .ltx_font_medium { font-weight: normal; } /* Font Shapes => font-style or font-variant */ .ltx_font_italic { font-style: italic; font-variant:normal; } .ltx_font_upright { font-style: normal; font-variant:normal; } .ltx_font_slanted { font-style: oblique; font-variant:normal; } .ltx_font_smallcaps { font-variant: small-caps; font-style:normal; } .ltx_font_oldstyle { font-variant: oldstyle-nums; /* experimental css3 ? Doesn't seem to work!*/ font-style:normal; -moz-font-feature-settings: "onum"; -ms-font-feature-settings: "onum"; -webkit-font-feature-settings: "onum"; font-variant-numeric: oldstyle-nums; } .ltx_font_mathcaligraphic { font-family: "Lucida Calligraphy", "Zapf Chancery","URW Chancery L"; } /* .ltx_font_mathscript { ? } */ cite { font-style: normal; } .ltx_red { color:red; } /*.ltx_centering { text-align:center; margin:auto; }*/ /*.ltx_inline-block.ltx_centering,*/ /* Hmm.... is this right in general? */ .ltx_centering { display:block; margin:auto; text-align:center; } /* Dubious stuff */ .ltx_hflipped { display:inline-block; -moz-transform: scaleX(-1); -o-transform: scaleX(-1); -webkit-transform: scaleX(-1); transform: scaleX(-1); filter: FlipH; -ms-fliter: "FlipH"; } .ltx_vflipped { display:inline-block; -moz-transform: scaleY(-1); -o-transform: scaleY(-1); -webkit-transform: scaleY(-1); transform: scaleY(-1); filter: FlipV; -ms-fliter: "FlipV"; } /* .ltx_phantom handled in xslt */ /* A somewhat amsart-like style */ .ltx_title_document { text-transform:uppercase; font-size: 150%; text-align:center; } .ltx_title_section { font-family:serif; font-weight:normal; font-variant:small-caps; font-size: 120%; text-align:center; } .ltx_authors { text-align:center; } .ltx_author_before,.ltx_author_after, .ltx_role_author .ltx_personname { text-transform:uppercase; } .ltx_abstract { margin-bottom:1em; } .ltx_title_abstract { font-family:serif; font-weight:normal; font-variant:small-caps;} .ltx_title_keywords, .ltx_title_classification { display:inline; font-size:100%; font-weight:normal; font-style:italic; } .ltx_role_dedicatory { font-size:100%; font-style:italic; text-align:center; margin:1em;} .ltx_figure .ltx_tag, .ltx_table .ltx_tag { font-family:serif; font-variant:small-caps;} .ltx_title_subsection, .ltx_title_subsubsection { font-family:serif; font-weight:bold;} .ltx_title_subsection:after, .ltx_title_subsubsection:after, .ltx_title_paragraph:after { content:"."; } /* abstract, subsection, subsubsection and paragraph titles are runin */ .ltx_abstract .ltx_title, .ltx_abstract .ltx_title + .ltx_p { display:inline; } .ltx_subsection .ltx_title, .ltx_subsection .ltx_title + .ltx_para, .ltx_subsection .ltx_title + .ltx_para > .ltx_p { display:inline; } .ltx_subsubsection .ltx_title, .ltx_subsubsection .ltx_title + .ltx_para, .ltx_subsubsection .ltx_title + .ltx_para > .ltx_p { display:inline; } .ltx_paragraph .ltx_title, .ltx_paragraph .ltx_title + .ltx_para, .ltx_paragraph .ltx_title + .ltx_para > .ltx_p { display:inline; } .ltx_title +.ltx_para > .ltx_p {text-indent:2em; } .ltx_itemize, .ltx_enumerate, .ltx_description { margin-left: clamp(1em, 3%, 2em); } .ltx_title_document { font-size:170%; text-align:center; margin:0.5em 0 0.5em 0; } .ltx_authors, .ltx_role_author { text-align:center; margin:0.5em 0 0.5em 0; } .ltx_role_author .ltx_personname { font-size: 120%; } .ltx_date { text-align:center; font-size: 120%; margin:0.5em 0 0.5em 0; } .ltx_subtitle { text-align:center; font-size: 120%; padding-left:0.2em; margin-left:-0.5em; } .ltx_title_abstract { text-align:center; font-size: 100%; font-weight:bold; } .ltx_abstract { margin-left:4em; margin-right:4em; } .ltx_title_acknowledgements, .ltx_title_keywords, .ltx_title_classification { text-align:left; font-size: 100%; font-weight:bold; margin:0.5 0 0 0; } .ltx_appendix, .ltx_section, .ltx_subsection, .ltx_subsubsection { margin-top:1.5em; } .ltx_paragraph, .ltx_subparagraph { margin-top:1.0em; } .ltx_title_appendix, .ltx_title_section, .ltx_title_bibliography { font-size:140%; font-weight:bold; margin-bottom:1em; } .ltx_title_subsection { font-size:120%; font-weight:bold; margin-bottom:1em; } .ltx_title_subsubsection { font-size:100%; font-weight:bold; margin-bottom:1em; } /* Paragraph & Subparagraph titles should be runin! */ .ltx_title_paragraph { font-size:100%; font-weight:bold; display:inline; margin-right:1em; } .ltx_paragraph .ltx_title, .ltx_paragraph .ltx_title + .ltx_para, .ltx_paragraph .ltx_title + .ltx_para > .ltx_p { display:inline; } .ltx_title_subparagraph { font-size:100%; font-weight:bold; display:inline; margin-left:2em; margin-right:1em; } .ltx_subparagraph .ltx_title, .ltx_subparagraph .ltx_title + .ltx_para, .ltx_subparagraph .ltx_title + .ltx_para > .ltx_p { display:inline; } .ltx_figure { text-align:center; margin:auto; margin:0.5em; } .ltx_table { text-align:center; margin:auto; margin:0.5em; } /* experimental: flex model, to center short captions, justify long ones */ .ltx_table, .ltx_figure { display:flex; flex-direction:column; align-items:center; } .ltx_table .ltx_caption, .ltx_figure .ltx_caption { text-align:justify; } /* first p in para gets indented */ .ltx_para > .ltx_p:first-child { text-indent:2em; } /* except the initial in a section */ section > .ltx_title +.ltx_para > .ltx_p, section > .ltx_title +.ltx_date +.ltx_para > .ltx_p {text-indent:0em; } .ltx_title_abstract + .ltx_p {text-indent:2em; } .ltx_itemize, .ltx_enumerate { margin-left:1em; } .ltx_theorem { margin-top:0.5em; margin-bottom:0.5em; } .ltx_theorem .ltx_title { margin-bottom:0.1em; } .ltx_theorem .ltx_title + .ltx_para, .ltx_theorem .ltx_title + .ltx_para .ltx_p, .ltx_theorem .ltx_title + .ltx_p { margin-top:0em; } Improving the rate of convergence of some Newton-Like methods for the solution of nonlinear equations containing A nondifferentiable term

Improving the rate of convergence of some Newton-Like methods for the solution of nonlinear equations containing A nondifferentiable term

Ioannis K. Argyros, Emil Cătinaş, Ion Păvăloiu

$s u b s o l :$ AMS (MOS) Subject Classification: 65J15, 65G99, 47H15, 49D15.

1 Introduction

In this study we are concerned with the problem of approximating a locally unique solution $x^{*}$ of a nonlinear equation

F (x) + G (x) = 0

(1)

where $F, G$ are defined on a closed convex subset $D$ of a Banach space $E_{1}$ with values in a Banach space $E_{2} .$ The operator $F$ is Fréchet-differentiable on $D$ whereas $G$ is only continuous there.

We use the Newton-Like method given by

x_{n + 1} = x_{n} + A_{n}^{- 1} (F (x_{n}) + G (x_{n})) (n \geq 0)

(2)

to generate a sequence ${x_{n}} (n \geq 0)$ converging to $x^{*} .$ Here $A_{n}$ is a linear operator approximating $F^{'} (x_{n}) (n \geq 0) .$ Sufficient conditions for the convergence of (2) to $x^{*}$ have given by several authors ([1], [2], [3], [4], [5],[7], [8], [11] and [12]). Recently, Cătinaş in [5] has used (2) for

A_{n} = F^{'} (x_{n}) + [x_{n - 1}, x_{n}; G] (n \geq 1),

(3)

where $[x, y; G]$ denotes a divided difference of order one of $G$ on $D$ for $x, y \in E_{1} .$ This way Cătinaş has managed to show that the order of convergence denoted by $λ$ lies in $[\frac{1 + \sqrt{5}}{2}, 2] .$ Cătinaş has also showed that iteration (2) is faster than iterations appearing in [1],[2], [4], [5], [7], [8],[11] and [12] for choices of $A_{n}$ other than the one given by (3).

In [3] we used (2) for

A_{n} = [x_{n}, x_{n - 1}; F] + [x_{n - 2}, x_{n}; F] - [x_{n - 2}, x_{n - 1}; F] + [x_{n - 1}, x_{n}; G] (n \geq 0) .

(4)

We showed that $λ \in [\frac{1 + \sqrt{5}}{2}, 1.839 \dots] .$ Sufficient conditions were also provided under which our error bounds are sharper than all previous ones (in particular those in [5]). We also note that our method is cheaper to use than that in [5].

In this study we have made a further attempt to improve the rate of convergence of iteration (2) by choosing $A_{n}$ appropriately. Sufficient convergence conditions as well as an error analysis have been provided.

Finally we show that our error bounds are smaller than all earlier ones ([1], [2], [3], [4], [5], [7], [8], [9], [10], [11] and[12]).

2 Convergence analysis

We need the following definitons on divided differences [4], [9], [10].

Definition 1

An operator denoted by $[x_{0}, y_{0}; H]$ belonging to the space $L (D, E_{2}), D \subseteq E_{1}$ (the Banach space of bounded linear operators from $E_{1}$ to $E_{2}$ is called the first order divided difference of the operator $H : D \to E_{2}$ at the points $x_{0}, y_{0} \in D$ if the following hold:

(a)

[x_{0}, y_{0}; H] (y_{0} - x_{0}) = H (y_{0}) - H (x_{0}), f o r x_{0} \neq y_{0} .

(5)

(b)

If $H$ is Fréchet-differentiable at $x_{0} \in D,$ then $[x_{0}, x_{0}; H] = H^{'} (x_{0}) .$

Definition 2

An operator denoted by $[x_{0}, y_{0}, z_{0}; H]$ belonging to the space $L (D, L (D, E_{2}))$ is called the second order divided difference of the operator $H : D \subseteq E_{1} \to E_{2}$ at the points $x_{0}, y_{0}, z_{0} \in D$ if the following hold:

(a)

[x_{0}, y_{0}, z_{0}; H] (z_{0} - x_{0}) = [y_{0}, z_{0}; H] - [x_{0}, y_{0}; H] .

(6)

(b)

If $H$ is twice Fréchet-differentiable at $x_{0} \in D,$ then

[x_{0}, x_{0}, x_{0}; H] = \frac{1}{2} H^{''} (x_{0}) .

We can prove the following semilocal result concerning the convergence of iteration (2).

Theorem 3

Assume that there exist points $x_{0}, x_{1} \in D$ and nonnegative real numbers $R$ , $ε$ and $m$ such that:

(a)

$U (x_{1}, R) = {x \in E_{1} | ‖ x - x_{1} ‖ \leq R} \subseteq D;$

(b)

the operators $F, G$ have divided differences of order one denoted by $[x, y; F]$ and $[x, y; G]$ respectively for all $x, y \in U (x_{1}, R);$

(c)

the linear operators $A_{n}$ are invertible for all $n \geq 0$ and

‖ A_{n}^{- 1} B_{n} ‖ \leq p_{n} ‖ x_{n - 1} - x_{n - 2} ‖ + q ‖ x_{n} - x_{n - 1} ‖ = ε_{n} (n \geq 1),

for some nonnegative sequences ${p_{n}},$ ${q_{n}} (n \geq 1)$ with

p_{n} + q_{n} \leq ε (n \geq 1),

where

B_{n} = [x_{n - 1}, x_{n}; F] + [x_{n - 1}, x_{n}; G] - A_{n - 1} (n \geq 1),

(d)

the points $x_{0}, x_{1}$ satisfy $‖ x_{1} - x_{0} ‖ \leq m;$

(e)

the following conditions hold:

‖ x_{2} - x_{1} ‖ \leq ‖ x_{1} - x_{2} ‖,

(7)

with $x_{2}$ given by (2) for $n = 1,$

r = m ε < 1,

(8)

R \geq \frac{m}{r} \sum_{k = 1}^{\infty} r_{k},

(9)

r_{k} = r^{s_{k}}, (k \geq 0),

(10)

where ${s_{k}}$ is the Fibonacci’s sequence

s_{0} = s_{1} = 1, s_{k + 1} = s_{k} + s_{k - 1}, (k \geq 0) .

(11)

Then

(i)

the sequence ${x_{n}} (n \geq 0),$ generated by (2) is well defined, remains in $U (x_{0}, R)$ and converges to a solution $x^{*} \in U (x_{0}, R)$ of the equation $F (x) + G (x) = 0;$

(ii)

the following a priori error estimated hold

‖ x^{*} - x_{n} ‖ \leq \frac{m t_{n}}{r (1 - t_{n}^{ℓ - 1})}, (n \geq 1),

(12)

where

t_{n} = r^{\frac{ℓ}{\sqrt{5}}}, (n \geq 1) and ℓ = \frac{1 + \sqrt{5}}{2};

(13)

(iii)

moreover, if there exists a nonnegative number $g$ such that

‖ A_{n}^{- 1} B_{n}^{*} ‖ \leq g < 1, (n \geq 1),

(14)

where

B_{n}^{*} = [y^{*}, x_{n}; F] + [y^{*}, x_{n}; G] - A_{n}, (n \geq 1),

(15)

with $y^{*}$ satisfying

F (y^{*}) + G (y^{*}) = 0 and y^{*} \in U (x_{0}, R),

then

x^{*} = y^{*} .

Proof. We shall show by induction that, for all $n \geq 2$

x_{n} \in U (x_{1}, R),

(16)

‖ x_{n} - x_{n - 1} ‖ \leq ‖ x_{n - 1} - x_{n - 2} ‖

(17)

and

‖ x_{n} - x_{n - 1} ‖ \leq \frac{m}{r} r_{n - 1} .

(18)

For $n = 2$ relations (16)-(18) follow from hypothses (d) and (e). Suppose relations (16)-(18) hold for $n = 2, 3, \dots k,$ where $k \geq 2 .$ Since $x_{k}, x_{k - 1} \in U (x_{1}, R)$ and $A_{k}$ is invertible, via (2) we can compute $x_{k + 1} .$ Using (2), we can obtain the approximation

$F (x_{k}) + G (x_{k})$	$= F (x_{k}) + G (x_{k}) - F (x_{k - 1}) - G (x_{k - 1}) - A_{k - 1} (x_{k} - x_{k - 1}) =$	(19)
	$= ([x_{k - 1}, x_{k}; F] + [x_{k - 1}, x_{k}; G] - A_{k - 1}) (x_{k} - x_{k - 1}) =$
	$= B_{k} (x_{k} - x_{k - 1}) by (16))$

By (2), (7??), (8???) and (19) we get

‖ x_{k + 1} - x_{k} ‖ \leq ε_{k} ‖ x_{k} - x_{k - 1} ‖ \leq ε ‖ x_{k - 1} - x_{k - 2} ‖ \cdot ‖ x_{k} - x_{k - 1} ‖ .

(20)

From the induction hypothese, (20) give on the one hand, that

‖ x_{n + 1} - x_{k} ‖ \leq \frac{ε}{r} r_{k - 2} m ‖ x_{k} - x_{k - 1} ‖ = r_{k - 2} ‖ x_{k} - x_{k - 1} ‖ < ‖ x_{k} - x_{k - 1} ‖,

that is, (17) for $n = k + 1,$ and, on the other hand

‖ x_{k + 1} - x_{k} ‖ \leq r_{k - 2} ‖ x_{k} - x_{k - 1} ‖ \leq r_{k - 2} r_{k - 1} \frac{m}{r} = \frac{m}{r} r_{k},

which shows (18) for $n = k + 1$ .

We must also show that $x_{k + 1} \in U (x_{1}, R) .$ Indeed, from the induction hypotheses and the triangle inequality we get

‖ x_{k + 1} - x_{1} ‖ \leq ‖ x_{2} - x_{1} ‖ + ‖ x_{3} - x_{2} ‖ + \dots + ‖ x_{k + 1} - x_{k} ‖ \leq \frac{m}{r} \sum_{j = 1}^{k} r_{k} < R .

We must show that sequence ${x_{n}} (n \geq 0)$ is Cauchy. We have that the Fibonacci’s sequence ${s_{k}} (k \geq 0)$ given by (11) can also be written as

s_{k} = \frac{1}{\sqrt{5}} [{(\frac{1 + \sqrt{5}}{2})}^{k + 1} - (\frac{1 - \sqrt{5}}{2})] \geq \frac{1}{\sqrt{5}} (\frac{1 + \sqrt{5}}{2}) κ = \frac{ℓ_{k}}{\sqrt{5}} (k \geq 1) .

Therefore, for any $k \geq 1, j \geq 1$ we get

	$‖ x_{k + j} - x_{k} ‖$	$\leq ‖ x_{k + 1} - x_{k} ‖ + ‖ x_{k + 2} - x_{k + 1} ‖ + \dots + ‖ x_{k + j} - x_{k + j - 1} ‖ \leq$
		$\leq \frac{m}{r} \sum_{j = k}^{k + j - 1} r_{i} \leq \frac{m}{r} \sum_{i = k}^{k + j - 1} r^{\frac{ℓ^{l}}{\sqrt{5}}} .$

Moreover, by Bernoulli’s inequality we get

$‖ x_{k + j} - x_{k} ‖$	$\leq \frac{m}{r} r^{\frac{ℓ^{k}}{5}} [1 + r^{\frac{ℓ^{k + 1} - ℓ^{k}}{\sqrt{5}}} + r^{\frac{ℓ^{k + 2 - ℓ}}{5}} + \dots + r^{\frac{ℓ^{k + j - 1} - ℓ^{k}}{\sqrt{5}}}] \leq$	(21)
	$\leq \frac{m}{r} r^{\frac{ℓ_{k}}{\sqrt{5}}} [1 + r^{\frac{ℓ^{l} (ℓ - 1)}{\sqrt{5}}} + r^{\frac{ℓ_{k} [1 + 2 (ℓ - 1) - 1]}{\sqrt{5}}} + \dots + r^{\frac{[1 + (j - 1) (ℓ - 1) - 1]}{\sqrt{5}}}]$
	$= \frac{m}{r} r^{\frac{ℓ^{k}}{\sqrt{5}}} \frac{1 - r^{\frac{v^{k} ℓ - 1}{\sqrt{5}} j}}{1 - r^{\frac{ℓ^{l} (ℓ - 1)}{\sqrt{5}}}} (k \geq 1) .$

By (8)and (21) it follows that the sequence ${x_{n}} (n \geq 0)$ is Caucchy in a Banch space $E_{1},$ and so it converges to some point $x^{*} \in U (x_{1}, R)$ (since $U (x_{1}, R)$ is a closed set). By letting $n \to \infty$ in (2), we obtain $F (x^{*}) + G (x^{*}) = 0;$ that is, $x^{*} \in U (x_{1}, R)$ is a solution of equation (1). Moreover, by letting $j \to \infty$ in (21) we obtain (12).

Furhermore, to show that $x^{*}$ is the unique solution of equation (1) in $U (x_{1}, R),$ let us assume that $y^{*} \in U (x_{1}, R)$ is a solution of equation (1) too.

$x_{k + 1} - y^{*}$	$= x_{k} - y^{*} - X_{k}^{- 1} (F (x_{k}) + G (x_{k})) =$	(22)
	$= - A_{k}^{- 1} [F (x_{k}) - F (y^{}) + G (x_{k}) - G y^{} - A_{k} (x_{k} - y^{*})] =$
	$= - A_{k}^{- 1} B_{k}^{} (x_{k} - y^{}),$

and hypothesis (14), we get

	$‖ x_{k + 1} - y^{*} ‖$	$\leq ‖ A_{k}^{- 1} B_{k}^{} ‖ ‖ x_{k} - y^{} ‖ \leq$		(23)
		$\leq g ‖ x_{k} - y^{} ‖ \leq \dots \leq g^{k} ‖ x_{1} - y^{} ‖ \leq g^{k} R .$

Since $0 \leq g < 1,$ by letting $k \to \infty$ in (23) we get $lim_{k \to \infty} x_{k} = y^{*} .$ But we have also showed that $lim_{k \to \infty} x_{k} = x^{*} .$ Hence, we deduce $x^{*} = y^{*} .$

That completes the proof of the Theorem.

Remark 4

Let us consider some special choices for the linear operators $A_{n} .$ Set

A_{n} = [x_{n}, y_{n}; F] + [h_{n}, z_{n}; F] - [h_{n}, z_{n - 1}; F] + [v_{n}, x_{n}; G] (n \geq 0),

(24)

where the sequences ${y_{n}},$ ${z_{n}},$ ${v_{n}},$ ${h_{n}} \in U (x_{1}, R) (n \geq 0)$ are given by

	$y_{n}$	$= x_{n} + α_{n} (x_{n - 1} - x_{n}), z_{n} = z_{n - 1} β_{n} (x_{n - 1} - x_{n}), z_{- 1} = x_{- 1} \in U (x_{1}, R),$
	$v_{n}$	$= x_{n} + y_{n} (x_{n - 1} - x_{n}) (n \geq 0),$

for some linear operator sequence ${α_{n}},$ ${β_{n}}$ and ${γ_{n}} (n \geq 0)$ with $γ_{n} \neq 0$ $(n \geq 0) .$ Assume that there exist nonnegative numbers $a, b, c$ and a real sequence ${a_{n}}$ $(n \geq 0)$ such that for all $x, y, v, w, z \in U (x_{1}, R)$

‖ A_{0}^{- 1} ([x, y; F] - [v, w; F]) ‖ \leq v (‖ x - v ‖ + ‖ y - w ‖)

(25)

‖ A_{n}^{- 1} A_{0} ‖ \leq a_{n} \leq a (n \geq 0),

(26)

and

‖ A_{n}^{- 1} [x, y, z; G] ‖ \leq c,

(27)

where $[x, y, z; G]$ is the divided difference of order two of $G$ on $U (x_{1}, R) .$ Then form the approximation

	$G (x_{n}) - G (x_{n - 1}) - [v_{n - 1}, x_{n - 1}; G] (x_{n} - x_{n - 1}) =$
	$= ([x_{n - 1}; G] - [v_{n - 1}, x_{n - 1}; G]) (x_{n} - x_{n - 1}) =$
	$= [v_{n - 1}, x_{n - 1}, x_{n}; G] (x_{n} v_{n - 1}) (x_{n} - x_{n - 1}),$

hypotheses (25), (26) and (27) we get

	$‖ A_{0}^{- 1} (G (x_{n}) - G (x_{n - 1}) - [v_{n - 1}, x_{n - 1}; G] (x_{n} - x_{n - 1})) ‖ \leq$
	$\leq c ‖ x_{n} - v_{n - 1} ‖ ‖ x_{n} - x_{n - 1} ‖ \leq$
	$\leq c (‖ x_{n} - v_{n - 1} ‖ + ‖ γ_{n} ‖ ‖ x_{n - 1} - x_{n - 2} ‖) ‖ x_{n} - x_{n - 1} ‖ (n \geq 0) .$

The sequence ${a_{n}} (n \geq 1)$ can be computed as follows. Let us assume that there exist $\bar{c} \geq 0$ such that

‖ A_{0}^{- 1} ([x, y; G] - [v_{0}, x_{0}; G]) ‖ \leq \bar{c} (‖ x - v_{0} ‖ + ‖ y - x_{0} ‖),

(28)

for all $x, y, v_{0}, x_{0} \in U (x_{1}, R) .$ Then from the approximation

	$A_{0}^{- 1} (A_{n} - A_{0}) = A_{0}^{- 1} {[x_{n}, y_{n}; F] + [h_{n}, z_{n}; G] - [h_{n}, z_{n - 1}; G] +$
	$+ [v_{n}, x_{n}; G] - [x_{0}, y_{0}; F] - [h_{0}, z_{0}; F] + [h_{0}, z_{- 1}; F] - [v_{0}, x_{0}; G]},$

we can get as before

‖ A_{0}^{- 1} (A_{n} - A_{0}) ‖ \leq {\bar{a}}_{n} (n \geq 1),

where

	${\bar{a}}_{n}$	$= b (‖ x_{n} - x_{0} ‖ + ‖ y_{n} - y_{0} ‖ + 2 ‖ h_{n} - h_{0} ‖ + ‖ z_{n} - z_{0} ‖ + ‖ z_{n - 1} - z_{- 1} ‖) +$
		$+ \bar{c} (‖ v_{n} - x_{0} ‖ + ‖ x_{n} - x_{0} ‖) (n \geq 1),$

and ${\bar{a}}_{n} < 1$ if the function $a (r) = (11 b + 2 \bar{c}) r + (b + 2 \bar{c}) m$ satisfies

a (R) < 1,

(29)

since

{\bar{a}}_{n} \leq a (R) (n \geq 1) .

It follows from the Banach lemma on invertible operators that $A_{n}^{- 1}$ exists $(n \geq 1)$ and

‖ A_{n}^{- 1} A_{0} ‖ \leq {(1 - {\bar{a}}_{n})}^{- 1} .

We can now set $a_{n} = (1 - {\bar{a}}_{n}) (n \geq 1)$ and $a = {(1 - (R))}^{- 1} .$ Moreover, from the approximation

	$F (x_{n}) - F (x_{n - 1}) -$
	$- ([x_{n - 1}, y_{n - 1}; F] + [h_{n - 1}, z_{n - 1}; F] - [h_{n - 1}, z_{n - 1}; F]) (x_{n} - x_{n - 1})$
	$= ([x_{n - 1}, x_{n}; F] - [x_{n - 1}, y_{n - 1}; F] - [x_{n - 1}, z_{n - 1}; F]) + [x_{n - 1}, z_{n - 2}; F]) (x_{n} - x_{n - 1}),$

hypothese (25), (26), and (27) and (28) we also get

	$\| \| A_{0}^{- 1} {F (x_{n}) - F (x_{n - 1}) -$		(30)
	$- ([x_{n - 1}, y_{n - 1} : F] + [h_{n - 1}, z_{n - 1}; F] - [h_{n - 1}, z_{n - 2}; F]) (x_{n} - x_{n - 1})} \| \|$
	$\leq b (‖ x_{n} - y_{n - 1} ‖ + ‖ z_{n - 1} - z_{n - 2} ‖) ‖ x_{n} - x_{n - 1} ‖ \leq$
	$\leq b (‖ x_{n} - x_{n - 1} ‖ + ‖ α_{n - 1} + β_{n - 1} ‖ ‖ x_{n - 2} - x_{n - 1} ‖) ‖ x_{n} - x_{n - 1} ‖ (n \geq 1) .$

Define the sequences ${d_{n}},$ ${δ_{n}}$ $(n \geq 1)$ by

d_{n} = (b + c) a_{n} and δ_{n} = a_{n} (c | | γ_{n} | | + b (| | α_{n} | | + | | β_{n} | |)] (n \geq 0) .

(31)

‖ x_{n + 1} - x_{n} ‖ \leq (d_{n} ‖ x_{n} - x_{n - 1} ‖ + δ_{n} ‖ x_{n - 1} - x_{n - 2} ‖) ‖ x_{n} - x_{n - 1} ‖ (n \geq 1) .

(32)

Hence we can set

p_{n} = δ_{n} and d_{n} = q_{n} (n \geq 0) .

(33)

We can impose additional conditions on the sequences ${α_{n}},$ ${β_{n}}$ and ${γ_{n}}$ $(n \geq 0) t h a t$ will guarantee that ${y_{n}},$ ${z_{n}},$ ${v_{n}} \in U (x_{1}, R) .$ Let us assume that there exist nonnegative numbers $α,$ $β,$ and $γ$ such that

‖ α_{n} ‖ \leq a, ‖ β_{n} ‖ \leq β and ‖ γ_{n} ‖ \leq γ (n \geq 0) .

Then, from the approximations

	$y_{n} - x_{1}$	$= (x_{n} - x_{1}) + α_{n} (x_{n - 1} - x_{n})$
	$v_{n} - x_{1}$	$= (x_{n} - x_{1}) + γ_{n} (x_{n - 1} - x_{n})$
	$z_{n} - x_{1}$	$= (x_{n - 1} - x_{1}) + β_{n} (x_{n - 1} - x_{n}),$

we can have

‖ x_{n} - x_{1} ‖ + ‖ α_{n} ‖ ‖ x_{n - 1} - x_{n} ‖ \leq \frac{m}{r} \sum_{i = 1}^{n - 1} r_{i} + \frac{m α}{r} r_{n - 1},

(34)

‖ v_{n} - x_{1} ‖ + ‖ γ_{n} ‖ ‖ x_{n - 1} - x_{n} ‖ \leq \frac{m}{r} \sum_{i = 1}^{n - 1} r_{i} + \frac{m γ}{r} r_{n - 1},

(35)

and

‖ z_{n} - x_{1} ‖ \leq ‖ z_{- 1} - x_{1} ‖ + β \sum_{i = 0}^{n - 1} ‖ x_{i} - x_{i - 1} ‖ \leq ‖ z_{- 1} - x_{1} ‖ + \frac{β m}{r} \sum_{i = 0}^{n - 1} r_{i} .

(36)

Hence $y_{n}, v_{n}, z_{n} \in U (x_{1}, R) (n \geq 0)$ if the right hand sider of the last three inequalities are respectively bounded above by $R$ .

Finally, the uniqueness of the solution $x^{*}$ can be extended in the ball $U (x_{1}, R_{1})$ for $R_{1} \geq R$ provided that the following inequality holds

g = (5 (b + \bar{c}) R + (b + c) R_{1} + 2 m \bar{c}) {(1 - a (R))}^{- 1} < 1 .

(37)

Indeed, as in (22) we get

	$B_{n}^{*}$	$= ([y^{}, x_{n}; F] - [x_{n}, y_{n}; F]) + ([y^{}, x_{n}; G] - [x_{0}, v_{0}; G]) +$
		$+ ([x_{0}, v_{0}; G] - [v_{n}, x_{n}; G]) + ([h_{n}, z_{n - 1}; F] - [h_{n}, z_{n}; F]) .$

Composing both sides of the above approximation by $A_{0}^{- 1},$ we easily deduce that $A_{0}^{- 1} B_{k}^{*}$ (in norm) is bounded above by the expression in the bracket of inequality (37). Hence, as in the proff of the Theorem, we deduce $x^{*} = y^{*} .$

Concluding, we note that we have showed: if hypotheses (c) of the Theorem are replaced by (24), (25), (26), (27), (28) and (37), then the conclusions of the Theorem hold in the ball $U (x_{1}, R_{1}) .$

Remark 5

Iteration (2) reduces to (4) considered in [5] if the linear operators ${A_{n}} (n \geq 0)$ are given by (24) for $α_{n} = 0,$ $γ_{n} = I,$ $β_{n} = 0,$ $z_{n} = 0$ $(n \geq 0) .$

Using the notation introduced in [5], we can set

ε_{n}^{1} = M K ‖ x_{n - 1} - x_{n - 2} ‖ + M (\frac{ℓ}{2} + K) ‖ x_{n} - x_{n - 1} ‖ (n \geq 2) .

(38)

Hence our error bounds (12) will be smaller than those in [5] , say if (see also (33)).

p_{n} \leq M K and q_{n} \leq (M \frac{ℓ}{2} + K) (n \geq 2)

(39)

and our initial error bounds $‖ x_{1} - x_{0} ‖$ are not greater than those in [5]. The choice of $p_{n}, q_{n}$ given by (33) shows that conditions (39) will be true if $a_{n}, α_{n}, β_{n}$ and $γ_{n} (n \geq 0)$ are ”small” enough.

Remark 6

Moreover, iteration (2) reduces to (1) considered in [11] if the linear operators ${A_{n}} (n \geq 0)$ are given by (24) for $G = 0,$ $α_{n} = I,$ $β_{n} = I,$ $z_{n} = x_{n}$ and $h_{n} = x_{n - 2} (n \geq 0) .$ Using the notation introduced in [11] we can set

ε_{n}^{2} = q_{0} ‖ x_{n - 3} - x_{n - 1} ‖ ‖ x_{n - 2} - x_{n - 1} ‖ + p_{0} ‖ x_{n} - x_{n - 1} ‖ (n \geq 1) .

(40)

Hence our error bounds (12) will be smaller in this case, say if

p_{n} \leq ‖ x_{n - 3} - x_{n - 1} ‖ and q_{n} \leq p_{0} (n \geq 1) .

(41)

Observations similar to those made at the end of Remark 2 can now follows.

Remark 7

Furthermore, iteration (2) reduces to (5) considered in [3] if the linear operators ${A_{n}} (n \geq 0)$ are given by (24) for

α_{n} = I, β_{n} = 0, γ_{n} = I, z_{n} = x_{n}, h_{n} = x_{n - 2} (n \geq 0) .

Using the notation introduced in [3], we can set

ε_{n}^{3} = (c_{4} + c_{2} ‖ x_{n - 3} - x_{n - 1} ‖) ‖ x_{n - 1} - x_{n - 2} ‖ + (x_{1} + c_{4}) ‖ x_{n} - x_{n - 1} ‖ (n \geq 1) .

(42)

Hence our bounds will be smaller in this case say if

p_{n} \leq c_{4} + c_{2} ‖ x_{n - 3} - x_{n - 1} ‖ and q_{n} \leq c_{1} + c_{4} (n \geq 1) .

(43)

Remark 8

Our results extend to included perturbed Newton-like methods of the form

x_{n + 1} = x_{n} - A_{n}^{- 1} (F (x_{n}) + G (x_{n})) - w_{n} (n \geq 0) .

(44)

The points ${w_{n}} (n \geq 0)$ are determined in such a way that interation ${x_{n}}$ $(n \geq 0)$ converges to a solution $x^{*}$ of equation (1). The import;ance of studying perturbed Newton-like methods comes from the fact that many commonly used variants of Newton’s method can be considered procedures of this type. Indeed, approximation (44) characterizes any iterative process in which corrections are taken as approximate solutions of Newton equations. We also note that if, for example, an equation on the real is solved, $F (x_{n}) > 0 (n \geq 0),$ and $A (x_{n}) (n \geq 0)$ overestimates the derivative $x_{n} - A_{n}^{- 1} F (x_{n}),$ is always larger than the corresponding Newton iterate. In such cases, a positive $w_{n} (n \geq 0)$ correction term is appropriate. Let us assume that there exists a real sequence ${u} (n \geq 0)$ such that

‖ A_{n} (w_{n}) - A_{n} (w_{n - 1}) ‖ \leq u_{n} (n \geq 1) .

(45)

Moreover, there exist real sequences ${ℓ_{n}},$ ${m_{n}}$ $(n \geq 0)$ such that

u_{n} \leq (ℓ_{n} ‖ x_{n - 1} - x_{n - 2} ‖ + m_{n} ‖ x_{n} - x_{n - 1} ‖) ‖ x_{n} - x_{n - 1} ‖ (n \geq 1) .

(46)

Set

{\bar{p}}_{n} = p_{n} + ℓ_{n} and {\bar{q}}_{n} = q_{n} + m_{n} (n \geq 1) .

Furthermore, assume that sequence ${w_{n}} (n \geq 0)$ is null. Finally, assume that the rest of the hypotheses of the Theorem are true with ${\bar{p}}_{n}, {\bar{q}}_{n}$ replacing $p_{n},$ $q_{n}$ $(n \geq 1),$ respectively. Then it can easily be seen that the conclusions of the Theorem will hold for the perturbed Newton-Like method generated by (44). Indeed, for example, approximation (19) will read

F (x_{k}) + G (x) + A_{k} (w_{k}) = [B_{k} + (A_{k} (w_{k}) - A_{k - 1} (w_{k - 1}))] (x_{k} - x_{k - 1}) (n \geq 1)

and by using the proof of the theorem, (45) and (46) we can arrive at (20). The rest is left to the motivated reader.

Remark 9

The selection of the points ${y_{n}},$ ${z_{n}},$ ${v_{n}}$ $(n \geq 0)$ can be generalized to include a wider range of problems. Let $T_{1},$ $T_{2},$ $T_{3} : D \subseteq E_{1} \to E_{2}$ begiven operators. Define for all $n \geq 0$ $y_{n} = T_{1} (x_{n}),$ $z_{n} - z_{n - 1} = T_{2} (x_{n}),$ $z_{- 1} = x_{- 1} \in U (x_{1}, R) \subseteq D$ and $v_{n} = T_{3} (x_{n}) .$ For this choice of $T_{1}, T_{2}$ and $T_{3}$ iteration (2)becomes a Steffensen-like method ([8], [9] and [10]). Moreover, operators $T_{1},$ $T_{2}$ and $T_{3}$ must be chosen so that estimte (7??) be true. See how this is done, for example, in Remark 1.

References

[1] I. K. Argyros, On the solution of equations with nondifferentiable operators and the Ptak error estimates, BIT, 30 (1990), 752-754.
[2] I. K. Argyros, On the solution of nonlinear equations with a nodifferentiable term, Rev. Anal. Numér. Théorie Approximation 22, 2(1993), 125-135.
[3] I. K. Argyros, On some iterative methods for solving nonlinear equations with a nondifferentiable term of order between 1.618…and 1.839…(sumitted to this journal).
[4] I. K. Argyros, and F. Szidarovszky, The Theory and Application of Iteration Methods, C.R.C., Press, Inc., Boca Raton, Florida, 1993.
[5] E. Cătinaş, On some iterative method for solving nonlinear equations, Rev. Anal. Numér. Theorie Approximation 23, 1 (1994), 47-53.
[6] G. Goldner, and M. Balazs, Remarks on divided differences and method of chords, Rev. Anal. Numér Theorie Approximation 3, 1 (1974), 19-30.
[7] L. V. Kantorovich, The method of succesive approximation for functional equations, Acta. Math. 71 (1939), 63-97.
[8] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
[9] I. Păvăloiu, Sur une généralisation de la méthod de Steffensen, Rev. Anal. Numér. Theorie Approximation 21, 1 (1972), 59-65.
[10] I. Păvăloiu, A convergence theorem concerning the chord methods, Rev. Anal. Numér. Theorie Approximation 22, 1 (1993), 83-85.
[11] F. A. Potra, On an iterative algorithm of order 1.839…for solving nonlinear equations, Numer. Funct. Anal. Optimiz. 7, 1 (1984-1985), 75-106.
[12] T. Yamamoto and X. Chen, Convergence domains of certain iterative methods for solving nonlinear equations, Numer. Funct. Anal. Optimix. 10, 1 and 2 (1989), 37-48.

Received August 10, 1996 Ioannis K. Argyros

Cameron University

Department of Mathematics

Lawton, OK 73505, U.S.A.

Emil Cătinaş and Ion Păvăloiu

Institutul de Calcul ”Tiberiu Popoviciu”

Str. Republicii Nr.37

C.P. 68, 3400 Cluj-Napoca, Romania

1998

On the Misra-Prigogine-Courbage Theory of Irreversibility

October 1, 2018

Upon the convergence of subconvex sequences

July 1, 2025

Approximation properties of a generalization of Bleimann, Butzer and Hahn operators

March 2, 2023

Improving the rate of convergence of some Newton-like methods for the solution of nonlinear equations containing a nondifferentiable term

Abstract

Authors

Keywords

Cite this paper as:

PDF

About this paper

Journal

Publisher Name

Paper on the journal website

Print ISSN

Online ISSN

MR

ZBL

Google Scholar citations

Paper (preprint) in HTML form

Improving the rate of convergence of some Newton-Like methods for the solution of nonlinear equations containing A nondifferentiable term

1 Introduction

2 Convergence analysis

Definition 1

Definition 2

Theorem 3

Remark 4

Remark 5

Remark 6

Remark 7

Remark 8

Remark 9

References

Related Posts