In order to approximate the solutions of nonlinear systems \[F(x)=0,\] with \(F:D\subseteq {\mathbb R}^n \rightarrow {\mathbb R}^n\), \(n\in {\mathbb N}\), we consider the method
\begin{align*} x_{k+1} & =x_{k}-A_{k}F(x_{k})\\A_{k+1} & =A_{k}\big(2I-F^{\prime}(x_{k+1}\big)A_{k}),\;k=0,1,…, \,A_{0}\in M_{n}({\Bbb R}), x_0 \in D,\end{align*} and we study its local convergence.
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
nonlinear systems of equations in R^n; Schulz method.
I. Păvăloiu, Local convergence of some Newton type methods for nonlinear systems, Rev. Anal. Numér. Théor. Approx., 33 (2004) 2, pp. 209-213. https://doi.org/10.33993/jnaat332-778
Cătinaş, E. and Păvăloiu, I., On approximating the eigenvalues and eigenvectors of linear continuous operators, Rev. Anal. Numér. Théor. Approx., 26, nos. 1-2, pp. 19-27, 1997, https://ictp.acad.ro/jnaat/journal/article/view/1997-vol26-nos1-2-art3
Diaconu, A., On the convergence of an iterative proceeding of Chebyshev type, Rev. Anal. Numér. Théor. Approx., 24, nos. 1-2, pp. 19-27, 1995, https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art9
Diaconu, A. and Păvăloiu, I., Sur quelques méthodes itératives pour la résolution des equations opérationelles, Rev. Anal. Numér. Théor. Approx., 1, no. 1, pp. 45-61, 1972, https://ictp.acad.ro/jnaat/journal/article/view/1972-vol1-art3
Lazăr, I., On a Newton type method, Rev. Anal. Numér. Théor. Approx., 23, no. 2, pp. 167-174, 1994, https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no2-art5
Păvăloiu, I., Introduction in the Theory of Approximating the Solutions of Equations, Ed. Dacia, Cluj-Napoca, 1976 (in Romanian).
Ulm, S., On the iterative method with simultaneous approximation of the inverse of the operator, Izv. Nauk. Estonskoi S.S.R., 16, no. 4, pp. 403-411, 1967.
Zehnder, J. E., A remark about Newton’s method, Comm. Pure Appl. Math., 37, pp. 361-366, 1974 https://doi.org/10.1002/cpa.3160270305.
/*======================================================================
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; }
Local convergence of some Newton-type methods for nonlinear systems∗
Local convergence of some Newton-type methods
for nonlinear systems∗
Ion Păvăloiu†
Dedicated to Professor Elena Popoviciu on the occasion
of her 80th birthday.
(Date: January 21, 2004)
Abstract.
In order to approximate the solutions of nonlinear systems
with ,
, we consider the method
|
|
|
|
|
|
|
|
and we study its local convergence.
Key words and phrases:
Nonlinear systems of equations.
1991 Mathematics Subject Classification:
65H10.
∗This work has been supported by the Romanian Academy under grant GAR 16/2004.
†“T. Popoviciu” Institute of Numerical Analysis, O.P
1, C.P. 68, Cluj-Napoca, and North University Baia Mare, Romania, e-mail: pavaloiu@ictp.acad.ro.
1. Introduction
Most of the problems regarding the methods of approximating the
solutions of nonlinear equations in abstract spaces lead, as is
well known, to solving of linear systems in ,
From here, the importance of the study of the
methods of approximating the solutions of linear and nonlinear
problems in
For the linear systems there have been studied several methods
which offer very good results for some relatively large classes of
problems. In the case of the nonlinear systems, the number of the
methods is limited and the possibility of their applications
differs from system to system.
One of the most used methods is the Newton method. The application of this
method to nonlinear systems requires at each step the approximate solution of
a linear system. It is well known that the matrix of this linear system is
given by the Jacobian of the nonlinear system.
The approximate computation of the elements of the matrix may introduce not
only truncation errors, but also rounding errors, and therefore the resulted
linear system at each iteration step is a perturbed one. Further errors may
appear from the algorithm used for solving the linear system.
A method which can overcome a part of the above mentioned difficulties seems
to be the one which at each iteration step uses an approximation for the
inverse of the Jacobian.
More precisely, let and the equation
We denote by the set of the square matrices of
order , having real elements.
Let and Consider the
sequences
and given by
| (2) |
|
|
|
|
|
|
|
|
where is the unity matrix.
The matrices approximate, as we shall see, the inverse of the Jacobian matrix.
In this note we show that locally, the -convergence order of
the sequence is the same as for the sequence
given by the Newton method, i.e., 2. This result completes those
obtained in [2]–[8].
2. Local convergence
The following results have maybe a smaller practical importance,
but they prove that, theoretically, the Newton method and method
(2) offer the same results regarding the local
convergence. Method (2) presents the advantage that
eliminates the algorithm errors mentioned above. However, the
truncation and the rounding errors cannot be avoided by this
algorithm. This algorithm may be easily programmed.
Regarding the function we make the following assumptions:
-
a)
system (1) has a solution ;
-
b)
-
c)
Lemma 1.
If verifies assumptions and there exists such
that:
-
i.
-
ii.
where and ,
then for any there exists and
Proof.
Let and consider the difference
| (4) |
|
|
|
It follows that
| (5) |
|
|
|
The finite increment formula [6]
leads to relation
|
|
|
from which, by (5) we get
|
|
|
It follows by the Banach lemma that
is invertible and
|
|
|
Further,
|
|
|
whence, by taking norms we obtain (3).
∎
Regarding the convergence of the iteration (2) we obtain the
following result.
Theorem 2.
If obeys the hypothesis of Lemma 1 and the initial approximations
and are taken such that
| (6) |
|
|
|
|
| (7) |
|
|
|
|
| (8) |
|
|
|
|
where and obey
| (9) |
|
|
|
|
| (10) |
|
|
|
|
where
Then for all , we have and
| (11) |
|
|
|
|
| (12) |
|
|
|
|
Proof.
From (8) and (6) it follows and therefore From the identity
|
|
|
taking into account (2) for we get
| (13) |
|
|
|
|
|
|
|
|
From the Taylor formula [6], it follows
| (14) |
|
|
|
|
| (15) |
|
|
|
|
By (14), (15), (3) and (13) it results
|
|
|
From the above relation, denoting by and taking into account (6), (7)
and (9),
|
|
|
Hence, by (8) it results that and, moreover,
(11) holds for
The second relation in (2) for implies that
| (16) |
|
|
|
The finite increment formula and the first
relation in (2) imply
|
|
|
If we take into account (15) and the above relation, denoting
we are lead to
| (17) |
|
|
|
Further, we have
|
|
|
Using the above relation, by (17) and (10) it results
|
|
|
|
|
|
|
|
|
|
|
|
i.e., (12) for
Assuming now that (11) and (12) are verified for
then, proceeding as above, one can show that these relations also
hold for
From (11), taking into account that it follows that Since we have admitted that is
continuous on by (12) it follows that
∎
Relations (11) and (12) show us that convergence order of
method (2) is at least 2, i.e., is not smaller than that of the
Newton method.
References
-
[1]
-
[2]
Cătinaş, E. and Păvăloiu, I.,
On approximating the eigenvalues and eigenvectors of linear
continuous operators, Rev. Anal. Numér. Théor. Approx.,
26, nos. 1–2, pp. 19–27, 1997.
-
[3]
Diaconu, A., On the convergence of an iterative
proceeding of Chebyshev type, Rev. Anal. Numér. Théor.
Approx., 24, nos. 1–2, pp. 19–27, 1995.
-
[4]
Diaconu, A. and Păvăloiu, I., Sur
quelques méthodes itératives pour la résolution des
equations opérationelles, Rev. Anal. Numér. Théor.
Approx., 1, no. 1, pp. 45–61, 1972.
-
[5]
Lazăr, I., On a Newton type method, Rev. Anal.
Numér. Théor. Approx., 23, no. 2, pp. 167–174,
1994.
-
[6]
Păvăloiu, I., Introduction in the Theory of
Approximating the Solutions of Equations, Ed. Dacia, Cluj-Napoca,
1976 (in Romanian).
-
[7]
Ulm, S., On the iterative method with simultaneous
approximation of the inverse of the operator, Izv. Nauk.
Estonskoi S.S.R., 16, no. 4, pp. 403–411, 1967.
-
[8]
Zehnder, J. E., A remark about Newton’s method,
Comm. Pure Appl. Math., 37, pp. 361–366, 1974.
-
[9]
