Abstract
The Chebyshev method for approximating the solutions of polynomial operator equations of degree 2 is presented. The convergence of the Chebyshev method is studied.
Author
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
polynomial equation of degree 2; Chebyshev method.
Cite this paper as:
I. Păvăloiu, On the Chebyshev method for approximating the solutions of polynomial operator equations of degree 2, Bul. Ştiinţ. Univ. Baia Mare, 16 (2000) no. 2, pp. 219-224.
About this paper
Journal
Publisher Name
Universitatea Baia Mare
JSTOR permalink
Print ISSN
Not available yet.
Online ISSN
Not available yet.
References
?
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; }
On the Chebyshev Method for Approximating the Solutions of Polynomial Operator Equations of Degree 2
Bul. Ştiinţ. Univ, Baia Mare, Ser. B,
Matematică-Informatică, Vol. XVI (2000) Nr. 2, 219–224
Dedicated to Maria S. Pop on her 60th anniversary
On the Chebyshev Method for Approximating the Solutions of Polynomial Operator Equations of Degree 2
Abstract.
In this paper, the Chebyshev method for approximating the solutions of polynomial operator equations of degree 2 is presented. The convergence of the Chebyshev method is studied.
MSC 2000: 47H60.
Keywords: polynomial operator, Chebyshev method.
1. Introduction
The polynomial operator equations represent an important class of operator equations [1]. Among them, the polynomial operator equations of degree 2 have a special importance because the convergence hypotheses for the usual methods (Newton method, chord method, Steffensen method, Chebyshev method, etc.) are much simplified compared to the general case [6]. In this note we shall study the convergence of the Chebyshev method for the mentioned equations.
Let be a Banach space and consider a mapping . We remind that the mapping is a polynomial operator of degree two if
-
a)
is three times differentiable;
-
b)
where is the trilinear null operator.
2. The convergence of the Chebyshev method
Consider the equation
| (2.1) |
where is the zero element. Given an initial approximation of a solution of the above equation, the Chebyshev method generates the sequence by
where .
Consider and denote . Since , it is clear that does not depend on , so we may take
From the Taylor formula we obtain
| (2.2) | ||||
| (2.3) |
By (2.3) it follows
which implies
| (2.4) |
Similarly (2.2), leads to
| (2.5) |
We shall make the following notations:
| (2.6) |
| (2.7) |
| (2.8) |
where
| (2.9) |
With the above notations, the following result holds:
Theorem 1.
If for some and the mapping satisfies
Proof.
First we shall show that hypothesis implies the existence of the application for all and, moreover, . Indeed, one has
Applying the Banach Lemma and taking into account relation it follows the existence of for all and, moreover,
| (2.10) |
Denote by the mapping given by
| (2.11) |
where .
It can be easily seen that for all , the following identity holds:
whence
| (2.12) |
Since from the Taylor formula we get
and by (2.2),
| (2.13) |
provided that . Since one obtains
i.e.
Suppose now that the following relations hold:
It is easy to show that
| (2.16) |
and, since , it follows that the sequence is Cauchy, so it converges. Denoting , it is clear that . Letting in (2.16) leads us to . ∎
The Chebyshev method may be applied with the aid of the following algorithm:
Let be an arbitrary approximation of the solution of (2.1), and which satisfies the hypotheses of Theorem 1. The next approximation may be obtained by
1. Solve the linear operator equation
2. Solve the linear operator equation
3. Compute
References
- [1] Argyros, I.K., Polynomial Operator Equations in Abstract Spaces and Applications, CRC Press, Boca Raton, Boston (1998).
- [2] Ciarlet, P.G., Introduction à l’Analyse Numérique Matricielle et à l’Optimisation, Mason, Paris (1990).
- [3] Chatelin, F., Valeur Propres de Matrices, Mason, Paris (1998).
- [4] Collatz, L., Functionalanalysis und Numerische Mathematick, Springer-Verlag, Berlin (1964).
- [5] Kartîşov, V.S., Iuhno, F.L., O nekotoryh k Modifikatsiah Method Niutona dlea Resenia Nelineinoi Spektralnoi Zadaci, J. Vîcisl. matem. i mamem. fiz. (33) 9, (1973), 1403-1409.
- [6] ††margin: clickable Păvăloiu, I. Sur les procédées itératifs à un ordre élevé de convergence, Mathematica, 12 (35) 2 (1970), 309–324.
Received 18.05.2000
”T. Popoviciu” Institute of Numerical Analysis
Str. Gh. Bilaşcu nr.37
C.P. 68, O.P. 1
3400 Cluj-Napoca