We consider the iterative methods for solving nonlinear equations in \(\mathbb{R}\) and the class of iterative methods obtained by inverse interpolation of Taylor type; we determine the iterative method having the greatest efficiency index. We show that this method is the Chebyshev method given by the second degree interpolatory polynomial for the inverse function.
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
I. Păvăloiu, On the efficiency of the computations for approximating the solutions of equations, Bul. Ştiinţ. Univ. Baia Mare, Ser. B Mat.-Inf., 14 (1998) no. 1, pp. 59-64.
/*======================================================================
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 efficiency of the computations for approximating the solutions of equations
Bul. Ştiinţ. Univ. Baia Mare. Ser. B
Matematică-Informatică, vol. XIV (1998), no. 1, 59–64
Dedicated to prof. Iulian Coroian on his 60th anniversary.
On the efficiency of the computations for approximating the solutions of equations
Ion Păvăloiu
1. Introduction
It is well known that the Lagrange-Hermite type inverse interpolation offers a
lot of iterative methods for solving equations. The preference for one method
or for another in obtaining a suitable approximation of a solution must take
into account, among other facts, not only the convergence order of the chosen
method, but also the amount of calculations needed at each iteration step.
While the convergence order may be determined exactly, the amount of
calculations from each iteration may be difficult to determine exactly.
The problem we are interested in this note consists in considering a certain
class of iterative methods and in determining the methods of that class which
have an optimal efficiency index.
Concerning the efficiency index, we shall adopt the definition given by A.M.
Ostrowski in [6]. As we shall see in the following, the
essential elements in defining the efficiency index of an iterative method are
the convergence order and the number of function evaluations needed at each
iteration step. As it is well known, the Lagrange-Hermite type interpolatory
methods use the values of the considered function, as well as the values of
its derivatives up to a certain order. The efficiency index introduced by
Ostrowski, as we shall see, seems to be well suited to our purposes.
We shall restrict here to considering the class of methods given by the
Chebyshev-type methods, which was also studied in [8], [9].
We shall show that the results obtained in [8] and [9] remain valid under more general hypotheses than those considered in
those papers. More exactly, we shall
admit here that the amount of
calculations needed at each iteration step is proportional with the number of
functions whose values are needed.
The factor proportionality is considered constant for a given function.
Let be an interval of the
real axis and consider the equation
where . We assume that equation (1) has a
unique solution For solving this equation we consider
the function and we suppose that is a fixed
point for We also assume that the sequence given by the iterations
| (2) |
|
|
|
converges. Obviously, if is continuous at , then
Concerning the function we shall also assume the following.
-
a)
the function is differentiable at
-
b)
for any we have that for
some where is the divided difference of on
the points and
Definition 1.
The sequence has the
convergence order , if there exists the limit
|
|
|
and
The following lemma can be easily proved.
Lemma 2.
If the function satisfies a) and b), then the necessary
and sufficient condition for the sequence
given by (2) to have the convergence order ,
is that there exists the limit
|
|
|
and
We shall denote in the following by the number of functions whose
values must be computed at each step in the iterations (2).
Taking into account lemma 2 and according to the efficiency index
proposed by Ostrowski, we shall admit the following definition.
Definition 3.
The number is called the efficiency
index of the method (2) if there exists the limit
|
|
|
and
2. Chebyshev-type iterations methods
Let be a natural number, and assume that the function obeys
-
the function is times differentiable on
-
the derivative satisfies
for any
In the above hypotheses it follows that the function admits an inverse
where and the solution
of verifies
Moreover, the function is times differentiable on each point from
the interior of and the -th derivative, where is
given by
| (5) |
|
|
|
where and the above sum extends on all nonnegative
integer solutions of the system
|
|
|
|
|
|
Let be given. The class of Chebyshev-type iterative methods is
given by
| (6) |
|
|
|
where
For different values of relation (6) generates a class of
iterative methods of Chebyshev type. We shall denote by (CM) this class.
3. The optimal efficiency index.
Taking into account remarks 3 and 4, by (3)
we get the following expression for the efficiency index of the class (CM):
Considering the function , it can be easily
verified that it attains its maximum value at the unique point It is
clear then that has the maximum
value at The following result holds.
Theorem 4.
Considering the class (CM), the method with the highest
efficiency index is the Chebyshev method (7).
References
-
[1]
Brent, R., Winograd, S. and Wolfe, Ph.,
Optimal iterative processes for root-finding, Numer. Math. 20
(5), (1973), 327–341.
-
[2]
Casulli, V. and Trigiante, D., Sui
procedimenti iterative compositi, Calcolo, XIII. IV (1976), pp. 403–420.
-
[3]
Coman, Gh.,
††margin:
available soon,
refresh and click here
Some practical approximation methods for
nonlinear equations, Anal. Numér. Théor.
Approx., 11 (1982) nos. 1–2, pp. 41–48.
-
[4]
Kacewitez, B., An integral-interpolation
iterative method for the solution of scalar equations, Numer. Math.
26(4), (1976), pp.355–365.
-
[5]
Kung, H. T. and Traub, J, F., Optimal order and
efficiency for iterations with two evaluations. SIAM. J. Numer. Anal.
13, 1, (1976) pp. 84–99.
-
[6]
Ostrowski, M.A., Solution of Equation and
Systems of Equations. Academic Press. New York and London (1960).
-
[7]
Păvăloiu, I.,
††margin:
available soon,
refresh and click here
The Solution of the
Equations by Interpolation (Romanian) Ed. Dacia Cluj-Napoca (1981).
-
[8]
††margin:
clickable
Păvăloiu, I.,
Optimal problems concerning
interpolation methods of solution of equations, Publications de L’Institut
Mathématique Beograd 52 (66) (1991), pp. 113–126.
-
[9]
††margin:
clickable
Păvăloiu, I.,
On computational complexity
in solving equations by Steffensen-type methods, Rev. Anal. Numér.
Théor. Approx. 24 (1995) nos. 1–2, pp. 215–219.
-
[10]
††margin:
clickable
Păvăloiu, I.,
On computational
complexity in solving equations by interpolation methods, Rev. Anal.
Numér. Théor. Approx. 24 (1995) nos. 1–2, pp. 201–213.
-
[11]
Traub, J.F. Iterative Methods for the Solving of
Equations, Prentice-Hall, New York (1964).
-
[12]
Traub, J.F., Theory of optimal algorithms,
Preprint . Department of Computer Science, Carnegie-Mellon University (1973)
pp. 1–22.
-
[13]
Traub, J.F. and Wozniakowski, H.,
Optimal radius of convergence of interpolatory iterations for operator
equations, Aequationes Mathematicae, 21 (1980), pp.159–172.
-
[14]
Turowicz, A.B., Sur les dérivées d’ordre
supérieur d’une fonction inverse, Ann. Polon. Math. 8 (1960), pp. 265–269.
”T. Popoviciu” Institute of Numerical Analysis
