Abstract
We generalize an approximation formula which in some particular cases has been studied by [J.F. Traub 1964] and \ [R.M.Humel and C.S. Secbeck 1949]. Denote by \(I_{x}\) the closed interval determined by the distinct points \(x,x_{0} \in \mathbb{R}\). Consider the nonlinear mapping \(f:I_{x}\rightarrow \mathbb{R}\), which has derivatives up to the order \(2n+1\) on \(I_{x}\), and deonte by \(G\) the set of functions $$G=\big\{g:g(t) =f(x_0) + (t-x_0) \sum \limits_{i=1}^{n} a_i f'(x_0 + b_i(t-x_0) , \ a_i, b_i \in \mathbb{R}, i=1,n, t\in I_x\big\}$$ From the set \(G\) we determine a function \(\bar{g}\) with the properties \(f^{(i)}(x_0) = \bar{g}^{(i)}(x_0)\). We determine the coefficients \(a_{i},b_{i},\ i=1,\ldots,n\) and we also evaluate the remainder \(f(t) -\bar{g}(t)\), \(t\in I_{x}\).
Authors
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
Scanned paper: on the journal website.
Latex version of the paper.
Cite this paper as:
I. Păvăloiu, On an approximation formula, Rev. Anal. Numér. Théor. Approx., 26 (1997) nos. 1-2, pp. 179-184.
About this paper
Publisher Name
Article on journal website
Print ISSN
1222-9024
Online ISSN
2457-8126
References
[1] C. I. Berezin and N. Jidkov, Metody vychisleny, Fizmatgiz, Moscow (1962).
[2] P. M. Humel and C. L. Seebeck Jr., A generalization of Taylor’s expansion, Amer. Math. Monthly 56 (1949), pp. 243-247.
[3] A. Lupas Calculul valorilor unor functii elementare, Gazeta Matematica (Ser. A) VII, l (1986), pp. 15-26.
[4] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englowood Cliffs, N.J., 1964.
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 an approximation formula
On an approximation formula
1991 Mathematics Subject Classification:
65D05, 65D10.1. Introduction
This Note contains some remarks concerning an approximation formula for functions, which is a generalization of some interpolation formulae given in [3] and [5]. In particular, we shall show that only one of the formulae of this type, mentioned in [5], has a maximal degree of exactness. Some particular cases of such formulae were also mentioned in [5, p.163].
Denote by the closed interval determined by two distinct points in . For a -times derivable function and , consider the class of functions given by
| (1) | ||||
Consider the following problem: Find a function such that
| (2) |
In [5] this problem was solved in some particular cases. We shall show that, for this problem has a unique solution and we shall give a representation for the remainder.
2. Determination of the approximating function
For we are looking for a function in verifying conditions (2) and having a maximal degree of approximation.
It is easily seen that conditions (2) lead to the following system, having the real numbers as unknowns:
| (3) |
Consider now a continuous function and let
| (4) |
be a quadrature formula, having as knots and as coefficients. Asking that for , (4) becomes the classical Gauss quadrature formula. On the other hand, the conditions for , lead again to the system (3), implying that must be the roots of the Legendre polynomial of degree i.e., the roots of the equation
| (5) |
The coefficients are given by the following formula
| (6) |
(see [2, p.261]).
Now, it is clear that the following theorem holds:
3. Determination of the remainder
In the conditions of Theorem 2.1, it follows that
| (8) |
where
| (9) |
and is a number contained in the open interval determined by and
4. Particular cases
- a)
-
b)
In this case, and
One also obtains the evaluation
where
Remark.
Approximation formula of the type considered in this Note could be useful for the approximate calculation of the values of some functions having rational functions as derivatives.
References
- [1]
- [2] C. I. Berezin and N. Jidkov, Metody vychisleny, Fizmatgiz, Moscow (1962).
- [3] P. M. Humel and C. L. Seebeck Jr., A generalization of Taylor’s expansion, Amer. Math. Monthly 56 (1949), 243–247.
- [4] A. Lupaş, Calculul valorilor unor funcţii elementare, Gazeta Matematică (Ser. A) VII, 1 (1968), 15–26.
- [5] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
Received March 15, 1996
Institutul de Calcul ”Tiberiu Popoviciu”
Str. G. Bilaşcu, nr.37
C.P. 68, O.P. 1, 3400 Cluj-Napoca
România