Overlapped Newton-Cotes Quadratures

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paper is to build numerical integration quadratures for equidistant data which use function values outside interval of integration. </p><p>One of the main applications for such formulas is to improve accuracy of compound rules without increase in number of function evaluations. In particular we propose refined composite algorithm which deliver higher precision than Simpson鈥檚 3/8 using the same number of function values.</p><p>Reference formulas and their properties are included.</p></div><p>Simpson’s 3/8 composite rule is perhaps the most used method for numerical integration of equally-spaced data.<br /> It has fifth order accuracy and simple computational structure:</p><p><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-e1d0953d8e9f5d5a413e307ebdd5ecc2_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="O(h^5)" title="Rendered by QuickLaTeX.com" height="22" width="50" style="vertical-align: -5px;"/>:</p><p class="ql-center-displayed-equation" style="line-height: 58px;"><span class="ql-right-eqno"> (1) </span><span class="ql-left-eqno">   </span><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-5605e794ac90baac35e5ca0d71b174e4_l3.png" height="58" width="487" class="ql-img-displayed-equation quicklatex-auto-format" alt="\begin{equation*} \int_{x_0}^{x_{3m}} f(x) \mathrm{d}x \approx \frac{3\,h}{8}\sum_{k=1}^{m}\,\left[f_{3k-3}+3\,(f_{3k-2}+f_{3k-1})+f_{3k}\right] \end{equation*}" title="Rendered by QuickLaTeX.com"/></p><p>Composite scheme divides whole integration interval by non-overlapped <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-3c3b178e4ba68ec299d74f2fbc2480e3_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="m" title="Rendered by QuickLaTeX.com" height="9" width="17" style="vertical-align: 0px;"/> smaller intervals and uses simple Newton-Cotes rule for each of them.<br /> Although this technique gives good results for appropriately chosen <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-2c13c4834932d54785d8212b72d98a90_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="h" title="Rendered by QuickLaTeX.com" height="14" width="10" style="vertical-align: 0px;"/> it seems that composite rule doesn’t use full potential of numerous interior function samples. They could contribute more to the overall precision of the algorithm.</p><p>Our goal is to improve accuracy of Simpson’s 3/8 composite rule without increase of function evaluations.</p><p>With this purpose we have build special quadratures which use function values beyond integration interval. Lets denote total number of nodes in quadrature as <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-b7c8107204004a443faf479ae6af8525_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="N" title="Rendered by QuickLaTeX.com" height="14" width="18" style="vertical-align: 0px;"/> and outer function values as <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-c4082fadcf4ee605b3245bbcc4032661_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="L" title="Rendered by QuickLaTeX.com" height="14" width="13" style="vertical-align: 0px;"/>. Then we have</p><p class="ql-center-displayed-equation" style="line-height: 60px;"><span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-dcd86f943cae7466342db1621c5a7212_l3.png" height="60" width="240" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ \int_{x_L}^{x_{N-1-L}} f(x) \mathrm{d}x \approx \sum_{i=0}^{N-1}c_i\,f_i \]" title="Rendered by QuickLaTeX.com"/></p><p>Here are several quadratures of such type for <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-a60e2e8c91a0efd99a14df8b90880c1f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="L=\frac{N}{2}-2" title="Rendered by QuickLaTeX.com" height="25" width="92" style="vertical-align: -7px;"/>:</p><p class="ql-center-displayed-equation" style="line-height: 295px;"><span class="ql-right-eqno">   </span><span class="ql-left-eqno">   </span><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-70dcf8d55696a9bb399555b0d627d793_l3.png" height="295" width="730" class="ql-img-displayed-equation quicklatex-auto-format" alt="\[ \begin{tabular}{|r|c|l|c|} \hline $N$& Overlapped Newton-Cotes Quadratures & Error Term & $L_1$ Norm\\ \hline &&&\\ 6&$\displaystyle{\frac{3\,h}{160}\,\left( -f_{{0}}+23\,f_{{1}}+58\,f_{{2}}+58\,f_{{3}}+23\,f_{{4}}-f_{{5}} \right)}$&$\displaystyle{\frac{13\,{h}^{7}}{2240}\,f^{(6)}(\xi)}$&1.03 \\ &&&\\ 8&$\displaystyle{\frac{h}{4480}\, \left[ \begin{matrix} 13\,(f_{{0}}+f_{{7}})-149\,(f_{{1}}+f_{{6}})+\\ 2049\,(f_{{2}}+f_{{5}})+4807\,(f_{{3}}+f_{{4}}) \end{matrix} \right] }$&$\displaystyle{\frac{7\,{h}^{9}}{6400}\,f^{(8)}(\xi)}$&1.04\\ &&&\\ 10&$\displaystyle{\frac{h}{89600}\, \left[ \begin{matrix} -49\,(f_{{0}}+f_{{9}})+603\,(f_{{1}}+f_{{8}})\\ -3960\,(f_{{2}}+f_{{7}})+42352\,(f_{{3}}+f_{{6}})+\\ 95454\,(f_{{4}}+f_{{5}}) \end{matrix} \right] }$&$\displaystyle{\frac{443\,{h}^{11}}{1971200}\,f^{(10)}(\xi)}$&1.06\\ &&&\\ \hline \end{tabular} \]" title="Rendered by QuickLaTeX.com"/></p><p>Now, in composite scheme we can apply overlapped quadratures for approximating integral on <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-eb77c7281b6fa151fff656a0b1ad432d_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="k=2,\dots,m-1" title="Rendered by QuickLaTeX.com" height="18" width="142" style="vertical-align: -4px;"/> intervals instead of lower-order Simpson’s 3/8 rule.</p><p>Composite rules with enhanced precision without increase in function evaluations:</p><p><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-23db7bea5f1b400afe0a943a2c4d70fc_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="O(h^7)" title="Rendered by QuickLaTeX.com" height="22" width="50" style="vertical-align: -5px;"/>:<br /> <a name="id3064568056"></a></p><p class="ql-left-displayed-equation" style="line-height: 126px;"><span class="ql-right-eqno"> (2) </span><span class="ql-left-eqno">   </span><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-11e239bf144473f9c6e2d7080bb7f730_l3.png" height="126" width="717" class="ql-img-displayed-equation quicklatex-auto-format" alt=" \begin{align*} \int_{x_0}^{x_{3m}} f(x) \mathrm{d}x &\approx \frac{3\,h}{8}\sum_{k=1,\,m}\,\left[f_{3k-3}+3\,(f_{3k-2}+f_{3k-1})+f_{3k}\right]\\ &+\frac{3\,h}{160}\sum_{k=2}^{m-1}\,\left[-f_{3k-4}+23\,f_{3k-3}+58\,f_{3k-2}+58\,f_{3k-1}+23\,f_{3k}-f_{3k+1}\right] \end{align*} " title="Rendered by QuickLaTeX.com"/></p><p><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-a04321567d37ffffacaf7e34ec967bca_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="O(h^9)" title="Rendered by QuickLaTeX.com" height="22" width="50" style="vertical-align: -5px;"/>:<br /> <a name="id3142336634"></a></p><p class="ql-left-displayed-equation" style="line-height: 126px;"><span class="ql-right-eqno"> (3) </span><span class="ql-left-eqno">   </span><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-94c641248f6ed9860c21b4e9b65b2a4e_l3.png" height="126" width="606" class="ql-img-displayed-equation quicklatex-auto-format" alt=" \begin{align*} \int_{x_0}^{x_{3m}} f(x) \mathrm{d}x &\approx \frac{3\,h}{8}\sum_{k=1,\,m}\,\left[f_{3k-3}+3\,(f_{3k-2}+f_{3k-1})+f_{3k}\right]\\ &+\frac{h}{4480}\sum_{k=2}^{m-1}\,{ \left[ \begin{matrix} 13\,(f_{{3k-5}}+f_{{3k+2}})-149\,(f_{{3k-4}}+f_{{3k+1}})+\\ 2049\,(f_{{3k-3}}+f_{{3k}})+4807\,(f_{{3k-2}}+f_{{3k-1}}) \end{matrix} \right]} \end{align*} " title="Rendered by QuickLaTeX.com"/></p><p><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-30d6d6eb4e817ffdb2daaf499d9fcb0b_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="O(h^{11})" title="Rendered by QuickLaTeX.com" height="22" width="58" style="vertical-align: -5px;"/>:<br /> <a name="id1203249426"></a></p><p class="ql-left-displayed-equation" style="line-height: 139px;"><span class="ql-right-eqno"> (4) </span><span class="ql-left-eqno">   </span><img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-77ad45025db6aaa59af1d947734a8d76_l3.png" height="139" width="655" class="ql-img-displayed-equation quicklatex-auto-format" alt=" \begin{align*} \int_{x_0}^{x_{3m}} f(x) \mathrm{d}x &\approx \frac{3\,h}{8}\sum_{k=1,\,m}\,\left[f_{3k-3}+3\,(f_{3k-2}+f_{3k-1})+f_{3k}\right]\\ &+\frac{h}{89600}\sum_{k=2}^{m-1}\,{ \left[ \begin{matrix} -49\,(f_{{3k-6}}+f_{{3k+3}})+603\,(f_{{3k-5}}+f_{{3k+2}})\\ -3960\,(f_{{3k-4}}+f_{{3k+1}})+42352\,(f_{{3k-3}}+f_{{3k}})+\\ 95454\,(f_{{3k-2}}+f_{{3k-1}}) \end{matrix} \right]} \end{align*} " title="Rendered by QuickLaTeX.com"/></p><p>Notice that we use extra nodes from neighboring intervals for estimation on every step <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-eb77c7281b6fa151fff656a0b1ad432d_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="k=2,\dots,m-1" title="Rendered by QuickLaTeX.com" height="18" width="142" style="vertical-align: -4px;"/>. This allows us to improve precision on interior intervals (first and last intervals still have <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-e1d0953d8e9f5d5a413e307ebdd5ecc2_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="O(h^5)" title="Rendered by QuickLaTeX.com" height="22" width="50" style="vertical-align: -5px;"/>). Most importantly, <strong>proposed high precision composite rules use the same number of nodes as Simpson’s 3/8 does</strong>. They can be plugged in place of Simpson’s 3/8 rules seamlessly in any application.</p><p>Special attention should be payed to the stability of the proposed formulas. As you see overlapped quadratures have negative coefficients and <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-a1f94881358a4082af4473376e47c04f_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="L_1" title="Rendered by QuickLaTeX.com" height="17" width="21" style="vertical-align: -3px;"/> norm is bigger than 1.0 as a result. This usually indicates possible instability of the quadratures in some cases. However their magnitude responses show that they are stable:</p><p><img class="aligncenter" src="http://www.holoborodko.com/pavel/downloads/overlapped_newton_cotes.png" alt="" /></p><p>Noisy high frequencies (usual indicators of instability) are suppressed, there are no other abnormalities. Besides we have tested proposed & Simpson’s 3/8 composite rules on 100+ test functions for integration from [1]. <strong>Overlapped quadratures gave more accurate result in 75% of cases without special tuning of the step size <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-2c13c4834932d54785d8212b72d98a90_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="h" title="Rendered by QuickLaTeX.com" height="14" width="10" style="vertical-align: 0px;"/></strong>.</p><p>This combined with the fact that actually norm is only slightly higher than 1.0 allows us to recommend refined composite schemes for applications.</p><h3>References</h3><p>I have used many excellent sources, but only one for the moment:<br /> [1] H. Engels, Numerical Quadrature and Cubature, Academic Press, London, 1980.</p><div> <br> <span style="float: left;"><div id="post-ratings-2822" class="post-ratings" itemscope itemtype="http://schema.org/Article" data-nonce="8be4995285"><img id="rating_2822_1" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="1 Star" title="1 Star" onmouseover="current_rating(2822, 1, '1 Star');" onmouseout="ratings_off(5, 0, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_2822_2" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="2 Stars" title="2 Stars" onmouseover="current_rating(2822, 2, '2 Stars');" onmouseout="ratings_off(5, 0, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_2822_3" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="3 Stars" title="3 Stars" onmouseover="current_rating(2822, 3, '3 Stars');" onmouseout="ratings_off(5, 0, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_2822_4" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="4 Stars" title="4 Stars" onmouseover="current_rating(2822, 4, '4 Stars');" onmouseout="ratings_off(5, 0, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /><img id="rating_2822_5" src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/stars_crystal/rating_on.gif" alt="5 Stars" title="5 Stars" onmouseover="current_rating(2822, 5, '5 Stars');" onmouseout="ratings_off(5, 0, 0);" onclick="rate_post();" onkeypress="rate_post();" style="cursor: pointer; border: 0px;" /> (<strong>4</strong> votes, average: <strong>5.00</strong>)<br /><span class="post-ratings-text" id="ratings_2822_text"></span><meta itemprop="name" content="Overlapped Newton-Cotes Quadratures" /><meta itemprop="headline" content="Overlapped Newton-Cotes Quadratures" /><meta itemprop="description" content="The goal of this paper is to build numerical integration quadratures for equidistant data which use function values outside interval of integration.  One of the main applications for such formulas ..." /><meta itemprop="datePublished" content="2011-03-28T20:09:39+09:00" /><meta itemprop="dateModified" content="2011-05-25T17:58:35+09:00" /><meta itemprop="url" content="http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/" /><meta itemprop="author" content="Pavel Holoborodko" /><meta itemprop="mainEntityOfPage" content="http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/" /><div style="display: none;" itemprop="publisher" itemscope itemtype="https://schema.org/Organization"><meta itemprop="name" content="Pavel Holoborodko" /><meta itemprop="url" content="http://www.holoborodko.com/pavel" /><div itemprop="logo" itemscope itemtype="https://schema.org/ImageObject"><meta itemprop="url" content="" /></div></div><div style="display: none;" itemprop="aggregateRating" itemscope itemtype="http://schema.org/AggregateRating"><meta itemprop="bestRating" content="5" /><meta itemprop="worstRating" content="1" /><meta itemprop="ratingValue" content="5" /><meta itemprop="ratingCount" content="4" /></div></div><div id="post-ratings-2822-loading" class="post-ratings-loading"><img src="http://www.holoborodko.com/pavel/wp-content/plugins/wp-postratings/images/loading.gif" width="16" height="16" class="post-ratings-image" />Loading...</div> </span> <span style="float: right;"> </span></div></div></div><div id="comments"><div id="comments-list" class="comments"><h3><span>7</span> Comments</h3><ol><li id="comment-5887" class="comment c c-y2011 c-m05 c-d26 c-h02 alt depth-1"><div class="comment-author vcard"> <span class="fn n">renato</span></div><div class="comment-meta">Posted May 25, 2011 at 5:51 pm <span class="meta-sep">|</span> <a href="#comment-5887" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Pavel, thank you for replying! As you suggested, I’ll try overlapped formulae also. Just a question : in (2)-(4) is the first summation running for k=0,m or perhaps for k=1,m?<br /> If k=0,m, wouldn’t it require 3 additional evaluations (f(-3),f(-2),f(-1)) ?</p><p>thank you,</p><p>renato</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=5887#respond' data-commentid="5887" data-postid="2822" data-belowelement="comment-5887" data-respondelement="respond" aria-label='Reply to renato'>Reply</a></div><ul class="children"><li id="comment-5888" class="comment c1 byuser comment-author-admin bypostauthor c-y2011 c-m05 c-d26 c-h03 depth-2"><div class="comment-author vcard"> <span class="fn n"><a href="http://www.holoborodko.com/pavel/" rel='external nofollow ugc' class='url url'>Pavel Holoborodko</a></span></div><div class="comment-meta">Posted May 25, 2011 at 6:00 pm <span class="meta-sep">|</span> <a href="#comment-5888" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Yep, this is mistake – already fixed. Thank you.</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=5888#respond' data-commentid="5888" data-postid="2822" data-belowelement="comment-5888" data-respondelement="respond" aria-label='Reply to Pavel Holoborodko'>Reply</a></div></li></ul></li><li id="comment-9070" class="comment c2 c-y2013 c-m04 c-d10 c-h05 alt depth-1"><div class="comment-author vcard"> <span class="fn n">Bernard</span></div><div class="comment-meta">Posted April 9, 2013 at 8:25 pm <span class="meta-sep">|</span> <a href="#comment-9070" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Hi Pavel,</p><p>I implemented overlapped Newton Cotes formula(3),but after extensive testing only for 1000 points I’m getting satisfactionary results.I suppose either wrong initialization of the accumulated sum or wrong indices of for loop contributed to this.<br /> Here is my code:</p><p>double QuadratureComposite8Points(double(*func)(double),double a,double b,int n)<br /> {<br /> if(func == NULL || n <= 0)<br /> return 0.0;</p><p> double term,x1,x2,sum,h,k;<br /> int i,ii;<br /> h = (b-a)/(3*n-1);</p><p> sum = FUNC(a) + 3.0*(FUNC(a+h) + FUNC(a+2*h)) + FUNC(a+3*h);<br /> //sum = FUNC(-a) + FUNC(b);<br /> printf("Sum = %.17f\n",sum);<br /> for(i = 1;i < n;i++){<br /> x1 = a+3*h*i;<br /> sum += FUNC(x1) +3.0*(FUNC(x1+h) + FUNC(x1+2*h)) + FUNC(x1+3*h);<br /> }</p><p> term = 3*h/8;</p><p> term*sum;<br /> //sum += 13.0*(FUNC(a) + FUNC(a+7*h)) – 149.0*(FUNC(a+h) + FUNC(a+6*h)) + 2049.0*(FUNC(a+2*h) + FUNC(a+5*h)) + 4807.0*(FUNC(a+3*h) + FUNC(a+4*h));<br /> for(ii = 2;ii< n-1;ii++){<br /> x2 = a + 3*h*ii;<br /> sum += 13.0*(FUNC(x2) + FUNC(x2+7*h)) – 149.0*(FUNC(x2+h) + FUNC(x2+6*h)) + 2049.0*(FUNC(x2+2*h) + FUNC(x2+5*h)) + 4807.0*(FUNC(x2+3*h) + FUNC(x2+4*h));</p><p> }</p><p> return (h*sum)/4480;</p><p> }</p><p>Thanks in advance:)</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=9070#respond' data-commentid="9070" data-postid="2822" data-belowelement="comment-9070" data-respondelement="respond" aria-label='Reply to Bernard'>Reply</a></div><ul class="children"><li id="comment-9081" class="comment c3 byuser comment-author-admin bypostauthor c-y2013 c-m04 c-d11 c-h01 depth-2"><div class="comment-author vcard"> <span class="fn n"><a href="http://www.holoborodko.com/pavel/" rel='external nofollow ugc' class='url url'>Pavel Holoborodko</a></span></div><div class="comment-meta">Posted April 10, 2013 at 4:40 pm <span class="meta-sep">|</span> <a href="#comment-9081" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Hi Bernard,</p><p>You don’t need the first loop. By <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-9990fd3baee77f894b94fd0cad9cfa37_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="k=1,m" title="Rendered by QuickLaTeX.com" height="14" width="58" style="vertical-align: -3px;"/> I meant that <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-c35e91af1d29f0b68304443cc1f8d8b2_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="k" title="Rendered by QuickLaTeX.com" height="11" width="8" style="vertical-align: 0px;"/> has only two values <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-c39c6b16cd3cd1d6da54ea6b96cc6731_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="1" title="Rendered by QuickLaTeX.com" height="11" width="6" style="vertical-align: 0px;"/> and <img src="http://www.holoborodko.com/pavel/wp-content/ql-cache/quicklatex.com-7f2f5581cb4809daecd239028d7fe538_l3.png" class="ql-img-inline-formula quicklatex-auto-format" alt="m" title="Rendered by QuickLaTeX.com" height="7" width="14" style="vertical-align: 0px;"/>, not a range.<br /> So you don’t need first loop.</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=9081#respond' data-commentid="9081" data-postid="2822" data-belowelement="comment-9081" data-respondelement="respond" aria-label='Reply to Pavel Holoborodko'>Reply</a></div></li></ul></li><li id="comment-9086" class="comment c4 c-y2013 c-m04 c-d11 c-h03 alt depth-1"><div class="comment-author vcard"> <span class="fn n">Bernard</span></div><div class="comment-meta">Posted April 10, 2013 at 6:28 pm <span class="meta-sep">|</span> <a href="#comment-9086" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Hi Pavel,</p><p>I really appreciate your help thank you.<br /> Bernard</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=9086#respond' data-commentid="9086" data-postid="2822" data-belowelement="comment-9086" data-respondelement="respond" aria-label='Reply to Bernard'>Reply</a></div></li><li id="comment-9087" class="comment c5 c-y2013 c-m04 c-d11 c-h03 depth-1"><div class="comment-author vcard"> <span class="fn n">Bernard</span></div><div class="comment-meta">Posted April 10, 2013 at 6:34 pm <span class="meta-sep">|</span> <a href="#comment-9087" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Pavel ,</p><p>so proper implementation should initialize the sum variable to this FUNC(a) + 3.0*(FUNC(a+h) + FUNC(a+2*h)) + FUNC(a+3*h); and calculate only the second loop which should start from k = 2 to m-1 ?</p><p>P.S<br /> I was confused by the sigma in the 3/8 composite rule.</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=9087#respond' data-commentid="9087" data-postid="2822" data-belowelement="comment-9087" data-respondelement="respond" aria-label='Reply to Bernard'>Reply</a></div></li><li id="comment-9090" class="comment c6 c-y2013 c-m04 c-d11 c-h06 alt depth-1"><div class="comment-author vcard"> <span class="fn n">Bernard</span></div><div class="comment-meta">Posted April 10, 2013 at 9:25 pm <span class="meta-sep">|</span> <a href="#comment-9090" title="Permalink to this comment"><strong>#</strong></a></div><div class="comment-content"><p>Hi Pavel,</p><p>Can you post the results obtained by you when you tested Composite Rule (4) on probability density function.Unfortunately my results differ by large margin when compared to composite simpson rule.Is this possible to post also your code implementation.</p><p>Thanks in advance<br /> Bernard</p></div><div class="comment-reply-link"><a rel='nofollow' class='comment-reply-link' href='http://www.holoborodko.com/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/?replytocom=9090#respond' data-commentid="9090" data-postid="2822" data-belowelement="comment-9090" data-respondelement="respond" aria-label='Reply to Bernard'>Reply</a></div></li></ol><div id="comments-nav-below" class="comment-navigation"><div class="paginated-comments-links"></div></div></div><div id="respond"><h3>Post a Comment</h3><div id="cancel-comment-reply"><a rel="nofollow" id="cancel-comment-reply-link" href="/pavel/numerical-methods/numerical-integration/overlapped-newton-cotes-quadratures/#respond" style="display:none;">Click here to cancel reply.</a></div><div class="formcontainer"><form id="commentform" action="http://www.holoborodko.com/pavel/wp-comments-post.php" method="post"><p 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Overlapped Newton-Cotes Quadratures