MaplePrimes - comments on Post, One more algorithm to find all roots of one-dimensional function

<rss xmlns:itunes="http://www.itunes.com/dtds/podcast-1.0.dtd" version="2.0">
  <channel>
    <title>MaplePrimes - comments on Post, One more algorithm to find all roots of one-dimensional function</title>
    <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots</link>
    <language>en-us</language>
    <copyright>2026 Maplesoft, A Division of Waterloo Maple Inc.</copyright>
    <generator>Maplesoft Document System</generator>
    <lastBuildDate>Fri, 12 Jun 2026 11:44:15 GMT</lastBuildDate>
    <pubDate>Fri, 12 Jun 2026 11:44:15 GMT</pubDate>
    <itunes:subtitle />
    <itunes:summary />
    <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
    <image>
      <url>http://www.mapleprimes.com/images/mapleprimeswhite.jpg</url>
      <title>MaplePrimes - comments on Post, One more algorithm to find all roots of one-dimensional function</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots</link>
    </image>
    <item>
      <title>Test</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots?ref=Feed:MaplePrimes:One more algorithm to find all roots of one-dimensional function:Comments#comment130887</link>
      <itunes:summary>&lt;p&gt;Can you kindly test that with exp(-1/x^2)*sin(1/x) instead of sin(1/x)?&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
      <guid>130887</guid>
      <pubDate>Sat, 18 Feb 2012 22:46:08 Z</pubDate>
      <itunes:author>Markiyan Hirnyk</itunes:author>
      <author>Markiyan Hirnyk</author>
    </item>
    <item>
      <title>Thank you</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots?ref=Feed:MaplePrimes:One more algorithm to find all roots of one-dimensional function:Comments#comment130892</link>
      <itunes:summary>&lt;p&gt;Thank you for question. Finally i'm ready to answer.&lt;/p&gt;
&lt;p&gt;&lt;a href="/view.aspx?sf=130892/430896/1dim_roots.zip"&gt;1dim_roots.zip&lt;/a&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
      <guid>130892</guid>
      <pubDate>Sun, 19 Feb 2012 05:32:07 Z</pubDate>
      <itunes:author>icegood</itunes:author>
      <author>icegood</author>
    </item>
    <item>
      <title>Comparison</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots?ref=Feed:MaplePrimes:One more algorithm to find all roots of one-dimensional function:Comments#comment130897</link>
      <itunes:summary>&lt;p&gt;Compare with the output of&lt;br&gt;&lt;br&gt;&amp;gt; with(RootFinding):&lt;br&gt;&amp;gt; st := time():Analytic(exp(-1/x^2)*sin(1/x), x, re = 0.1e-1 .. 1.0, im = -0.1e-1 .. 0.1e-1, digits = 20); time()-st;&lt;br&gt;&lt;br&gt;0.31830988618379067154, 0.15915494309189533577, 0.079577471545947667880,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.10610329539459689051, 0.063661977236758134305, 0.053051647697298445255,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.045472840883398667363, 0.039788735772973833942, 0.035367765131532296838,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.031830988618379067154, 0.028937262380344606504, 0.026525823848649222628,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.024485375860291590119, 0.022736420441699333681, 0.021220659078919378103,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.019894367886486916971, 0.018724110951987686561, 0.017683882565766148419,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.016753151904410035344, 0.015915494309189533578, 0.015157613627799555788,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.014468631190172303259, 0.013839560268860463980, 0.013262911924324611314,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.012732395447351626862, 0.012242687930145795059, 0.011789255043844098946,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.011368210220849666841, 0.010976202971854850743, 0.010610329539459689051,&lt;br&gt;&lt;br&gt;&amp;nbsp; 0.010268060844638408759&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; 466.203&lt;br&gt;&lt;br&gt;&lt;br&gt;on my ancient comp. It would be interesting to compare the algorithms too.&lt;br&gt;&lt;br&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
      <guid>130897</guid>
      <pubDate>Sun, 19 Feb 2012 11:52:40 Z</pubDate>
      <itunes:author>Markiyan Hirnyk</itunes:author>
      <author>Markiyan Hirnyk</author>
    </item>
    <item>
      <title>Last command</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots?ref=Feed:MaplePrimes:One more algorithm to find all roots of one-dimensional function:Comments#comment130969</link>
      <itunes:summary>&lt;p&gt;&lt;a href="http://www.mapleprimes.com/posts/130877-One-More-Algorythm-To-Find-All-Roots#comment130897"&gt;@Markiyan Hirnyk&lt;/a&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;which version of maple do you use? After waiting &amp;gt;30 min on my machine kernel crashes because all &amp;gt;1.5Gb available free of memory was wasted.&lt;/p&gt;
&lt;p&gt;While my one still can be better. After taking x_(n+1)=x_n,2 i.e.&lt;/p&gt;
&lt;p&gt;&lt;img src="data:image/png;base64,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" alt=""&gt;&lt;/p&gt;
&lt;p&gt;and with majorant&lt;/p&gt;
&lt;p&gt;proc(u::algebraic,v::algebraic) &lt;br&gt;&amp;nbsp; local x:=1/u, y:=1/v, mx:=floor(2*x/Pi), my:=floor(2*y/Pi),dd;&lt;br&gt;&amp;nbsp; #&lt;br&gt;&amp;nbsp; if (mx=my) then&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; if (mx mod 2=0) then&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; exp(-y^2)*(x^2*abs(cos(y))+2*x^3*abs(sin(x)));&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; else&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; exp(-y^2)*(x^2*abs(cos(x))+2*x^3*abs(sin(y)));&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; end if;&lt;br&gt;&amp;nbsp; elif (x-y&amp;nbsp;&amp;nbsp;&amp;nbsp; dd:=evalf(sin(max(abs(x-mx*Pi/2),abs(y-mx*Pi/2))));&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; if (mx mod 2=0) then&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; exp(-y^2)*(x^2+2*x^3*dd);&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; else&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp;&amp;nbsp; exp(-y^2)*(x^2*dd+2*x^3);&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; end if;&lt;br&gt;&amp;nbsp; else&lt;br&gt;&amp;nbsp;&amp;nbsp;&amp;nbsp; exp(-y^2)*(x^2+2*x^3);&lt;br&gt;&amp;nbsp; end if;&lt;br&gt;end proc&lt;br&gt;&lt;br&gt;&lt;/p&gt;
&lt;p&gt;i found all roots in ~40 sec. Unlike to odd ones i cannot say that choose of x_(n+1)=x_n,k with even k is legally but it works in this example. While with such majorant both approach to roots (from left) and escaping from them (from right) has near to quadratical convergency/divergency.&lt;/p&gt;
&lt;p&gt;Ideally if limit(M(u,v), v-&amp;gt;u)=|F'(u)| - then convergence will be really quadratical.&lt;/p&gt;
&lt;p&gt;&lt;a href="/view.aspx?sf=130969/431018/1dim_roots_best_fitt.mw"&gt;1dim_roots_best_fitt.mw&lt;/a&gt;&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
      <guid>130969</guid>
      <pubDate>Tue, 21 Feb 2012 01:08:20 Z</pubDate>
      <itunes:author>icegood</itunes:author>
      <author>icegood</author>
    </item>
    <item>
      <title>With Maple 13</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots?ref=Feed:MaplePrimes:One more algorithm to find all roots of one-dimensional function:Comments#comment130973</link>
      <itunes:summary>&lt;p&gt;Using Maple 13, I have just repeated it (see&amp;nbsp; &lt;a href="/view.aspx?sf=130973/431024/analytic.mw"&gt;analytic.mw&lt;/a&gt;&amp;nbsp; ) in 47M, 478.312 s .&lt;/p&gt;
&lt;p&gt;PS. The vote up is mine.&lt;/p&gt;
&lt;p&gt;PPS. I have submitted&amp;nbsp; SCR concerning it.&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
      <guid>130973</guid>
      <pubDate>Tue, 21 Feb 2012 02:00:11 Z</pubDate>
      <itunes:author>Markiyan Hirnyk</itunes:author>
      <author>Markiyan Hirnyk</author>
    </item>
    <item>
      <title>Damn</title>
      <link>http://www.mapleprimes.com/posts/130877-One-More-Algorithm-To-Find-All-Roots?ref=Feed:MaplePrimes:One more algorithm to find all roots of one-dimensional function:Comments#comment130975</link>
      <itunes:summary>&lt;p&gt;&lt;a href="http://www.mapleprimes.com/posts/130877-One-More-Algorythm-To-Find-All-Roots#comment130973"&gt;@Markiyan Hirnyk&lt;/a&gt;&amp;nbsp;&lt;/p&gt;
&lt;p&gt;Software regression detected :( I thought so, that vote is yours. But still want have rather ideal solution. Was really surprised, that can use x_n,2. It's essential speedup not only in this example but in half of general case too (in other half x_n,1 should be taken). Need counterexamples or proof. Still learning &lt;a href="http://en.wikipedia.org/wiki/Taylor%27s_theorem"&gt;Taylor&lt;/a&gt; theorem with G(x) term. That M is monotonous by 2nd parameter. It could be essential.&lt;/p&gt;</itunes:summary>
      <description>The latest comments added to the Post, One more algorithm to find all roots of one-dimensional function</description>
      <guid>130975</guid>
      <pubDate>Tue, 21 Feb 2012 02:47:16 Z</pubDate>
      <itunes:author>icegood</itunes:author>
      <author>icegood</author>
    </item>
  </channel>
</rss>