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<p>Import words via API</p> <p><b>New page</b></p><div>==== valsi ====<br /> cnanlagau<br /> ==== type ====<br /> fu&#039;ivla<br /> ==== creator ====<br /> [[personal/krtisfranks|krtisfranks]]<br /> ==== time entered ====<br /> Sun Apr 1 01:47:34 2018<br />  <br /> == English ==<br /> === Definition #70581 - Preferred ===<br />  <br /> ==== definition ====<br /> x&lt;sub&gt;1&lt;/sub&gt; is the generalized arithmetic-geometric mean of the elements of the 2-element set x&lt;sub&gt;2&lt;/sub&gt; (set; cardinality must be 2) of order x&lt;sub&gt;3&lt;/sub&gt; (either single extended-real number xor an unordered pair/2-element set of extended-real numbers).<br /> ==== notes ====<br /> <br /> Elements of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; must have the same units/dimensionality; the result has the same units/dimensionality as them. If &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt; = &lt;i&gt;p&lt;/i&gt;&lt;/span&gt; for a single extended-real number &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;p&lt;/i&gt;&lt;/span&gt;, then MATH<br /> $x_3=(p,p-1)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt; = (&lt;i&gt;p&lt;/i&gt;, &lt;i&gt;p&lt;/i&gt; - 1)&lt;/span&gt; also. If MATH<br /> $x_3 = (p,q)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt; = (&lt;i&gt;p&lt;/i&gt;, &lt;i&gt;q&lt;/i&gt;)&lt;/span&gt; then the algorithm uses the &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;p&lt;/i&gt;&lt;/span&gt;th-power mean ([[cnanfadi|cnanfadi]]) and the &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;q&lt;/i&gt;&lt;/span&gt;th-power mean; thus MATH<br /> $x_3 = (1,0)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt; = (1, 0)&lt;/span&gt; corresponds to the standard arithmetic-geometric mean, MATH<br /> $x_3 = (0, -1)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt; = (0, - 1)&lt;/span&gt; corresponds to the geometric-harmonic mean. A poor choice of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt;&lt;/span&gt; will lead to non-convergence of the sequences produced by the algorithm and, thus, leave &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; undefined (NAN error). &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; is the value to which each of the sequences produced by the algorithm converge (iff these values are mutually equal). Let &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;M&lt;/i&gt;&lt;sub&gt;i&lt;/sub&gt;&lt;/span&gt; denote the unweighted &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;i&lt;/i&gt;&lt;/span&gt;th-power mean for all &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;i&lt;/i&gt;&lt;/span&gt; and let MATH<br /> $x_2 = (a_0,g_0),$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt; = (&lt;i&gt;a&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;, &lt;i&gt;g&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;),&lt;/span&gt; MATH<br /> $x_3 = (p,q)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt; = (&lt;i&gt;p&lt;/i&gt;, &lt;i&gt;q&lt;/i&gt;)&lt;/span&gt;; then the algorithm produces potentially infinite sequences MATH<br /> $a = (a_0, a_1, ...), g = (g_0, g_1, ...)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;a&lt;/i&gt; = (&lt;i&gt;a&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;, &lt;i&gt;a&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;,...), &lt;i&gt;g&lt;/i&gt; = (&lt;i&gt;g&lt;/i&gt;&lt;sub&gt;0&lt;/sub&gt;, &lt;i&gt;g&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;,...)&lt;/span&gt;, where MATH<br /> $a_n = M_p(a_&lt;a class=&quot;undefined&quot; href=&quot;../dict/(n-1)?bg=1;langidarg=2&quot;&gt;(n-1)&lt;/a&gt;, g_&lt;a class=&quot;undefined&quot; href=&quot;../dict/(n-1)?bg=1;langidarg=2&quot;&gt;(n-1)&lt;/a&gt;), g_n = M_q(a_&lt;a class=&quot;undefined&quot; href=&quot;../dict/(n-1)?bg=1;langidarg=2&quot;&gt;(n-1)&lt;/a&gt;, g_&lt;a class=&quot;undefined&quot; href=&quot;../dict/(n-1)?bg=1;langidarg=2&quot;&gt;(n-1)&lt;/a&gt;)$<br /> <br /> &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;a&lt;/i&gt;&lt;sub&gt;n&lt;/sub&gt; = &lt;i&gt;M&lt;/i&gt;&lt;sub&gt;p&lt;/sub&gt;(&lt;i&gt;a&lt;/i&gt;&lt;sub&gt;(n-1)&lt;/sub&gt;, &lt;i&gt;g&lt;/i&gt;&lt;sub&gt;(n-1)&lt;/sub&gt;), &lt;i&gt;g&lt;/i&gt;&lt;sub&gt;n&lt;/sub&gt; = &lt;i&gt;M&lt;/i&gt;&lt;sub&gt;q&lt;/sub&gt;(&lt;i&gt;a&lt;/i&gt;&lt;sub&gt;(n-1)&lt;/sub&gt;, &lt;i&gt;g&lt;/i&gt;&lt;sub&gt;(n-1)&lt;/sub&gt;)&lt;/span&gt; for all positive integers &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;n&lt;/i&gt;&lt;/span&gt;. Notice that this word is symmetric (&lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;1&lt;/sub&gt;&lt;/span&gt; remains constant and none of the sumti change in meaning) under an internal permutation of the elements/entries of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;2&lt;/sub&gt;&lt;/span&gt; and/or of &lt;span class=&quot;MATH&quot;&gt;&lt;i&gt;x&lt;/i&gt;&lt;sub&gt;3&lt;/sub&gt;&lt;/span&gt; (when taken as a pair or set), separately. See also: [[gau&#039;i&#039;o|gau&#039;i&#039;o]].<br /> <br /> ==== gloss words ====<br /> <br /> &lt;ul&gt;&lt;li&gt;[[natlang/en/AGM|AGM ; arithmetic-geometric mean]]&lt;/li&gt;&lt;li&gt;[[natlang/en/arithmetic-geometric mean|arithmetic-geometric mean]]&lt;/li&gt;&lt;/ul&gt;<br /> <br /> ==== created by ====<br /> [[personal/krtisfranks|krtisfranks]]<br /> ==== vote information ====<br /> 1<br /> <br /> ==== time ====<br /> Sun Apr 1 02:55:45 2018<br /> <br /> <br /> <br /> &lt;br/&gt;&lt;font size=&quot;+1&quot;&gt;Examples&lt;/font&gt;<br /> &lt;hr/&gt;<br /> &lt;dl&gt;<br /> &lt;/dl&gt;<br /> <br /> === Etymology ===</div>

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