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Previous: <a href="maxima_186.html#ggf_002dpkg" accesskey="p" rel="previous">ggf-pkg</a>, Up: <a href="maxima_186.html#ggf_002dpkg" accesskey="u" rel="up">ggf-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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<a name="Functions-and-Variables-for-ggf-1"></a>
<h3 class="section">49.1 Functions and Variables for ggf</h3>
<a name="Item_003a-ggf_002fdefvr_002fGGFINFINITY"></a><dl>
<dt><a name="index-GGFINFINITY"></a>Option variable: <strong>GGFINFINITY</strong></dt>
<dd><p>Default value: 3
</p>
<p>This is an option variable for function <code>ggf</code>.
</p>
<p>When computing the continued fraction of the
generating function, a partial quotient having a degree
(strictly) greater than <var>GGFINFINITY</var> will be discarded and
the current convergent will be considered as the exact value
of the generating function; most often the degree of all
partial quotients will be 0 or 1; if you use a greater value,
then you should give enough terms in order to make the
computation accurate enough.
</p>
<p>See also <code><a href="#ggf">ggf</a></code>.
</p>
</dd></dl>
<a name="Item_003a-ggf_002fdefvr_002fGGFCFMAX"></a><dl>
<dt><a name="index-GGFCFMAX"></a>Option variable: <strong>GGFCFMAX</strong></dt>
<dd><p>Default value: 3
</p>
<p>This is an option variable for function <code>ggf</code>.
</p>
<p>When computing the continued fraction of the
generating function, if no good result has been found (see
the <var>GGFINFINITY</var> flag) after having computed <var>GGFCFMAX</var> partial
quotients, the generating function will be considered as
not being a fraction of two polynomials and the function will
exit. Put freely a greater value for more complicated
generating functions.
</p>
<p>See also <code><a href="#ggf">ggf</a></code>.
</p>
</dd></dl>
<a name="ggf"></a><a name="Item_003a-ggf_002fdeffn_002fggf"></a><dl>
<dt><a name="index-ggf"></a>Function: <strong>ggf</strong> <em>(<var>l</var>)</em></dt>
<dd><p>Compute the generating function (if it is a fraction of two
polynomials) of a sequence, its first terms being given. <var>l</var>
is a list of numbers.
</p>
<p>The solution is returned as a fraction of two polynomials.
If no solution has been found, it returns with <code>done</code>.
</p>
<p>This function is controlled by global variables <var>GGFINFINITY</var> and <var>GGFCFMAX</var>. See also <var>GGFINFINITY</var> and <var>GGFCFMAX</var>.
</p>
<p>To use this function write first <code>load("ggf")</code>.
</p>
<div class="example">
<pre class="example">(%i1) load("ggf")$
(%i2) makelist(fib(n),n,0,10);
(%o2) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
(%i3) ggf(%);
x
(%o3) - ----------
2
x + x - 1
(%i4) taylor(%,x,0,10);
2 3 4 5 6 7 8 9 10
(%o4)/T/ x + x + 2 x + 3 x + 5 x + 8 x + 13 x + 21 x + 34 x + 55 x
+ . . .
(%i5) makelist(2*fib(n+1)-fib(n),n,0,10);
(%o5) [2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123]
(%i6) ggf(%);
x - 2
(%o6) ----------
2
x + x - 1
(%i7) taylor(%,x,0,10);
2 3 4 5 6 7 8 9
(%o7)/T/ 2 + x + 3 x + 4 x + 7 x + 11 x + 18 x + 29 x + 47 x + 76 x
10
+ 123 x + . . .
</pre></div>
<p>As these examples show, the generating function does create a function
whose Taylor series has coefficients that are the elements of the
original list.
</p>
</dd></dl>
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