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<p><a id="X826D8334845549EC" name="X826D8334845549EC"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X826D8334845549EC">3 <span class="Heading">Polynomials</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7BDA2E7C85ECC18C">3.1 <span class="Heading">The Floats pseudo-field</span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AB8BC957952B662">3.1-1 FLOAT_PSEUDOFIELD</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X788CDC24834012D7">3.2 <span class="Heading">Roots of polynomials</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C9EF7E27EFA3288">3.3 <span class="Heading">Finding integer relations</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85DEB4B584870F67">3.3-1 PSLQ</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X83445BFB7901B88F">3.4 <span class="Heading">LLL lattice reduction</span></a>
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<h3>3 <span class="Heading">Polynomials</span></h3>

<p><a id="X7BDA2E7C85ECC18C" name="X7BDA2E7C85ECC18C"></a></p>

<h4>3.1 <span class="Heading">The Floats pseudo-field</span></h4>

<p>Polynomials with floating-point coefficients may be manipulated in <strong class="pkg">GAP</strong>; though they behave, in subtle ways, quite differently than polynomials over rings. A "pseudo-field" of floating-point numbers is available to create them using the standard <strong class="pkg">GAP</strong> syntax.</p>

<p><a id="X7AB8BC957952B662" name="X7AB8BC957952B662"></a></p>

<h5>3.1-1 FLOAT_PSEUDOFIELD</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FLOAT_PSEUDOFIELD</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>The "pseudo-field" of floating-point numbers, containing all floating-point numbers in the current implementation.</p>

<p>Note that it is not really a field, e.g. because addition of floating-point numbers is not associative. It is mainly used to create indeterminates, as in the following example:</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">x := Indeterminate(FLOAT_PSEUDOFIELD,"x");</span>
x
<span class="GAPprompt">gap></span> <span class="GAPinput">2*x^2+3;</span>
2.0*x^2+3.0
<span class="GAPprompt">gap></span> <span class="GAPinput">Value(last,10);</span>
203.0
</pre></div>

<p><a id="X788CDC24834012D7" name="X788CDC24834012D7"></a></p>

<h4>3.2 <span class="Heading">Roots of polynomials</span></h4>

<p>The Jenkins-Traub algorithm has been implemented, in arbitrary precision for MPFR and MPC.</p>

<p>Furthermore, CXSC can provide complex enclosures for the roots of a complex polynomial.</p>

<p><a id="X7C9EF7E27EFA3288" name="X7C9EF7E27EFA3288"></a></p>

<h4>3.3 <span class="Heading">Finding integer relations</span></h4>

<p>The PSLQ algorithm has been implemented by Steve A. Linton, as an external contribution to <strong class="pkg">Float</strong>. This algorithm receives as input a vector of floats <span class="SimpleMath">\(x\)</span> and a required precision <span class="SimpleMath">\(\epsilon\)</span>, and seeks an integer vector <span class="SimpleMath">\(v\)</span> such that <span class="SimpleMath">\(|x\cdot v|<\epsilon\)</span>. The implementation follows quite closely the original article <a href="chapBib_mj.html#biBMR1836930">[BB01]</a>.</p>

<p><a id="X85DEB4B584870F67" name="X85DEB4B584870F67"></a></p>

<h5>3.3-1 PSLQ</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSLQ</code>( <var class="Arg">x</var>, <var class="Arg">epsilon</var>[, <var class="Arg">gamma</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PSLQ_MP</code>( <var class="Arg">x</var>, <var class="Arg">epsilon</var>[, <var class="Arg">gamma</var>[, <var class="Arg">beta</var>]] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: An integer vector <span class="SimpleMath">\(v\)</span> with <span class="SimpleMath">\(|x\cdot v|<\epsilon\)</span>.</p>

<p>The PSLQ algorithm by Bailey and Broadhurst (see <a href="chapBib_mj.html#biBMR1836930">[BB01]</a>) searches for an integer relation between the entries in <span class="SimpleMath">\(x\)</span>.</p>

<p><span class="SimpleMath">\(\beta\)</span> and <span class="SimpleMath">\(\gamma\)</span> are algorithm tuning parameters, and default to <span class="SimpleMath">\(4/10\)</span> and <span class="SimpleMath">\(2/\sqrt(3)\)</span> respectively.</p>

<p>The second form implements the "Multi-pair" variant of the algorithm, which is better suited to parallelization.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">PSLQ([1.0,(1+Sqrt(5.0))/2],1.e-2);
</span>
[ 55, -34 ] # Fibonacci numbers
<span class="GAPprompt">gap></span> <span class="GAPinput">RootsFloat([1,-4,2]*1.0);
</span>
[ 0.292893, 1.70711 ] # roots of 2x^2-4x+1
<span class="GAPprompt">gap></span> <span class="GAPinput">PSLQ(List([0..2],i->last[1]^i),1.e-7);
</span>
[ 1, -4, 2 ] # a degree-2 polynomial fitting well
</pre></div>

<p><a id="X83445BFB7901B88F" name="X83445BFB7901B88F"></a></p>

<h4>3.4 <span class="Heading">LLL lattice reduction</span></h4>

<p>A faster implementation of the LLL lattice reduction algorithm has also been implemented. It is accessible via the commands <code class="code">FPLLLReducedBasis(m)</code> and <code class="code">FPLLLShortestVector(m)</code>.</p>


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