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<head><title>track(List,List,List) -- track a user homotopy</title>
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<div><h1>track(List,List,List) -- track a user homotopy</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>solsT = track(S,T,solsS)</tt></div>
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<li><span>Function: <a href="_track_lp__List_cm__List_cm__List_rp.html" title="track a user homotopy">track</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>S</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> of polynomials in the start system</span></li>
<li><span><tt>T</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> of polynomials in the target system</span></li>
<li><span><tt>solsS</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, <a href="../../Macaulay2Doc/html/___List.html" title="the class of all lists -- {...}">List</a> of start solutions</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, <tt>solsT</tt>, solutions of <tt>T=0</tt> obtained by continuing <tt>solsS</tt></span></li>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>gamma => </tt><span><span>default value 1</span>, a parameter in the homotopy: H(t)=(1-t)^tDegree S + gamma t^tDegree T</span></span></li>
<li><span><tt>tDegree => </tt><span><span>default value 1</span>, a parameter in the homotopy: H(t)=(1-t)^tDegree S + gamma t^tDegree T</span></span></li>
<li><span><tt>tStep => </tt><span><span>default value .05</span>, initial step size</span></span></li>
<li><span><tt>tStepMin => </tt><span><span>default value .000001</span>, minimal step size</span></span></li>
<li><span><tt>stepIncreaseFactor => </tt><span><span>default value 2</span>, determine how step size is adjusted</span></span></li>
<li><span><tt>numberSuccessesBeforeIncrease => </tt><span><span>default value 4</span>, determine how step size is adjusted</span></span></li>
<li><span><tt>Predictor => </tt><span><span>default value RungeKutta4</span>, a method to predict the next point on the homotopy path: choose between <a href="_track_lp__List_cm__List_cm__List_rp.html" title="track a user homotopy">RungeKutta4</a>, <a href="_track_lp__List_cm__List_cm__List_rp.html" title="track a user homotopy">Tangent</a>, <a href="_track_lp__List_cm__List_cm__List_rp.html" title="track a user homotopy">Euler</a>, <a href="_track_lp__List_cm__List_cm__List_rp.html" title="track a user homotopy">Secant</a>, <a href="_track_lp__List_cm__List_cm__List_rp.html" title="track a user homotopy">ProjectiveNewton</a>. The latter provides certified tracking.</span></span></li>
<li><span><tt>maxCorrSteps => </tt><span><span>default value 3</span>, max number of steps corrector takes before a failure is declared</span></span></li>
<li><span><tt>CorrectorTolerance => </tt><span><span>default value .000001</span>, corrector succeeds if the relative error does not esceed this tolerance</span></span></li>
<li><span><tt>EndZoneFactor => </tt><span><span>default value .05</span>, size of `end zone'</span></span></li>
<li><span><tt>InfinityThreshold => </tt><span><span>default value 100000</span>, paths are truncated if the norm of the approximation exceeds the threshold</span></span></li>
<li><span><tt>Projectivize => </tt><span><span>default value false</span>, if true then the system is homogenized and projective tracker is executed</span></span></li>
<li><span><tt>Normalize => </tt><span><span>default value false</span>, normalize the start and target systems w.r.t. Bombieri-Weyl norm</span></span></li>
<li><span><tt>NoOutput => </tt><span><span>default value false</span>, if true, no output is produced (useful in combination with <a href="_get__Solution_lp__Z__Z_rp.html" title="get various attributes of the specified solution">getSolution</a>)</span></span></li>
<li><span><a href="_track_lp__List_cm__List_cm__List_rp.html">AffinePatches => ...</a>, -- track a user homotopy</span></li>
<li><span><a href="_track_lp__List_cm__List_cm__List_rp.html">MultistepDegree => ...</a>, -- track a user homotopy</span></li>
<li><span><a href="_track_lp__List_cm__List_cm__List_rp.html">SLP => ...</a>, -- track a user homotopy</span></li>
<li><span><a href="_track_lp__List_cm__List_cm__List_rp.html">SLPcorrector => ...</a>, -- track a user homotopy</span></li>
<li><span><a href="_track_lp__List_cm__List_cm__List_rp.html">SLPpredictor => ...</a>, -- track a user homotopy</span></li>
<li><span><a href="_solve__System_lp..._cm_sp__Software_sp_eq_gt_sp..._rp.html">Software => ...</a>, -- specify software for the solver</span></li>
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</li>
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<div class="single"><h2>Description</h2>
<div>Polynomial homotopy continuation techniques are used to obtain solutions of the target system given a start system.<br/><table class="examples"><tr><td><pre>i1 : R = CC[x,y];</pre>
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<tr><td><pre>i2 : S = {x^2-1,y^2-1};</pre>
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<tr><td><pre>i3 : T = {x^2+y^2-1, x*y};</pre>
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<tr><td><pre>i4 : solsS = {(1,-1),(1,1),(-1,1),(-1,-1)};</pre>
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<tr><td><pre>i5 : track(S,T,solsS) / first
o5 = {{-3.8337e-16, -1}, {1, 1.17962e-15}, {3.8337e-16, 1}, {-1,
------------------------------------------------------------------------
-1.17962e-15}}
o5 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div>Predictor=>ProjectiveNewton works only with Software=>M2 and Normalize=>true</div>
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