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<head><title>goodInitialPair -- a good (conjectured by Shub and Smale) initial pair</title>
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<div><h1>goodInitialPair -- a good (conjectured by Shub and Smale) initial pair</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>(S,sol) = goodInitialPair T</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>T</tt>, a list of polynomials</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>S</tt>, list of polynomials</span></li>
<li><span><tt>sol</tt>, a list containing (one) solution of S</span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>GeneralPosition => </tt><span><span>default value false</span>, make a random unitary change of coordinates</span></span></li>
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<div class="single"><h2>Description</h2>
<div>Generates a start system S that is conjectured to have good complexity when used in linear homotopy with target system T leading to one solution. <table class="examples"><tr><td><pre>i1 : T = randomSd {2,3};</pre>
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<tr><td><pre>i2 : (S,solsS) = goodInitialPair T
2
o2 = ({1.41421x1*x3, 1.73205x2*x3 }, {{0, 0, 1}})
o2 : Sequence</pre>
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<tr><td><pre>i3 : M = track(S,T,solsS,gamma=>0.6+0.8*ii,Software=>M2)
o3 = {{{-.265576+.344521*ii, -.0232196+.0598427*ii, .890497+.116918*ii},
------------------------------------------------------------------------
#steps => 13, status => REGULAR, last t => 1, cond#^{-1} => .582332}}
o3 : List</pre>
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<div class="waystouse"><h2>Ways to use <tt>goodInitialPair</tt> :</h2>
<ul><li>goodInitialPair(List)</li>
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