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<head><title>refine(List,List) -- refine numerical solutions to a system of polynomial equations</title>
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<div><h1>refine(List,List) -- refine numerical solutions to a system of polynomial equations</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>solsR = refine(T,sols)</tt></div>
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<li><span>Function: <a href="_refine_lp__List_cm__List_rp.html" title="refine numerical solutions to a system of polynomial equations">refine</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>T</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, polynomials of the system</span></li>
<li><span><tt>sols</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, solutions (lists of coordinates)</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>solsR</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, refined solutions</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>Iterations => </tt><span><span>default value null</span>, number of refining iterations of Newton's methods</span></span></li>
<li><span><tt>Bits => </tt><span><span>default value 300</span>, number of bits of precision</span></span></li>
<li><span><tt>ErrorTolerance => </tt><span><span>default value 1e-10</span>, a bound on the desired estimated error</span></span></li>
<li><span><tt>ResidualTolerance => </tt><span><span>default value 0</span>, a bound on desired residual</span></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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<div class="single"><h2>Description</h2>
<div>Uses Newton's method to correct the given solutions so that the resluting approximation has its estimated relative error bound by <tt>Tolerance</tt>; the number of iterations is at most <tt>maxCorrSteps</tt>.<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 : sols = { {1.1_CC,0.1}, {-0.1,1.2} };</pre>
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<tr><td><pre>i5 : refine(T, sols, Software=>M2, ErrorTolerance=>.001, Iterations=>10)
o5 = {{1, 4.66507e-9}, {-8.37549e-8, 1}}
o5 : List</pre>
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<div class="single"><h2>Caveat</h2>
<div>If option <tt>Software=>M2engine</tt> is specified, then the refinement happens in the M2 engine and it is assumed that the last path tracking procedure took place with the same option and was given the same target system. Any other value of this option would launch an M2-language procedure.</div>
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