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POIs this a bug in NSolve in mathematica?
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copied!<p>One would expect and hope that if you ask <code>Mathematica</code> to find the roots of a polynomial, it should give the same (approximate) answers whether you do this symbolically, then find numerical approximations to these exact answers, or whether you do it numerically. Here's an example which (in <code>Mathematica 7</code>, running on OS X) where this fails badly:</p> <pre><code>poly = -112 + 1/q^28 + 1/q^26 - 1/q^24 - 6/q^22 - 14/q^20 - 25/q^18 - 38/q^16 - 52/q^14 - 67/q^12 - 81/q^10 - 93/q^8 - 102/q^6 - 108/ q^4 - 111/q^2 - 111 q^2 - 108 q^4 - 102 q^6 - 93 q^8 - 81 q^10 - 67 q^12 - 52 q^14 - 38 q^16 - 25 q^18 - 14 q^20 - 6 q^22 - q^24 + q^26 + q^28; Total[q^4 /. NSolve[poly == 0, q]] - Total[q^4 /. N[Solve[poly == 0, q]]] </code></pre> <p>(Note: this is actually a Laurent polynomial, and if you multiply through by a large power of <code>q</code> the problem goes away.)</p> <p>The last line here is just a demonstration that the solutions found are very different; in fact it's the quantity we were trying to compute in the problem we were working on.</p> <p>If you look closely at the output of <code>NSolve[poly == 0, q]</code> and of <code>N[Solve[poly == 0, q]</code>, you'll see that NSolve only gives <code>54</code> roots instead of the expected <code>56</code>. It's not that it just missed a repeated root or anything; it's missing the two largest roots in magnitude (approximately <code>+/- 1.59</code>)</p> <blockquote> <p>Is this a bug in Mathematica? Does anyone have an explanation for why this is happening?</p> </blockquote>

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