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<h4 class="subsection">15.5.2 Nth Root</h4>
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<a name="index-Nth-root-algorithm"></a>
<p>Integer Nth roots are taken using Newton’s method with the following
iteration, where <em>A</em> is the input and <em>n</em> is the root to be taken.
</p>
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<pre class="example"> 1 A
a[i+1] = - * ( --------- + (n-1)*a[i] )
n a[i]^(n-1)
</pre></div>
<p>The initial approximation <em>a[1]</em> is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just above the
true root. The iteration converges quadratically when started from a good
approximation. When <em>n</em> is large more initial bits are needed to get
good convergence. The current implementation is not particularly well
optimized.
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