Fast scheme for the evaluation $x^k$ for $k\in[0,1]$

Question

In numerical analysis, we have the scheme to evaluate $x^{1/2}$ or more generally the scheme to evaluate $x^{1/n}$.

For $x^k$ with $k\in[0,1]$, is there a fast scheme to evaluate it? My naive implement is writting $k$ into fration like $k=\frac{\alpha}{\beta}$, but it seems that this method will not work for irrational number and the computation is expensive. I am wondering in programming like mathematica or matlab, how is $x^k$ calculated?

Not sure we'd use rhe word "scheme" in English. "Algorithm" might be used, or, less formally, "method." — Thomas Andrews, Mar 23 '13 at 19:12
There's no such thing as "irrational numbers" in finite-precision arithmetic... ;) so, find the continued fraction approximant that is nearest to $k$ and proceed as in the rational case... — J. M. ain't a mathematician, Mar 23 '13 at 19:13
The quick and dirty approach would be to take a sufficiently close rational number, as J.M. said. — Potato, Mar 23 '13 at 19:14

score 2 · Accepted Answer · answered Mar 23 '13 at 19:10

2

Do $x^\alpha = \exp(\alpha \ln x)$, use a CORDIC algorithm for the functions.

answered Mar 23 '13 at 19:10

vonbrand

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Fast scheme for the evaluation $x^k$ for $k\in[0,1]$

1 Answers1