Interval Power Function

Interval arithmetic is a powerful tool for reliable computing.

The definition of powers with real base and real exponent is a problematic task. I propose three differently extensive versions of a real power function for use in interval arithmetic: pow1, a function which is defined for positive real bases only; pow2, a function which additionally defines powers with negative base and integral exponent; pow3, a function which even allows powers with negative base and some rational exponent.

$\begin{array}{rl}\mathrm{pow1}:& \mathrm{ℝ⁺}\times ℝ\to ℝ\\ & (x,y)\mapsto \exp (y\cdot \log x)\\ \mathrm{pow2}:& (\mathrm{ℝ⁺}\times ℝ)\cup \left(\right\{0\}\times \mathrm{ℝ⁺})\cup (\mathrm{ℝ⁻}\times \mathbb{Z})\to ℝ\\ & (x,y)\mapsto \{\begin{array}{ll}\exp (y\cdot \log x)& \text{if }x\text{ positive,}\\ 0& \text{if }x\text{ zero,}\\ \exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y\text{ even,}\\ -\exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y\text{ odd.}\end{array}\\ \mathrm{pow3}:& (\mathrm{ℝ⁺}\times ℝ)\cup \left(\right\{0\}\times \mathrm{ℝ⁺})\cup (\mathrm{ℝ⁻}\times \{\frac{m}{n}|m,n\in \mathbb{Z}\text{, }n\text{ odd}\})\to ℝ\\ & (x,y)\mapsto \{\begin{array}{ll}\exp (y\cdot \log x)& \text{if }x\text{ positive,}\\ 0& \text{if }x\text{ zero,}\\ \exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y=\frac{m}{n}\text{ with }m\text{ even and }n\text{ odd,}\\ -\exp (y\cdot \log |x\left|\right)& \text{if }x\text{ negative and }y=\frac{m}{n}\text{ with }m\text{ odd and }n\text{ odd.}\end{array}\end{array}$

2011 — I substantiate above definitions and develop interval extensions of the functions, as well as reverse operations. The original implementation is based on Matlab and the INTLAB toolbox as part of my thesis. All resulting interval functions can be tested at the exploration page.

2013 — Above functions are published in the journal articel Variants of the general interval power function in Soft Computing, August 2013, Volume 17, Issue 8, pp 1357-1366.

2014 — The proposed IEEE standard for interval arithmetic, IEEE P1788, enters the sponsor balloting phase. The function pow in the standard document equals my proposal pow1 plus limit values. I am a member of the balloting commitee.

2015 — I release the initial version of the GNU Octave interval package. It contains the functions pow and power. The latter is the general power function and equals my proposal pow2.

June 11, 2015 — The IEEE standard for interval arithmetic, IEEE Std 1788-2015, gains final appoval.