8

I started to learn GNU Octave today, and tried the first expression given in the manual

exp(i*pi)

The result is 

ans = -1.0000e+000 + 1.2246e-016i

And it seems GNU Scientific Library gives similar results too.

So is this a Octave bug, or general problems of numeric analysis software (symbolic evaluation software will definitely give an answer of exact one)?

Improve this question
4
  • 2
    Seems octave is primaly intended for numerical analysis. Mathematica would definitely give you a better answer... :P just joking... Try searching for a GNU symbolical analysis solution
    – gd1
    Commented Jul 27, 2011 at 15:21
  • @Giacomo: I know that. I just wonder if all numeric analytic software evaluates to such number, or just GNU Octave.
    – Siyuan Ren
    Commented Jul 27, 2011 at 16:37
  • @Karsus Ren this isn't actually a software bug, but a hardware one. It's an inherent limitation of trying to evaluate expressions with irrational numbers on hardware with a limited amount of storage for a single number.
    – Breakthrough
    Commented Aug 3, 2011 at 17:22
  • download.oracle.com/docs/cd/E19957-01/806-3568/…
    – James
    Commented Aug 8, 2011 at 16:11

1 Answer 1

Reset to default
8

This isn't a bug with either, but due to the way computers perform floating point operations. There is a limited amount of precision any computer can operate with, and so you will sometimes see anomalies like this. While it is possible to write software that can handle this, it would take a lot more computation time and drastically increase memory requirements.

If you look at it, e^(i*pi) returns -1 + 1.2x10^-16i. As you can see, the imaginary component is extremely small (most would consider it negligible, since it's 16 orders of magnitude smaller then the real part). This component is introduced by rounding and precision errors, both with the calculation itself, as well as the stored value of pi since it's irrational (see this link for another example dealing with irrational numbers).

If this calculation error is unacceptable, you should look into math packages which perform symbolic rather then numerical analysis, or ones that use high-precision floating point numbers. The caveats of these are that they will drastically increase your memory requirements, and symbolic analysis is often much slower. Also, higher precision numbers will just shrink the magnitude of the rounding/precision errors, not eliminate them.

Improve this answer
3
  • 1
    I just need a confirmation that this is the common behavior of numerical analysis software.
    – Siyuan Ren
    Commented Aug 4, 2011 at 10:15
  • @Karsus Ren it's actually behaviour of this kind of software with high enough precision. I know that seems counter-intuitive, but lower precision numbers don't return these anomalies as often. See this article from Microsoft for more details, but remember, this issue is a combination of both the source code and the compiler/target architecture. Also, this Wikipedia article has some good background information on the problem.
    – Breakthrough
    Commented Aug 4, 2011 at 10:40
  • 1
    The underlying problem is that Octave's constant pi is not the mathematical constant π, but a close floating-point approximation to it. The exp function adds another small error to that. A system that works with symbolic expressions could compute exp(i*pi) exactly; Octave is not such a system.
    – Keith Thompson
    Commented Jul 11, 2012 at 8:08

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .