[Date Prev][Date Next][Thread Prev][Thread Next][Date Index][Thread Index]
## Re: fractional powers

**From**: |
Mario Storti |

**Subject**: |
Re: fractional powers |

**Date**: |
Tue, 12 Nov 1996 17:29:12 -0300 |

>* I'm somewhat disturbed by the following behaviour on octave. Perhaps it*
>* is standard and I shouldn't be worried but it surprised me. Consider the*
>* following question: what is the cube root of -1? Clearly the answer*
>* should be -1. Now ask octave*
>
>* octave:1> x = (-1)^(1/3)*
>* x = 0.50000 + 0.86603i*
>
>* it gets wierder if you now cube that number*
>
>* octave:2> x^3*
>* ans = -1.0000e+00 + 1.2246e-16i*
>
>* This is pretty close to the truth but still strange to my way of thinking.*
>* Similar wierdness shows up with other fractional powers: 1/5, 1/7, etc.*
>
>* Any thoughts?*
>
>* Heber Farnsworth | Department of Finance*
>* Univerity of Washington | Box 353200*
>* tele: (206) 528-0793 home | Seattle, WA 98195-3200*
>* tele: (206) 543-4773 finance web: http://weber.u.washington.edu/~heberf*
>* fax: (206) 685-9392 email: address@hidden*
>
I think that Octave, Matlab and all the others (even Fortran! I tried
it right now!!.) performs z^(1/3) as:
z^(1/3)=exp(1/3*log(z))
log(z) is multi-valued and the standard definition is log(z) real over
the positive real axis and with a branch-cut in the negative real
axis. For z strictly on the negative real axis it gives:
log(z)=log(|z|) + i*pi
and then z^(1/3) is computed as:
z^(1/3)=exp(1/3*(log(|z|) + i*pi))
=|z|^(1/3)* exp(i*pi/3)
which is the solution given by Octave. If you add a small negative
imaginary part, then you cross the branch-cut and then:
log(z)=log(|z|) - i*pi
and the result will be:
z^(1/3)=exp(1/3*(log(|z|) - i*pi))
=|z|^(1/3)* exp(-i*pi/3)
>* octave:9> (-1-i*1e-15)^(1/3)*
>* ans = 0.50000 - 0.86603i*
Mario
%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%%%%<>%%%
Mario Alberto Storti | Fax: (54)(42) 55.09.44 |
Grupo de Tecnologia Mecanica | Tel: (54)(42) 55.91.75 |
INTEC, Guemes 3450 - 3000 Santa Fe | http://venus.unl.edu.ar/gtm-eng.html |
Argentina | Home: Gob. Vera 3161 |
Reply: address@hidden | (54)(42) 55.00.23 |