3 ms·

Author here: At the time of writing, I was not aware that the matrix form and the closed form are same, because you can derive the golden ratio based formula f

by abhi9u 3y ago

Author here: At the time of writing, I was not aware that the matrix form and the closed form are same, because you can derive the golden ratio based formula from the matrix formula. Full disclaimer: I am not a math major, and I can make such mistakes :) Although, I am curious to learn about exponentiation of real numbers. While writing this article, I was investigating how different libraries/languages implement exponentiation. While for matrix exponentiation, numpy used the algorithm I showed in the article. I also looked at the pow() function in libm of FreeBSD and it used the form you provided. I've been trying to look at an explanation behind this. Please share a pointer if you have any references.

SuchAnonMuchWow 3y ago
The main issue with floating point number is that it doesn't act like a ring (https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics)) because of rounding errors. For example the equality x*(y*z) = (x*y)*z doesn't hold because each multiplication introduce small rounding errors. And this kind of equations are required for the exponentiation-by-squaring method (consider for example how x**5 is computed: x**5 = (((x*x)*x)*x)*x) using a naive exponentiation approach, but also x**5 = x*(x*x)*(x*x) = x*(x*x)**2 using the exponentiation-by-squaring method). Integer operations and matrix operations are both rings, so the exponentiation-by-squaring approach works. For floating points, it doesn't give an exact result, but could still be used in practice to get an approximate result. The pow function doesn't use this approach however, because both arguments of the pow function are floating points numbers, and can thus compute things like pow(5, 0.5) = sqrt(5), or even 534.2**12.7. The exponentiation-by-squaring approach works only when the exponent is an integer, so we need other solutions. How exactly the pow function (or other math function like exp, log, cos, ...) are computed is a vast research subject in itself, but if you are interested, the Handbook of Floating-Point Arithmetic (https://perso.ens-lyon.fr/jean-michel.muller/Handbook.html https://perso.ens-lyon.fr/jean-michel.muller/Handbook.html) is both a good introduction to the domain (and the problems associated with the finite precision of floating point number), and provides a lot of techniques to work around or minimize the rounding errors that happens when manipulating floating point numbers.
- abhi9u 3y ago
  Thank you so much, both for providing more background on exponentiation, and for the reference. :-)

by abhi9u 3y ago

SuchAnonMuchWow 3y ago

The main issue with floating point number is that it doesn't act like a ring (https://en.wikipedia.org/wiki/Ring_(mathematics) https://en.wikipedia.org/wiki/Ring_(mathematics)) because of rounding errors. For example the equality x*(y*z) = (x*y)*z doesn't hold because each multiplication introduce small rounding errors. And this kind of equations are required for the exponentiation-by-squaring method (consider for example how x**5 is computed: x**5 = (((x*x)*x)*x)*x) using a naive exponentiation approach, but also x**5 = x*(x*x)*(x*x) = x*(x*x)**2 using the exponentiation-by-squaring method). Integer operations and matrix operations are both rings, so the exponentiation-by-squaring approach works. For floating points, it doesn't give an exact result, but could still be used in practice to get an approximate result. The pow function doesn't use this approach however, because both arguments of the pow function are floating points numbers, and can thus compute things like pow(5, 0.5) = sqrt(5), or even 534.2**12.7. The exponentiation-by-squaring approach works only when the exponent is an integer, so we need other solutions. How exactly the pow function (or other math function like exp, log, cos, ...) are computed is a vast research subject in itself, but if you are interested, the Handbook of Floating-Point Arithmetic (https://perso.ens-lyon.fr/jean-michel.muller/Handbook.html https://perso.ens-lyon.fr/jean-michel.muller/Handbook.html) is both a good introduction to the domain (and the problems associated with the finite precision of floating point number), and provides a lot of techniques to work around or minimize the rounding errors that happens when manipulating floating point numbers.

abhi9u 3y ago
Thank you so much, both for providing more background on exponentiation, and for the reference. :-)