If you would like a full description of octonion algebra, I suggest you visit my web site and pull the PDF file. www.octospace.com
I know nothing about Sage, can't help you there. Rick On Feb 16, 7:51 pm, Dan Christensen <[EMAIL PROTECTED]> wrote: > Eric Drechsel <[EMAIL PROTECTED]> writes: > > We want to compute withoctonionsin Sage, and discussed at our > > meeting last night possibly implementing such an algebra. > > > As I understand it, multiplication ofoctonionscan be implemented on > > top of the quaternions in a simple way using the Cayley-Dickson > > constructionhttp://en.wikipedia.org/wiki/Octonion#Cayley-Dickson_construction > > . > > If you define A_0 to be the reals and A_n to be the Cayley-Dickson > construction applied to A_{n-1}, then A_3 is theoctonions. More > generally, write A_n(F) for the Cayley-Dickson construction applied > to A_{n-1}(F), where A_0 = F. > > I've done lots of computations with A_4(float), A_5(float) and > A_6(float) in python, but not using Sage. I think the cleanest approach > is to define a function CayleyDickson that takes a class representing a > *-algebra and produces a new class representing the result of applying > the Cayley-Dickson construction. This allows you to change the base > "field" from float to int, Z, Z[x], high-precision floats, or whatever > you need for a particular application. > > Potential problems: > > - If I call CayleyDickson(float) twice, do I get a new class each time? > > - How to handle coercions between A_n(F) and A_m(F)? And between A_n(F) > and A_n(F')? (F and F' are ground rings with *.) > > - Efficiency. In my application, I was multiplying elements of > A_n(float) zillions of times, and this was *way* to slow using > the obvious recursive method in python. (If I recall correctly, > the time scales like 4**n.) > > I instead worked with a class that directly implemented A_n(float) > using numpy arrays, and used weave to compile some inline C code for > the multiplication operation. (Cython/Pyrex might be just as good for > this, but I'm not sure if they can directly access numpy arrays as C > arrays.) It falls back to using python for the multiplication for > "object" arrays, which allows for changing float to other classes, > but is awkward. > > I'm not sure how to get the best of both worlds. Maybe the first > way, with some clever Cython/Pyrex, could be made both flexible > and efficient? Or maybe the first way, with some special casing > to speed up common cases? > > Dan --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
