I’m pleased to announce the release of two new numeric packages, HemiplexNumbers and HemiplexFactorizations, and their accompanying manuscript Jordan algebras, hemiplex numbers, and the Cholesky decomposition of arbitrary symmetric matrices with my co-author Alan Edelman.
This is a fun example of Julia’s flexibility: we define a “new” (aka, rediscover an old) type of number, and then show that it’s useful in solving an apparently-unsolvable problem, computing the Cholesky factorization of indefinite matrices. The Cholesky factorization is one of linear algebra’s favorites, being highly stable and with excellent sparsity-preserving characteristics. The catch is that classically it is only applicable to positive-semidefinite matrices; obtaining similar properties for an indefinite factorization has proven challenging.
The hemiplex numbers provide a principled path to obtaining these advantages for all symmetric matrices. As implemented today, the packages provide both stability and performance on par with existing LAPACK factorizations, so they may be worth a try for your practical problem-solving.
For those with a research interest in topics like these, we also make it clear in the manuscript that the promise of hemiplex numbers has only been partly realized, and there may be substantial new gains awaiting enterprising algebraists.
I look forward to any feedback, positive or negative. Enjoy!
Absolutely fascinating! I wonder whether this publication paves the path for matrices similarly to what complex numbers provided with the fundamental theorem of algebra. Okay, the Cholesky factorization might not be as important as algebra, but nevertheless I look forward to see which possibilities this will open up. Congratulations to this major publication!
In the past I needed to use quite some workarounds to get a Cholesky factorization from not-really-positive-definite matrices to harvest the speed advantage they brought with them. This was a reoccurring pattern in my work: After mangling a suitably complicated matrix-based problem long enough to finally get it to Cholesky factorize, the problem was basically solved, because then the implementation was both fast enough and robust enough to be capable to handle safety-relevant real-time systems.
I would be really curious to test your hemiplex-based Cholesky factorization (or should we call it Holy-Edelman factorization?) on these problems.
P.S.
A really nice paper! Nevertheless, Fig. 6.3 would benefit from being larger and having the x-offset which is promised in the caption.
The idea of getting around numerical issues by inventing weird new number systems is very cool, like with dual numbers and differentiation.
I wonder how many potential applications of this concept there are.
Especially if you think about it and realize that in the end the computer is only performing rational computations. So all the complex/hemiplex operations are in the end broken down to their real (on the computer: rational) parts of their imaginary or what-not units.
So in the end all these number systems “only” help us to identify computational patterns which we were not able to identify otherwise. To me this is still mind-boggling, after all these years working with complex numbers, so I guess it will be even more so for hemiplex numbers.