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 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!