Forward rules for matrix factorizations

[Fredrik Bagge Carlson]

ChainRules contains some forward-mode rules for the most common factorizations like QR, Cholesky and Eigen, but several other factorizations are missing rules, notably

• schur
• hessenberg

Both of which are common in basic control-systems tools (Schur decomposition is performed as an initial numeric stabilization is most linear-algebra functions in control). Unfortunately, these factorizations are used on arbitrary matrices that can have eigenvalues anywhere, including on the imaginary axis, which might complicate matters slightly.

If suitable for including in this package, it would be nice to have rules for those! :)

[Songchen Tan]

We plan to plug into ChainRules's rule sets in the future, which means all rules they have written will be understandable in this package. This package itself isn't supposed to have comprehensive rules for linear algebra. Therefore you can contribute the rules to the ChainRules.jl package.

[Fredrik Bagge Carlson]

@ChrisRackauckas you encouraged me to open this issue here, did you have some particular reason in mind why this package would be a more suitable place for these rules than ChainRules.jl?