This package provides an overloaded `factorize`

and `\`

that work with hyperdual-valued arrays.

It uses the hyper dual type defined by the HyperDualNumbers.jl package. The idea is that for a hyperdual-valued matrix

its inverse is given by

Therefore, only the inverse of is required to evaluate the inverse of .
This package makes available a `HyperDualFactors`

type which containts the factors of and the non-real parts of , and overloads `factorize`

to create an instance of `HyperDualFactors`

, which can then be called with `\`

to efficiently solve hyperdual-valued linear systems of the type .

This package should be useful for autodifferentiation of functions that use `\`

.
Note that this package is the equivalent of the DualMatrixTools.jl package, but for hyperdual numbers instead of dual numbers.

Create your hyperdual-valued matrix

`M`

:`julia> M = A + ε₁ * B + ε₂ * C + ε₁ε₂ * D`

Factorize

`M`

:`julia> Mf = factorize(M)`

Apply

`\`

to solve systems of the type`M * x = b`

`julia> x = Mf \ b`

In the context of iterative processes with multiple factorizations and forward and back substitutions, you may want to propagate hyperdual-valued numbers while leveraging (potentially) the fact the real part of the matrices to be factorized remains the same throughout.
This package provides an in-place `factorize`

, with a flag to update (or not) the factors.
Usage is straightforward.
By default, `factorize`

does *not* update the factors

```
julia> factorize(Mf, M) # only Mf.B, Mf.C, and Mf.D is updated
```

If you want to update the real-valued factors too, use

```
julia> factorize(Mf, M, update_factors=true) # The factors in Mf.Af are also updated
```

If you use this package, please cite it! You can export the citation by first clicking on the DOI badge at the top, which links to the Zenodo record of the package, and then clicking on the citation format you want in the "Export" box at the bottom of the page.

11/09/2018

about 1 year ago

2 commits