DualMatrixTools is a Julia package for solving dual-valued linear systems.

Essentially, it provides an overload for Julia’s `factorize`

and `\`

functions that work with dual-valued arrays.
It uses the dual type defined by the DualNumbers.jl package.
The idea is that for a dual-valued matrix

$$M = A + \varepsilon B,$$

its inverse is given by

$$M^{-1} = (I - \varepsilon A^{-1} B) A^{-1}.$$

Therefore, only the inverse of $A$ is required to evaluate the inverse of $M$.
This package makes available a `DualFactors`

type which containts (i) the factors of $A$ and (ii) the non-real part, $B$.
It also overloads `factorize`

to create an instance of `DualFactors`

(when invoked with a dual-valued matrix), which can then be called with `\`

to efficiently solve dual-valued linear systems of the type $M \, x = B$.

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

.
Note the same idea extends to hyper dual numbers (see the HyperDualMatrixTools.jl package).

## Usage

First, create your dual-valued matrix `M`

:

```
julia> M = A + ε * B
```

Then, apply `\`

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

- without factorization:

```
julia> x = M \ b
```

- or better, with prior factorization:

```
julia> Mf = factorize(M)
julia> x = Mf \ b
```

(This is better in case you want to solve for another `b`

!)

## Advanced usage

In the context of iterative processes with multiple factorizations and forward and back substitutions, you may want to propagate dual-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 is updated
```

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

```
julia> factorize(Mf, M, update_factors=true) # Mf.B and Mf.Af are updated
```

## Citation

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