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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