In this algebra, it’s possible to compute on a mesh of arbitrary 5 dimensional conformal geometric algebra simplices, which can be represented by a bundle of nested dyadic tensors.

```
julia> using Grassmann, StaticArrays; basis"+-+++"
(⟨+-+++⟩, v, v₁, v₂, v₃, v₄, v₅, v₁₂, v₁₃, v₁₄, v₁₅, v₂₃, v₂₄, v₂₅, v₃₄, v₃₅, v₄₅, v₁₂₃, v₁₂₄, v₁₂₅, v₁₃₄, v₁₃₅, v₁₄₅, v₂₃₄, v₂₃₅, v₂₄₅, v₃₄₅, v₁₂₃₄, v₁₂₃₅, v₁₂₄₅, v₁₃₄₅, v₂₃₄₅, v₁₂₃₄₅)
julia> value(rand(Chain{V,1,Chain{V,1}}))
5-element StaticArrays.SArray{Tuple{5},Chain{⟨+-+++⟩,1,𝕂,253} where 253 where 𝕂,1,5} with indices SOneTo(5):
-0.33459594357756073v₁ - 0.3920064467082769v₂ - 0.575474920388841v₃ + 0.6150287650825268v₄ - 0.7568209093000915v₅
-0.7402635950699139v₁ - 0.9303076362461833v₂ + 0.9729806462365271v₃ - 0.8514563480551867v₄ + 0.09906887873006287v₅
-0.7456570397821101v₁ - 0.6497560949330929v₂ + 0.4756585550844967v₃ - 0.31169948016530347v₄ - 0.9355928499340793v₅
-0.4555014543082292v₁ + 0.712268225360094v₂ - 0.7500443783398549v₃ - 0.36349628003234713v₄ + 0.5005769037801056v₅
-0.07402971220645727v₁ + 0.19911765119918146v₂ - 0.4980618129231722v₃ - 0.7728564279829087v₄ + 0.9165735719353756v₅
julia> A = Chain{V,1}(rand(SMatrix{5,5}))
(0.9244801277294266v₁ + 0.029444337884018346v₂ + 0.745495522394158v₃ + 0.6695874677964055v₄ + 0.4998003712198389v₅)v₁ + (0.5423877012973404v₁ + 0.30112324458605655v₂ + 0.9530587650033631v₃ + 0.2706004745612134v₄ + 0.37762612797501616v₅)v₂ + (0.7730171467954035v₁ + 0.019660709510785912v₂ + 0.39119534821037494v₃ + 0.9403026278575068v₄ + 0.07545094732793833v₅)v₃ + (0.7184128110093908v₁ + 0.6295740775044767v₂ + 0.5179035493253021v₃ + 0.039081667453648716v₄ + 0.3719284661613145v₅)v₄ + (0.5033657705978616v₁ + 0.41183905359914386v₂ + 0.7761548051732969v₃ + 0.07635587137916744v₄ + 0.5582934197259402v₅)v₅
```

Additionally, in Grassmann.jl we prefer the nested usage of pure `ChainBundle`

parametric types for large re-usable global cell geometries, from which local dyadics can be selected.

Programming the `A\b`

method is straight forward with some Julia language metaprogramming and Grassmann.jl by first instantiating some Cramer symbols

```
Base.@pure function Grassmann.Cramer(N::Int)
x,y = SVector{N}([Symbol(:x,i) for i ∈ 1:N]),SVector{N}([Symbol(:y,i) for i ∈ 1:N])
xy = [:(($(x[1+i]),$(y[1+i])) = ($(x[i])∧t[$(1+i)],t[end-$i]∧$(y[i]))) for i ∈ 1:N-1]
return x,y,xy
end
```

These are exterior product variants of the Cramer determinant symbols (N! times N-simplex hypervolumes), which can be combined to directly solve a linear system:

```
@generated function Base.:\(t::Chain{V,1,<:Chain{V,1}},v::Chain{V,1}) where V
N = ndims(V)-1 # paste this into the REPL for faster eval
x,y,xy = Grassmann.Cramer(N)
mid = [:($(x[i])∧v∧$(y[end-i])) for i ∈ 1:N-1]
out = Expr(:call,:SVector,:(v∧$(y[end])),mid...,:($(x[end])∧v))
return Expr(:block,:((x1,y1)=(t[1],t[end])),xy...,
:(Chain{V,1}(getindex.($(Expr(:call,:./,out,:(t[1]∧$(y[end])))),1))))
end
```

Which results in the following highly efficient `@generated`

code for solving the linear system,

```
(x1, y1) = (t[1], t[end])
(x2, y2) = (x1 ∧ t[2], t[end - 1] ∧ y1)
(x3, y3) = (x2 ∧ t[3], t[end - 2] ∧ y2)
(x4, y4) = (x3 ∧ t[4], t[end - 3] ∧ y3)
Chain{V, 1}(getindex.(SVector(v ∧ y4, (x1 ∧ v) ∧ y3, (x2 ∧ v) ∧ y2, (x3 ∧ v) ∧ y1, x4 ∧ v) ./ (t[1] ∧ y4), 1))
```

Benchmarks with that algebra indicate a 3x faster performance than `SMatrix`

for applying `A\b`

to bundles of dyadic elements.

```
julia> @btime $(rand(SMatrix{5,5},10000)).\Ref($(SVector(1,2,3,4,5)));
2.588 ms (29496 allocations: 1.44 MiB)
julia> @btime $(Chain{V,1}.(rand(SMatrix{5,5},10000))).\$(v1+2v2+3v3+4v4+5v5);
808.631 μs (2 allocations: 390.70 KiB)
julia> @btime $(SMatrix(A))\$(SVector(1,2,3,4,5))
150.663 ns (0 allocations: 0 bytes)
5-element SArray{Tuple{5},Float64,1,5} with indices SOneTo(5):
-4.783720495603508
6.034887114999602
1.017847212237964
6.379374861538397
-4.158116538111051
julia> @btime $A\$(v1+2v2+3v3+4v4+5v5)
72.405 ns (0 allocations: 0 bytes)
-4.783720495603519v₁ + 6.034887114999605v₂ + 1.017847212237964v₃ + 6.379374861538393v₄ - 4.1581165381110505v₅
```

Such a solution is not only more efficient than Julia’s StaticArrays.jl method for `SMatrix`

, but is also useful to minimize allocations in Grassmann.jl finite element assembly.

Similarly, the Cramer symbols can also be manipulated to invert the linear system or for determining whether a point is within a simplex.

```
julia> using Grassmann; basis"3"
(⟨+++⟩, v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)
julia> T = Chain{V,1}(Chain(v1),v1+v2,v1+v3)
(1v₁ + 0v₂ + 0v₃)v₁ + (1v₁ + 1v₂ + 0v₃)v₂ + (1v₁ + 0v₂ + 1v₃)v₃
julia> barycenter(T) ∈ T, (v1+v2+v3) ∈ T
(true, false)
```

There are multiple equivalent ways of computing the same results using the `⋅`

and `:`

dyadic products.

```
julia> T\barycenter(T) == inv(T)⋅barycenter(T)
true
julia> sqrt(T:T) == norm(SMatrix(T))
true
```

The following Makie.jl streamplot was generated with the `Grassmann.Cramer`

method and interpolated from Nedelec edges of a Maxwell finite element solution.

More info about these examples is at https://grassmann.crucialflow.com/dev/tutorials/dyadic-tensors

Hermann Grassmann was the inventor of linear algebra as we know it today.