Dyadic tensor product ⊗

Dyadic tensors are represented with the Grassmann’s exterior product algebra or nested Chain{V,1,Chain{V,1}} elements, generating a $2^{2n}$-dimensional mother algebra with the direct sum of the $n$-dimensional vector space and its dual vector space. The product of the vector basis and covector basis elements form the $n^2$-dimensional bivector subspace of the full $\frac{(2n)!}{2(2n−2)!}$-dimensional bivector sub-algebra. The package Grassmann is working towards making the full extent of this number system available in Julia by using static compiled parametric type information to handle sparse sub-algebras, such as the dyadic (1,1)-tensor bivector algebra or mother algebra.

Nested dyadic algebra

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 Values{5, Chain{⟨+-+++⟩, 1}} with indices SOneTo(5):
    0.811309v₁ + 0.819754v₂ + 0.978929v₃ + 0.60966v₄ + 0.406656v₅
   0.936018v₁ - 0.140807v₂ + 0.597128v₃ + 0.405145v₄ - 0.551934v₅
 -0.106691v₁ + 0.875287v₂ + 0.509549v₃ + 0.0350532v₄ - 0.252754v₅
 -0.0387932v₁ - 0.376618v₂ + 0.989887v₃ + 0.595954v₄ + 0.490055v₅
 -0.237868v₁ - 0.647482v₂ + 0.37111v₃ - 0.00499849v₄ + 0.295921v₅
julia> A = Chain{V,1}(rand(SMatrix{5,5}))(0.585434v₁+0.0639376v₂+0.274858v₃+0.564351v₄+0.0310541v₅)v₁ + (0.158229v₁+0.895529v₂+0.0375865v₃+0.683585v₄+0.0928829v₅)v₂ + (0.374523v₁+0.959813v₂+0.183036v₃+0.465702v₄+0.114261v₅)v₃ + (0.0406361v₁+0.155372v₂+0.817529v₃+0.803852v₄+0.722683v₅)v₄ + (0.717076v₁+0.48188v₂+0.665802v₃+0.0641773v₄+0.268439v₅)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

julia> 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:

julia> @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 $3\times$ 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 ℝ3(⟨111⟩, 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)

Of course, there are multiple equivalent ways of computing the same results using the ⋅ and : dyadic products.

julia> T\barycenter(T) == inv(T)⋅barycenter(T)ERROR: DimensionMismatch: expected input array of length -1, got length 0
julia> sqrt(T:T) == norm(SMatrix(T))true

It is possible to generate a Makie.jl streamplot diagrams with the Grassmann.Cramer method for interpolated data of finite element solutions.

Mother algebra formalism

Note that Λ(ℝ3) gives the vector basis, and Λ(ℝ3)' gives the covector basis:

julia> Λ(ℝ3)DirectSum.Basis{⟨111⟩,8}(v, v₁, v₂, v₃, v₁₂, v₁₃, v₂₃, v₁₂₃)
julia> Λ(ℝ3)'DirectSum.Basis{⟨---⟩',8}(w, w¹, w², w³, w¹², w¹³, w²³, w¹²³)

The following command yields a local 2D vector and covector basis,

julia> @mixedbasis ℝ2(⟨++--⟩*, v, v₁, v₂, w¹, w², v₁₂, v₁w¹, v₁w², v₂w¹, v₂w², w¹², v₁₂w¹, v₁₂w², v₁w¹², v₂w¹², v₁₂w¹²)
julia> w1+2w21w¹ + 2w²
julia> ans(v1+v2)0v

The sum w1+2w2 is interpreted as a covector element of the dual vector space, which can be evaluated as a linear functional when a vector argument is input. Using these in the workspace, it is possible to use the Grassmann exterior $\wedge$-tensor product operation to construct elements ℒ of the dyadic (1,1)-bivector subspace of linear transformations from the mother algebra.

julia> ℒ = (v1+2v2)∧(3w1+4w2)0v₁₂ + 3v₁w¹ + 4v₁w² + 6v₂w¹ + 8v₂w² + 0w¹²

The element ℒ is a linear form which can be evaluated,

julia> ℒ(v1+v2)0v₁ + 0v₂ + 9w¹ + 12w²
julia> L = [1,2] * [3,4]'; L * [1,1]2-element Vector{Int64}:
  7
 14

which is a computation equivalent to a matrix computation.

The TensorAlgebra evalution is still a work in progress, and the API and implementation may change as more features and algebraic operations and product structure are added.

Importing the Leech lattice generator

In the example below, we define a constant Leech which can be used to obtain linear combinations of the Leech lattice,

julia> using Grassmann

julia> generator = [8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0;
       2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0;
       2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 0 0 0 0 0 0 0 0;
       2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 0 0 0 0 0 0 0 0;
       2 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 0 0 0 0 0 0 0 0;
       4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0;
       2 0 2 0 2 0 0 2 2 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0;
       2 0 0 2 2 2 0 0 2 0 2 0 0 0 0 0 2 0 2 0 0 0 0 0;
       2 2 0 0 2 0 2 0 2 0 0 2 0 0 0 0 2 0 0 2 0 0 0 0;
       0 2 2 2 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0 2 0 0 0;
       0 0 0 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 0 0 2 2 0 0;
       0 0 0 0 0 0 0 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0;
       -3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1]

julia> const E24,W24 = Λ(24), ℝ^24⊕(ℝ^24)';

julia> const Leech = Chain{Submanifold(W24),Float64}(generator./sqrt(8));

julia> typeof(Leech)
Chain{⟨++++++++++++++++++++++++------------------------⟩*,2,Float64,1128}

julia> ndims(Manifold(Leech))
48

The Leech generator matrix is contained in the 1128-dimensional bivector subalgebra of the space with 48 indices.

julia> Leech(E24.v1)
2.82842712474619v₁ + 0.0v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN

julia> Leech(E24.v2)
1.414213562373095v₁ + 1.414213562373095v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN

julia> Leech(E24.v3)
1.414213562373095v₁ + 0.0v₂ + 1.414213562373095v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN

...

Then a TensorAlgebra evaluation of Leech at an Integer linear combination would be

julia> Leech(E24.v1 + 2*E24.v2)
5.65685424949238v₁ + 2.82842712474619v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 0.0va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.0w² + 0.0w³ + 0.0w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN

julia> ans⋅ans
39.99999999999999v

julia> Leech(E24.v2 + E24.v5)
2.82842712474619v₁ + 1.414213562373095v₂ + 0.0v₃ + 0.0v₄ + 0.0v₅ + 0.0v₆ + 0.0v₇ + 0.0v₈ + 0.0v₉ + 0.0v₀ + 1.414213562373095va + 0.0vb + 0.0vc + 0.0vd + 0.0ve + 0.0vf + 0.0vg + 0.0vh + 0.0vi + 0.0vj + 0.0vk + 0.0vl + 0.0vm + 0.0vn + 0.0w¹ + 0.7071067811865475w² + 1.414213562373095w³ + 1.414213562373095w⁴ + 0.0w⁵ + 0.0w⁶ + 0.0w⁷ + 0.0w⁸ + 0.0w⁹ + 0.0w⁰ + 0.0wA + 0.0wB + 0.0wC + 0.0wD + 0.0wE + 0.0wF + 0.0wG + 0.0wH + 0.0wI + 0.0wJ + 0.0wK + 0.0wL + 0.0wM + 0.0wN

julia> ans⋅ans
7.499999999999998v

The Grassmann package is designed to smoothly handle high-dimensional bivector algebras with headroom to spare. Although some of these calculations may have an initial delay, repeated calls are fast due to built-in caching and pre-compilation.

In future updates, more emphasis will be placed on increased type-stability with more robust sparse output allocation in the computational graph and minimal footprint but maximal type-stability for intermediate results and output.