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improved robustness of algebra
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@chakravala
chakravala committed May 21, 2024
1 parent 5a4c30a commit 2523f25
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2 changes: 1 addition & 1 deletion Project.toml
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Expand Up @@ -16,7 +16,7 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
[compat]
julia = "1"
Leibniz = "0.2"
DirectSum = "0.8.1"
DirectSum = "0.8.12"
AbstractTensors = "0.8"
ComputedFieldTypes = "1"
Requires = "1"
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4 changes: 2 additions & 2 deletions src/Grassmann.jl
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Expand Up @@ -26,15 +26,15 @@ import AbstractTensors: Values, Variables, FixedVector, clifford, hodge, wedge,
export ⊕, ℝ, @V_str, @S_str, @D_str, Manifold, Submanifold, Signature, DiagonalForm, value
export @basis, @basis_str, @dualbasis, @dualbasis_str, @mixedbasis, @mixedbasis_str, Λ
export ℝ0, ℝ1, ℝ2, ℝ3, ℝ4, ℝ5, ℝ6, ℝ7, ℝ8, ℝ9, mdims, tangent, metric, antimetric
export hodge, wedge, vee, complement, dot, antidot, istangent
export hodge, wedge, vee, complement, dot, antidot, istangent, Values

import Base: @pure, ==, isapprox
import Base: print, show, getindex, setindex!, promote_rule, convert, adjoint
import DirectSum: V0, ⊕, generate, basis, getalgebra, getbasis, dual, Zero, One, Zero, One
import Leibniz: hasinf, hasorigin, dyadmode, value, pre, vsn, metric, mdims, gdims
import Leibniz: bit2int, indexbits, indices, diffvars, diffmask
import Leibniz: symmetricmask, indexstring, indexsymbol, combo, digits_fast
import DirectSum: antimetric
import DirectSum: antimetric, signbool

import Leibniz: hasconformal, hasinf2origin, hasorigin2inf
import AbstractTensors: valuetype, scalar, isscalar, trivector, istrivector, ⊗, complement
Expand Down
73 changes: 50 additions & 23 deletions src/algebra.jl
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Expand Up @@ -36,25 +36,33 @@ Geometric algebraic product: ω⊖η = (-1)ᵖdet(ω∩η)⊗(Λ(ω⊖η)∪L(ω
*(a::X,b::Y,c::Z...) where {X<:TensorAlgebra,Y<:TensorAlgebra,Z<:TensorAlgebra} = *(a*b,c...)

@pure function mul(a::Submanifold{V},b::Submanifold{V},der=derive_mul(V,UInt(a),UInt(b),1,true)) where V
ba,bb = UInt(a),UInt(b)
(diffcheck(V,ba,bb) || iszero(der)) && (return Zero(V))
A,B,Q,Z = symmetricmask(V,ba,bb)
pcc,bas,cc = (hasinf(V) && hasorigin(V)) ? conformal(V,A,B) : (false,A⊻B,false)
d = getbasis(V,bas|Q)
out = (typeof(V)<:Signature || count_ones(A&B)==0) ? (parity(a,b)⊻pcc ? Single{V}(-1,d) : d) : Single{V}((pcc ? -1 : 1)*parityinner(V,A,B),d)
diffvars(V)≠0 && !iszero(Z) && (out = Single{V}(getbasis(loworder(V),Z),out))
return cc ? (v=value(out);out+Single{V}(hasinforigin(V,A,B) ? -(v) : v,getbasis(V,conformalmask(V)⊻UInt(d)))) : out
if isdiag(V)
ba,bb = UInt(a),UInt(b)
(diffcheck(V,ba,bb) || iszero(der)) && (return Zero(V))
A,B,Q,Z = symmetricmask(V,ba,bb)
d = getbasis(V,(A⊻B)|Q)
out = if typeof(V)<:Signature || count_ones(A&B)==0
(parity(a,b) ? Single{V}(-1,d) : d)
else
Single{V}(signbool(parityinner(V,A,B)),d)
end
diffvars(V)≠0 && !iszero(Z) && (out = Single{V}(getbasis(loworder(V),Z),out))
return out
else
out = paritygeometric(V,UInt(a),UInt(b))
isempty(out) ? Zero(V) : +(Single{V}.(out)...)
end
end

function ⟑(a::Single{V},b::Submanifold{V}) where V
v = derive_mul(V,UInt(basis(a)),UInt(b),a.v,true)
bas = mul(basis(a),b,v)
order(a.v)+order(bas)>diffmode(V) ? Zero(V) : Single{V}(v,bas)
order(a.v)+order(bas)>diffmode(V) ? Zero(V) : v*bas
end
function ⟑(a::Submanifold{V},b::Single{V}) where V
v = derive_mul(V,UInt(a),UInt(basis(b)),b.v,false)
bas = mul(a,basis(b),v)
order(b.v)+order(bas)>diffmode(V) ? Zero(V) : Single{V}(v,bas)
order(b.v)+order(bas)>diffmode(V) ? Zero(V) : v*bas
end

export ∗, ⊛, ⊖
Expand Down Expand Up @@ -173,24 +181,43 @@ export ⋅
Interior (right) contraction product: ω⋅η = ω∨⋆η
"""
@pure function contraction(a::Submanifold{V},b::Submanifold{V}) where V
g,C,t,Z = interior(a,b)
(!t || iszero(derive_mul(V,UInt(a),UInt(b),1,true))) && (return Zero(V))
d = getbasis(V,C)
istangent(V) && !iszero(Z) && (d = Single{V}(getbasis(loworder(V),Z),d))
return isone(g) ? d : Single{V}(g,d)
iszero(derive_mul(V,UInt(a),UInt(b),1,true)) && (return Zero(V))
if isdiag(V) || hasconformal(V)
g,C,t,Z = interior(a,b)
(!t) && (return Zero(V))
d = getbasis(V,C)
istangent(V) && !iszero(Z) && (d = Single{V}(getbasis(loworder(V),Z),d))
return isone(g) ? d : Single{V}(g,d)
else
Cg,Z = parityinterior(V,UInt(a),UInt(b),Val(true))
#d = getbasis(V,C)
#istangent(V) && !iszero(Z) && (d = Single{V}(getbasis(loworder(V),Z),d))
return isempty(Cg) ? Zero(V) : +(Single{V}.(Cg)...)
end
end

function contraction(a::X,b::Y) where {X<:TensorTerm{V},Y<:TensorTerm{V}} where V
ba,bb = UInt(basis(a)),UInt(basis(b))
g,C,t,Z = interior(V,ba,bb)
!t && (return Zero(V))
v = derive_mul(V,ba,bb,value(a),value(b),AbstractTensors.dot)
if istangent(V) && !iszero(Z)
_,_,Q,_ = symmetricmask(V,ba,bb)
v = !(typeof(v)<:TensorTerm) ? Single{V}(v,getbasis(V,Z)) : Single{V}(v,getbasis(loworder(V),Z))
count_ones(Q)+order(v)>diffmode(V) && (return Zero(V))
if isdiag(V) || hasconformal(V)
g,C,t,Z = interior(V,ba,bb)
!t && (return Zero(V))
v = derive_mul(V,ba,bb,value(a),value(b),AbstractTensors.dot)
if istangent(V) && !iszero(Z)
_,_,Q,_ = symmetricmask(V,ba,bb)
v = !(typeof(v)<:TensorTerm) ? Single{V}(v,getbasis(V,Z)) : Single{V}(v,getbasis(loworder(V),Z))
count_ones(Q)+order(v)>diffmode(V) && (return Zero(V))
end
return Single{V}(g*v,getbasis(V,C))
else
Cg,Z = parityinterior(V,ba,bb,Val(true))
v = derive_mul(V,ba,bb,value(a),value(b),AbstractTensors.dot)
if istangent(V) && !iszero(Z)
_,_,Q,_ = symmetricmask(V,ba,bb)
v = !(typeof(v)<:TensorTerm) ? Single{V}(v,getbasis(V,Z)) : Single{V}(v,getbasis(loworder(V),Z))
count_ones(Q)+order(v)>diffmode(V) && (return Zero(V))
end
return isempty(Cg) ? Zero(V) : (+(Single{V}.(Cg)...))*v
end
return Single{V}(g*v,getbasis(V,C))
end

export ⨼, ⨽
Expand Down
4 changes: 3 additions & 1 deletion src/composite.jl
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Expand Up @@ -183,7 +183,8 @@ end
function Base.exp(t::T) where T<:TensorGraded
S,B,V = T<:Submanifold,T<:TensorTerm,Manifold(t)
if B && isnull(t)
return One(V)
vt = valuetype(t)
return Couple{V,basis(t)}(one(vt),zero(vt))
elseif isR301(V) && grade(t)==2 # && abs(t[0])<1e-9 && !options.over
u = sqrt(abs(abs2(t)[1]))
u<1e-5 && (return One(V)+t)
Expand Down Expand Up @@ -804,6 +805,7 @@ end

DirectSum.Λ(x::Chain{V,1,<:Chain{V,1}},G) where V = compound(x,G)
compound(x,G::T) where T<:Integer = compound(x,Val(G))
compound(x::Chain{V,1,<:Chain{V,1}},::Val{0}) where V = Chain{V,0}(Values(Chain{V,0}(1)))
@generated function compound(x::Chain{V,1,<:Chain{V,1}},::Val{G}) where {V,G}
Expr(:call,:(Chain{V,G}),Expr(:call,:Values,[Expr(:call,:∧,[:(@inbounds x[$i]) for i ∈ indices(j)]...) for j ∈ indexbasis(mdims(V),G)]...))
end
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128 changes: 122 additions & 6 deletions src/forms.jl
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Expand Up @@ -339,12 +339,12 @@ for pinor ∈ (:Spinor,:AntiSpinor)#,:Multivector)
end
end

display_matrix(m::Chain{V,G,<:TensorGraded{W,G}}) where {V,G,W} = vcat(transpose([V,chainbasis(V,G)...]),hcat(chainbasis(W,G),matrix(m)))
display_matrix(m::TensorGraded{V,G,<:Spinor{W}}) where {V,G,W} = vcat(transpose([V,evenbasis(V)...]),hcat(evenbasis(W),matrix(m)))
display_matrix(m::TensorGraded{V,G,<:AntiSpinor{W}}) where {V,G,W} = vcat(transpose([V,oddbasis(V)...]),hcat(oddbasis(W),matrix(m)))
display_matrix(m::TensorAlgebra{V,<:Multivector{W}}) where {V,W} = vcat(transpose([V,fullbasis(V)...]),hcat(fullbasis(W),matrix(m)))
display_matrix(m::Chain{V,G,<:TensorGraded{W,G}}) where {V,G,W} = vcat(transpose([Submanifold(V),chainbasis(V,G)...]),hcat(chainbasis(W,G),matrix(m)))
display_matrix(m::TensorGraded{V,G,<:Spinor{W}}) where {V,G,W} = vcat(transpose([Submanifold(V),evenbasis(V)...]),hcat(evenbasis(W),matrix(m)))
display_matrix(m::TensorGraded{V,G,<:AntiSpinor{W}}) where {V,G,W} = vcat(transpose([Submanifold(V),oddbasis(V)...]),hcat(oddbasis(W),matrix(m)))
display_matrix(m::TensorAlgebra{V,<:Multivector{W}}) where {V,W} = vcat(transpose([Submanifold(V),fullbasis(V)...]),hcat(fullbasis(W),matrix(m)))
for (pinor,bas) ∈ ((:Spinor,:evenbasis),(:AntiSpinor,:oddbasis),(:Multivector,:fullbasis))
@eval display_matrix(m::$pinor{V,<:TensorAlgebra{W}}) where {V,W} = vcat(transpose([V,$bas(V)...]),hcat($bas(W),matrix(m)))
@eval display_matrix(m::$pinor{V,<:TensorAlgebra{W}}) where {V,W} = vcat(transpose([Submanifold(V),$bas(V)...]),hcat($bas(W),matrix(m)))
end

export Projector, Dyadic, Proj
Expand Down Expand Up @@ -401,6 +401,15 @@ Chain(P::Dyadic{V}) where V = Chain{V}(P)

export TensorOperator, Endomorphism

#=struct TensorOperator{V,W,S<:TensorAlgebra{W},T<:TensorAlgebra{V,S}} <: TensorNested{V,S}
v::T
TensorOperator{V,W}(t::T) where {V,W,S<:TensorAlgebra{W},T<:TensorAlgebra{V,S}} = new{V,W,S,T}(t)
TensorOperator{V}(t::T) where {V,W,S<:TensorAlgebra{W},T<:TensorAlgebra{V,S}} = new{V,W,S,T}(t)
TensorOperator(t::T) where {V,W,S<:TensorAlgebra{W},T<:TensorAlgebra{V,S}} = new{V,W,S,T}(t)
end
Endomorphism{V,S<:TensorAlgebra{V},T<:TensorAlgebra{V,S}} = TensorOperator{V,V,S,T}=#

struct TensorOperator{V,W,T<:TensorAlgebra{V,<:TensorAlgebra{W}}} <: TensorNested{V,T}
v::T
TensorOperator{V,W}(t::T) where {V,W,T<:TensorAlgebra{V,<:TensorAlgebra{W}}} = new{V,W,T}(t)
Expand All @@ -412,6 +421,8 @@ Endomorphism{V,T<:TensorAlgebra{V,<:TensorAlgebra{V}}} = TensorOperator{V,V,T}

value(t::TensorOperator) = t.v
matrix(m::TensorOperator) = matrix(value(m))
compound(m::TensorOperator,g) = TensorOperator(compound(value(m),g))
compound(m::TensorOperator,g::Integer) = TensorOperator(compound(value(m),g))
getindex(t::TensorOperator,i::Int,j::Int) = value(value(t.v)[j])[i]

for op ∈ (:(Base.inv),)
Expand All @@ -432,6 +443,9 @@ function show(io::IO, ::MIME"text/plain", t::TensorOperator)
if isempty(X) && get(io, :compact, false)::Bool
return show(io, X)
end
show_matrix(io, t, X)
end
function show_matrix(io::IO, t, X)
# 0) show summary before setting :compact
summary(io, t)
isempty(X) && return
Expand Down Expand Up @@ -544,6 +558,15 @@ end
@inline @generated function matmul(x::Values{N,<:Multivector{V}},y::Values{N}) where {N,V}
Expr(:call,:Values,[Expr(:call,:+,[:(@inbounds y[$i]*value(x[$i])[$j]) for i ∈ list(1,N)]...) for j ∈ list(1,1<<mdims(V))]...)
end
@inline @generated function matmul(x::Values{N,<:Spinor{V}},y::Values{N}) where {N,V}
Expr(:call,:Values,[Expr(:call,:+,[:(@inbounds y[$i]*value(x[$i])[$j]) for i ∈ list(1,N)]...) for j ∈ list(1,1<<(mdims(V)-1))]...)
end
@inline @generated function matmul(x::Values{N,<:AntiSpinor{V}},y::Values{N}) where {N,V}
Expr(:call,:Values,[Expr(:call,:+,[:(@inbounds y[$i]*value(x[$i])[$j]) for i ∈ list(1,N)]...) for j ∈ list(1,1<<(mdims(V)-1))]...)
end

contraction(x::Spinor{W,<:Spinor{V},N},y::Spinor{V,T,N}) where {W,N,V,T} = Spinor{V}(matmul(value(x),value(y)))
contraction(x::AntiSpinor{W,<:AntiSpinor{V},N},y::AntiSpinor{V,T,N}) where {W,N,V,T} = AntiSpinor{V}(matmul(value(x),value(y)))

contraction(a::Dyadic{V,<:Chain{V,1,<:Chain},<:Chain{V,1,<:Chain}} where V,b::TensorGraded) = sum(value(a.x).⊗(value(a.y).⋅b))
contraction(a::Dyadic{V,<:Chain{V,1,<:Chain}} where V,b::TensorGraded) = sum(value(a.x).⊗(a.y.⋅b))
Expand Down Expand Up @@ -632,7 +655,7 @@ end
@pure fulldyad(V) = Multivector.(fullbasis(V))
@pure fullbasis(V) = Λ(V).b

@pure chaindyad(V,G) = Chain.(chainbasis(V,G))
@pure chaindyad(V,G=1) = Chain.(chainbasis(V,G))
@pure function chainbasis(V,G=1)
N = mdims(V)
r,b = binomsum(N,G),binomial(N,G)
Expand Down Expand Up @@ -661,3 +684,96 @@ gradedoperator(fun,V) = TensorOperator(Multivector{V}(fun.(Λ(V).b)))
end
evenoperator(t::TensorAlgebra{V}) where V = TensorOperator(Spinor{V}(evenbasis(V) .⊘ Ref(t)))
oddoperator(t::TensorAlgebra{V}) where V = TensorOperator(AntiSpinor{V}(oddbasis(V) .⊘ Ref(t)))

# metric tensor

antimetrictensor(V,G) = compound(metrictensor(V),grade(V)-G)
antimetrictensor(V,::Val{G}=Val(1)) where G = compound(metrictensor(V),Val(grade(V)-G))
metrictensor(V::TensorBundle) = TensorOperator(map(Chain,value(metricdyad(V))))
metrictensor(V::Int) = TensorOperator(map(Chain,value(metricdyad(V))))
metricdyad(V::TensorBundle) = metricdyad(Submanifold(V))
metricdyad(V::Int) = metricdyad(Submanifold(V))
function metricdyad(V)
if hasconformal(V)
N = mdims(V)
TensorOperator(Chain{V}(Single{V}((UInt(2),-1)),Single{V}((UInt(1),-1)),[Single{V}((UInt(1)<<i,1)) for i ∈ list(2,N-1)]...))
else
cayley(V,1,(x,y)->value(contraction(x,y)))
end
end

metricfull(V) = TensorOperator(Multivector{V}(vcat(value.(map.(Multivector,value.(metrictensor.(V,list(0,mdims(V))))))...)))
metriceven(V) = TensorOperator(Spinor{V}(vcat(value.(map.(Spinor,value.(metrictensor.(V,evens(0,mdims(V))))))...)))
metricodd(V) = TensorOperator(AntiSpinor{V}(vcat(value.(map.(AntiSpinor,value.(metrictensor.(V,evens(1,mdims(V))))))...)))

struct MetricTensor{n,ℙ,g,Vars,Diff,Name} <: TensorBundle{n,ℙ,g,Vars,Diff,Name}
@pure MetricTensor{N,M,S,F,D,L}() where {N,M,S,F,D,L} = new{N,M,S,F,D,L}()
end

export MetricTensor, metrictensor
@pure MetricTensor{N,M,S,F,D}() where {N,M,S,F,D} = MetricTensor{N,M,S,F,D,1}()
@pure MetricTensor{N,M,S}() where {N,M,S} = MetricTensor{N,M,S,0,0}()
@pure MetricTensor{N,M}(b::Values{N,<:Tuple}) where {N,M} = MetricTensor{N,M,metricsig(M,Values.(b))}()
@pure MetricTensor{N,M}(b::Values{N,<:Values}) where {N,M} = MetricTensor{N,M,metricsig(M,b)}()
@pure MetricTensor{N,M}(b::Values{N,<:Chain}) where {N,M} = MetricTensor{N,M,metricsig(M,value.(b))}()
@pure MetricTensor{N,M}(b::Values{N,<:AbstractVector}) where {N,M} = MetricTensor{N,M,metricsig(M,Values{N}.(b))}()
MetricTensor{N,M}(b::Vector) where {N,M} = MetricTensor{N,M}(Values(b...))
@pure MetricTensor(b::Tuple) = MetricTensor(Values(b))
@pure MetricTensor(b::Values{N}) where N = MetricTensor{N,0}(b)
MetricTensor(b::Values{N,<:Real}) where N = DiagonalForm(b)
MetricTensor(b::AbstractVector{<:Real}) = DiagonalForm(b)
MetricTensor(b::AbstractVector) = MetricTensor{length(b),0}(b)
MetricTensor(b::AbstractMatrix) = MetricTensor(getindex.(Ref(b),:,1:(size(b)[1])))
MetricTensor(b::Chain{V,G,<:Chain} where {V,G}) = MetricTensor(value(b))
MetricTensor(b::Chain) = DiagonalForm(value(b))
MetricTensor(b::TensorOperator) = MetricTensor(value(b))
MetricTensor(b...) = MetricTensor(b)

@pure Manifold(::Type{T}) where T<:MetricTensor = T()

@pure metrictensor(b::Submanifold{V}) where V = isbasis(b) ? metrictensor(V) : TensorOperator(map(b,b(value(metrictensor(V)))))
@pure metrictensor(V,G) = compound(metrictensor(V),G)
@pure function metrictensor(V::MetricTensor{N,M,S} where N) where {M,S}
out = Chain{Submanifold(V)}.(metrictensor_cache[S])
TensorOperator(Chain{Submanifold(V)}(isdual(V) ? SUB(out) : out))
end
const metrictensor_cache = Values[]
@pure function metricsig(M,b::Values)
a = dyadmode(M)>0 ? SUB(b) : b
if a ∈ metrictensor_cache
findfirst(x->x==a,metrictensor_cache)
else
push!(metrictensor_cache,a)
length(metrictensor_cache)
end
end

@pure DirectSum.getalgebra(V::MetricTensor) = DirectSum.getalgebra(Submanifold(V))

for t ∈ (Any,Integer)
@eval @inline getindex(s::MetricTensor{N,M,S} where {N,M},i::T) where {S,T<:$t} = value(value(metrictensor(s))[i])
end
@inline getindex(vs::MetricTensor,i::Vector) = [getindex(vs,j) for j ∈ i]
@inline getindex(vs::MetricTensor,i::UnitRange{Int}) = [getindex(vs,j) for j ∈ i]
@inline getindex(vs::MetricTensor{N,M,S} where M,i::Colon) where {N,S} = Vector(value(metrictensor(vs)))

@pure Signature(V::MetricTensor{N,M}) where {N,M} = Signature{N,M,UInt(0)}()

# anything array-like gets summarized e.g. 10-element Array{Int64,1}
Base.summary(io::IO, a::MetricTensor) = Base.array_summary(io, a, _axes(metrictensor(a)))

show(io::IO,M::MetricTensor) = Base.show(io,Submanifold(M))
function show(io::IO, ::MIME"text/plain", M::MetricTensor)
X = display_matrix(value(metrictensor(M)))
if isempty(X) && get(io, :compact, false)::Bool
return show(io, X)
end
show_matrix(io,M,X)
end

(M::MetricTensor)(b::Int...) = Submanifold{M}(b)
(M::MetricTensor)(b::T) where T<:AbstractVector{Int} = Submanifold{M}(b)
(M::MetricTensor)(b::T) where T<:AbstractRange{Int} = Submanifold{M}(b)

import LinearAlgebra: isdiag; export isdiag
isdiag(::MetricTensor) = false
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Registration pull request created: JuliaRegistries/General/107351

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