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composite.jl
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composite.jl
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# This file is part of Grassmann.jl
# It is licensed under the AGPL license
# Grassmann Copyright (C) 2019 Michael Reed
# _ _ _
# | | | | | |
# ___| |__ __ _| | ___ __ __ ___ ____ _| | __ _
# / __| '_ \ / _` | |/ / '__/ _` \ \ / / _` | |/ _` |
# | (__| | | | (_| | <| | | (_| |\ V / (_| | | (_| |
# \___|_| |_|\__,_|_|\_\_| \__,_| \_/ \__,_|_|\__,_|
#
# https://github.com/chakravala
# https://crucialflow.com
export exph, log_fast, logh_fast, pseudoexp, pseudolog, pseudodot, @pseudo
export pseudoabs, pseudoabs2, pseudosqrt, pseudocbrt, pseudoinv, pseudoscalar
export pseudocos, pseudosin, pseudotan, pseudocosh, pseudosinh, pseudotanh
## exponential & logarithm function
@inline Base.expm1(t::Submanifold{V,0}) where V = Single{V}(ℯ-1)
@inline Base.expm1(t::T) where T<:TensorGraded{V,0} where V = Single{Manifold(t)}(AbstractTensors.expm1(value(T<:TensorTerm ? t : scalar(t))))
Base.expm1(t::Chain) = expm1(multispin(t))
function Base.expm1(t::T) where T<:TensorAlgebra
V = Manifold(t)
if T<:Couple
B = basis(t); BB = value(B⟑B)
if BB == -1
return Couple{V,B}(expm1(t.v))
end
end
S,term,f = t,(t⟑t)/2,norm(t)
norms = FixedVector{3}(f,norm(term),f)
k::Int = 3
@inbounds while norms[2]<norms[1] || norms[2]>1
S += term
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term *= t/k
@inbounds norms .= (norms[2],norm(term),ns)
k += 1
end
return S
end
@eval @generated function Base.expm1(b::Multivector{V,T}) where {V,T}
loop = generate_loop_multivector(V,:term,:B,:*,:geomaddmulti!,geomaddmulti!_pre,:k)
return quote
B = value(b)
sb,nb = scalar(b),AbstractTensors.norm(B)
sb ≈ nb && (return Single{V}(AbstractTensors.expm1(value(sb))))
$(insert_expr(loop[1],:mvec,:T,Float64)...)
S = zeros(mvec(N,t))
term = zeros(mvec(N,t))
S .= B
out .= value(b⟑b)/2
norms = FixedVector{3}(nb,norm(out),norm(term))
k::Int = 3
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
S += out
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term .= out
# term *= b/k
$(loop[2])
@inbounds norms .= (norms[2],norm(out),ns)
k += 1
end
return Multivector{V}(S)
end
end
@eval @generated function Base.expm1(b::Spinor{V,T}) where {V,T}
loop = generate_loop_spinor(V,:term,:B,:*,:geomaddspin!,geomaddspin!_pre,:k)
return quote
B = value(b)
sb,nb = scalar(b),AbstractTensors.norm(B)
sb ≈ nb && (return Single{V}(AbstractTensors.expm1(value(sb))))
$(insert_expr(loop[1],:mvecs,:T,Float64)...)
S = zeros(mvecs(N,t))
term = zeros(mvecs(N,t))
S .= B
out .= value(b⟑b)/2
norms = FixedVector{3}(nb,norm(out),norm(term))
k::Int = 3
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
S += out
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term .= out
# term *= b/k
$(loop[2])
@inbounds norms .= (norms[2],norm(out),ns)
k += 1
end
return Spinor{V}(S)
end
end
function Base.exp(t::Multivector{V}) where V
st = scalar(t)
mt = t-st
sq = mt⟑mt
if isscalar(sq)
hint = value(scalar(sq))
isnull(hint) && (return AbstractTensors.exp(value(st))*(One(V)+t))
θ = unabs!(AbstractTensors.sqrt(AbstractTensors.abs(value(scalar(abs2(mt))))))
return AbstractTensors.exp(value(st))*(hint<0 ? AbstractTensors.cos(θ)+mt*(AbstractTensors.:/(AbstractTensors.sin(θ),θ)) : AbstractTensors.cosh(θ)+mt*(AbstractTensors.:/(AbstractTensors.sinh(θ),θ)))
else
return One(V)+expm1(t)
end
end
function Base.exp(t::Spinor{V}) where V
st = scalar(t)
mt = t-st
sq = mt⟑mt
if isscalar(sq)
hint = value(scalar(sq))
isnull(hint) && (return AbstractTensors.exp(value(st))*(One(V)+t))
θ = unabs!(AbstractTensors.sqrt(AbstractTensors.abs(value(scalar(abs2(mt))))))
return AbstractTensors.exp(value(st))*(hint<0 ? AbstractTensors.cos(θ)+mt*(AbstractTensors.:/(AbstractTensors.sin(θ),θ)) : AbstractTensors.cosh(θ)+mt*(AbstractTensors.:/(AbstractTensors.sinh(θ),θ)))
else
return One(V)+expm1(t)
end
end
function Base.exp(t::Multivector{V},::Val{hint}) where {V,hint}
st = scalar(t)
mt = t-st
sq = mt⟑mt
if isscalar(sq)
isnull(hint) && (return AbstractTensors.exp(value(st))*(One(V)+t))
θ = unabs!(AbstractTensors.sqrt(AbstractTensors.abs(value(scalar(abs2(mt))))))
return AbstractTensors.exp(value(st))*(hint<0 ? AbstractTensors.cos(θ)+mt*(AbstractTensors.:/(AbstractTensors.sin(θ),θ)) : AbstractTensors.cosh(θ)+mt*(AbstractTensors.:/(AbstractTensors.sinh(θ),θ)))
else
return One(V)+expm1(t)
end
end
Base.expm1(t::Phasor{V}) where V = exp(t)-One(V)
function Base.exp(t::Phasor{V,B}) where {V,B}
z = exp(angle(t))
Phasor{V,B}(exp(radius(t)+z.v.re),z.v.im)
end
function Base.exp(t::Couple{V,B}) where {V,B}
st,mt = scalar(t),imaginary(t)
if isscalar(B⟑B)
hint = value(scalar(B⟑B))
isnull(hint) && (return AbstractTensors.exp(value(st))*(One(V)+t))
θ = unabs!(AbstractTensors.sqrt(AbstractTensors.abs(value(scalar(abs2(mt))))))
return AbstractTensors.exp(value(st))*(hint<0 ? AbstractTensors.cos(θ)+mt*(AbstractTensors.:/(AbstractTensors.sin(θ),θ)) : AbstractTensors.cosh(θ)+mt*(AbstractTensors.:/(AbstractTensors.sinh(θ),θ)))
else
return One(V)+expm1(t)
end
end
function Base.expm1(t::PseudoCouple{V,B}) where {V,B}
if isscalar(B)
exp(t)-One(V)
else
expm1(multispin(t))
end
end
function Base.exp(t::PseudoCouple{V,B}) where {V,B}
if isscalar(B)
PseudoCouple{V,B}(value(exp(Couple{V,Submanifold(V)}(value(t)))))
else
exp(multispin(t))
end
end
@pure isR301(V::DiagonalForm) = DirectSum.diagonalform(V) == Values(1,1,1,0)
@pure isR301(::Submanifold{V}) where V = isR301(V)
@pure isR301(V) = false
@inline unabs!(t) = t
@inline unabs!(t::Expr) = (t.head == :call && t.args[1] == :abs) ? t.args[2] : t
function Base.exp(t::T) where T<:TensorGraded
S,B,V = T<:Submanifold,T<:TensorTerm,Manifold(t)
if B && isnull(t)
return One(V)
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)
v,cu,su = (t∧t)*(-0.5/u),cos(u),sin(u)
return (cu-v*su)+((su+v*cu)*t)*(inv(u)-v/(u*u))
end # need general inv(u+v) ~ inv(u)-v/u^2
i = B ? basis(t) : t
sq = i⟑i
if isscalar(sq)
hint = value(scalar(sq))
isnull(hint) && (return One(V)+t)
grade(t)==0 && (return Single{Manifold(t)}(AbstractTensors.exp(value(S ? t : scalar(t)))))
θ = unabs!(AbstractTensors.sqrt(AbstractTensors.abs(value(scalar(abs2(t))))))
hint<0 ? AbstractTensors.cos(θ)+t*(S ? AbstractTensors.sin(θ) : AbstractTensors.:/(AbstractTensors.sin(θ),θ)) : AbstractTensors.cosh(θ)+t*(S ? AbstractTensors.sinh(θ) : AbstractTensors.:/(AbstractTensors.sinh(θ),θ))
else
return One(V)+expm1(t)
end
end
function Base.exp(t::T,::Val{hint}) where T<:TensorGraded{V} where {V,hint}
S = T<:Submanifold
i = T<:TensorTerm ? basis(t) : t
sq = i⟑i
if isscalar(sq)
isnull(hint) && (return One(V)+t)
grade(t)==0 && (return Single{V}(AbstractTensors.exp(value(S ? t : scalar(t)))))
θ = unabs!(AbstractTensors.sqrt(AbstractTensors.abs(value(scalar(abs2(t))))))
hint<0 ? AbstractTensors.cos(θ)+t*(S ? AbstractTensors.sin(θ) : AbstractTensors.:/(AbstractTensors.sin(θ),θ)) : AbstractTensors.cosh(θ)+t*(S ? AbstractTensors.sinh(θ) : AbstractTensors.:/(AbstractTensors.sinh(θ),θ))
else
return One(V)+expm1(t)
end
end
qlog(b::PseudoCouple,x::Int=10000) = qlog(multispin(b),x)
qlog(b::AntiSpinor,x::Int=10000) = qlog(Multivector(b),x)
function qlog(w::T,x::Int=10000) where T<:TensorAlgebra
V = Manifold(w)
w2,f = w⟑w,norm(w)
prod = w⟑w2
S,term = w,prod/3
norms = FixedVector{3}(f,norm(term),f)
k::Int = 5
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ x
S += term
ns = norm(S)
@inbounds ns ≈ norms[3] && break
prod *= w2
term = prod/k
@inbounds norms .= (norms[2],norm(term),ns)
k += 2
end
return 2S
end # http://www.netlib.org/cephes/qlibdoc.html#qlog
qlog_fast(b::PseudoCouple,x::Int=10000) = qlog_fast(multispin(b),x)
qlog_fast(b::AntiSpinor,x::Int=10000) = qlog_fast(Multivector(b),x)
@eval @generated function qlog_fast(b::Multivector{V,T,E},x::Int=10000) where {V,T,E}
loop = generate_loop_multivector(V,:prod,:B,:*,:geomaddmulti!,geomaddmulti!_pre)
return quote
$(insert_expr(loop[1],:mvec,:T,Float64)...)
f = norm(b)
w2::Multivector{V,T,E} = b⟑b
B = value(w2)
S = zeros(mvec(N,t))
prod = zeros(mvec(N,t))
term = zeros(mvec(N,t))
S .= value(b)
out .= value(b⟑w2)
term .= out/3
norms = FixedVector{3}(f,norm(term),f)
k::Int = 5
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ x
S += term
ns = norm(S)
@inbounds ns ≈ norms[3] && break
prod .= out
# prod *= w2
$(loop[2])
term .= out/k
@inbounds norms .= (norms[2],norm(term),ns)
k += 2
end
S *= 2
return Multivector{V}(S)
end
end
@eval @generated function qlog_fast(b::Spinor{V,T,E},x::Int=10000) where {V,T,E}
loop = generate_loop_spinor(V,:prod,:B,:*,:geomaddspin!,geomaddspin!_pre)
return quote
$(insert_expr(loop[1],:mvec,:T,Float64)...)
f = norm(b)
w2::Spinor{V,T,E} = b⟑b
B = value(w2)
S = zeros(mvecs(N,t))
prod = zeros(mvecs(N,t))
term = zeros(mvecs(N,t))
S .= value(b)
out .= value(b⟑w2)
term .= out/3
norms = FixedVector{3}(f,norm(term),f)
k::Int = 5
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ x
S += term
ns = norm(S)
@inbounds ns ≈ norms[3] && break
prod .= out
# prod *= w2
$(loop[2])
term .= out/k
@inbounds norms .= (norms[2],norm(term),ns)
k += 2
end
S *= 2
return Spinor{V}(S)
end
end
Base.log(t::TensorTerm) = log(Couple(t))
Base.log(t::Phasor) = (r=radius(t); log(r)+angle(t))
Base.log1p(t::Phasor{V}) where V = log(One(V)+t)
Base.log(t::Couple{V,B}) where {V,B} = value(B*B)==-1 ? Couple{V,B}(log(t.v)) : log(radius(t))+angle(t)
Base.log1p(t::Couple{V,B}) where {V,B} = value(B*B)==-1 ? Couple{V,B}(log1p(t.v)) : log(One(V)+t)
Base.log(t::Quaternion{V}) where V = iszero(metric(V)) ? log(radius(t))+angle(t,r) : qlog((t-One(V))/(t+One(V)))
Base.log1p(t::Quaternion{V}) where V = iszero(metric(V)) ? log(One(V)+t) : qlog(t/(t+2))
@inline Base.log(t::T) where T<:TensorAlgebra{V} where V = qlog((t-One(V))/(t+One(V)))
@inline Base.log1p(t::T) where T<:TensorAlgebra = qlog(t/(t+2))
function Base.log(t::PseudoCouple{V,B}) where {V,B}
if isscalar(B)
PseudoCouple{V,B}(value(log(Couple{V,Submanifold(V)}(value(t)))))
else
log(multispin(t))
end
end
function Base.log1p(t::PseudoCouple{V,B}) where {V,B}
if isscalar(B)
PseudoCouple{V,B}(value(log1p(Couple{V,Submanifold(V)}(value(t)))))
else
log1p(multispin(t))
end
end
Base.exp(t::AntiSpinor) = exp(Multivector(t))
Base.expm1(t::AntiSpinor) = expm1(Multivector(t))
Base.log(t::AntiSpinor) = log(Multivector(t))
Base.log1p(t::AntiSpinor) = log1p(Multivector(t))
log_fast(t::AntiSpinor) = log_fast(Multivector(t))
logh_fast(t::AntiSpinor) = logh_fast(Multivector(t))
for op ∈ (:cosh,:sinh)
@eval begin
Base.$op(t::PseudoCouple) = $op(multispin(t))
Base.$op(t::AntiSpinor) = $op(Multivector(t))
end
end
for op ∈ (:log,:exp,:asin,:acos,:atan,:acot,:sinc,:cosc)
@eval @inline Base.$op(t::T) where T<:TensorGraded{V,0} where V = Single{V}($op(value(t)))
end
for op ∈ (:log,:log2,:log10,:asech,:acosh,:acos,:sinc)
@eval @inline Base.$op(::One{V}) where V = Zero(V)
end
for op ∈ (:atanh,:acoth)
@eval @inline Base.$op(::One{V}) where V = Infinity(V)
end
@inline Base.sinh(::Zero{V}) where V = Zero(V)
for op ∈ (:exp,:exp2,:exp10,:cosh,:sinc) # exp
@eval @inline Base.$op(::Zero{V}) where V = One(V)
end
for op ∈ (:asin,:atan,:asinh,:atanh,:cosc,:sqrt,:cbrt)
@eval @inline Base.$op(t::Zero) = t
end
for op ∈ (:tanh,:coth)
@eval @inline Base.$op(::Infinity{V}) where V = One(V)
end
for op ∈ (:acoth,:acot,:sinc,:cosc)
@eval @inline Base.$op(::Infinity{V}) where V = Zero(V)
end
for op ∈ (:exp,:exp2,:exp10,:log,:log2,:log10,:cosh,:sinh,:acosh,:asinh,:sqrt,:cbrt)
@eval @inline Base.$op(t::Infinity) = t
end
for (qrt,n) ∈ ((:sqrt,2),(:cbrt,3))
@eval begin
@inline function Base.$qrt(t::T) where T<:TensorAlgebra
isscalar(t) ? $qrt(scalar(t)) : exp(log(t)/$n)
end
@inline function Base.$qrt(t::Quaternion{V}) where V
iszero(metric(V)) ? $qrt(radius(t))*exp(angle(t)/$n) : exp(log(t)/$n)
end
@inline function Base.$qrt(t::Couple{V,B}) where {V,B}
value(B*B)==-1 ? Couple{V,B}($qrt(t.v)) :
$qrt(radius(t))*exp(angle(t)/$n)
end
@inline Base.$qrt(t::Phasor) = Phasor($qrt(radius(t)),angle(t)/$n)
@inline Base.$qrt(t::Submanifold{V,0} where V) = t
@inline Base.$qrt(t::T) where T<:TensorGraded{V,0} where V = Single{V}($Sym.$qrt(value(T<:TensorTerm ? t : scalar(t))))
end
end
## trigonometric
@inline Base.cosh(t::T) where T<:TensorGraded{V,0} where V = Single{Manifold(t)}(AbstractTensors.cosh(value(T<:TensorTerm ? t : scalar(t))))
function Base.cosh(t::T) where T<:TensorAlgebra
V = Manifold(t)
if T<:Couple
B = basis(t); BB = value(B⟑B)
if BB == -1
return Couple{V,B}(cosh(t.v))
end
end
τ = t⟑t
S,term = τ/2,(τ⟑τ)/24
f = norm(S)
norms = FixedVector{3}(f,norm(term),f)
k::Int = 6
@inbounds while norms[2]<norms[1] || norms[2]>1
S += term
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term *= τ/(k*(k-1))
@inbounds norms .= (norms[2],norm(term),ns)
k += 2
end
return One(V)+S
end
@eval @generated function Base.cosh(b::Multivector{V,T,E}) where {V,T,E}
loop = generate_loop_multivector(V,:term,:B,:*,:geomaddmulti!,geomaddmulti!_pre,:(k*(k-1)))
return quote
sb,nb = scalar(b),norm(b)
sb ≈ nb && (return Single{V}(AbstractTensors.cosh(value(sb))))
$(insert_expr(loop[1],:mvec,:T,Float64)...)
τ::Multivector{V,T,E} = b⟑b
B = value(τ)
S = zeros(mvec(N,t))
term = zeros(mvec(N,t))
S .= value(τ)/2
out .= value((τ⟑τ))/24
norms = FixedVector{3}(norm(S),norm(out),norm(term))
k::Int = 6
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
S += out
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term .= out
# term *= τ/(k*(k-1))
$(loop[2])
@inbounds norms .= (norms[2],norm(out),ns)
k += 2
end
@inbounds S[1] += One(V)
return Multivector{V}(S)
end
end
@eval @generated function Base.cosh(b::Spinor{V,T,E}) where {V,T,E}
loop = generate_loop_spinor(V,:term,:B,:*,:geomaddspin!,geomaddspin!_pre,:(k*(k-1)))
return quote
sb,nb = scalar(b),norm(b)
sb ≈ nb && (return Single{V}(AbstractTensors.cosh(value(sb))))
$(insert_expr(loop[1],:mvecs,:T,Float64)...)
τ::Spinor{V,T,E} = b⟑b
B = value(τ)
S = zeros(mvecs(N,t))
term = zeros(mvecs(N,t))
S .= value(τ)/2
out .= value((τ⟑τ))/24
norms = FixedVector{3}(norm(S),norm(out),norm(term))
k::Int = 6
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
S += out
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term .= out
# term *= τ/(k*(k-1))
$(loop[2])
@inbounds norms .= (norms[2],norm(out),ns)
k += 2
end
@inbounds S[1] += One(V)
return Spinor{V}(S)
end
end
@inline Base.sinh(t::T) where T<:TensorGraded{V,0} where V = Single{Manifold(t)}(AbstractTensors.sinh(value(T<:TensorTerm ? t : scalar(t))))
function Base.sinh(t::T) where T<:TensorAlgebra
V = Manifold(t)
if T<:Couple
B = basis(t); BB = value(B⟑B)
if BB == -1
return Couple{V,B}(sinh(t.v))
end
end
τ,f = t⟑t,norm(t)
S,term = t,(t⟑τ)/6
norms = FixedVector{3}(f,norm(term),f)
k::Int = 5
@inbounds while norms[2]<norms[1] || norms[2]>1
S += term
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term *= τ/(k*(k-1))
@inbounds norms .= (norms[2],norm(term),ns)
k += 2
end
return S
end
@eval @generated function Base.sinh(b::Multivector{V,T,E}) where {V,T,E}
loop = generate_loop_multivector(V,:term,:B,:*,:geomaddmulti!,geomaddmulti!_pre,:(k*(k-1)))
return quote
sb,nb = scalar(b),norm(b)
sb ≈ nb && (return Single{V}(AbstractTensors.sinh(value(sb))))
$(insert_expr(loop[1],:mvec,:T,Float64)...)
τ::Multivector{V,T,E} = b⟑b
B = value(τ)
S = zeros(mvec(N,t))
term = zeros(mvec(N,t))
S .= value(b)
out .= value(b⟑τ)/6
norms = FixedVector{3}(norm(S),norm(out),norm(term))
k::Int = 5
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
S += out
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term .= out
# term *= τ/(k*(k-1))
$(loop[2])
@inbounds norms .= (norms[2],norm(out),ns)
k += 2
end
return Multivector{V}(S)
end
end
@eval @generated function Base.sinh(b::Spinor{V,T,E}) where {V,T,E}
loop = generate_loop_spinor(V,:term,:B,:*,:geomaddspin!,geomaddspin!_pre,:(k*(k-1)))
return quote
sb,nb = scalar(b),norm(b)
sb ≈ nb && (return Single{V}(AbstractTensors.sinh(value(sb))))
$(insert_expr(loop[1],:mvecs,:T,Float64)...)
τ::Spinor{V,T,E} = b⟑b
B = value(τ)
S = zeros(mvecs(N,t))
term = zeros(mvecs(N,t))
S .= value(b)
out .= value(b⟑τ)/6
norms = FixedVector{3}(norm(S),norm(out),norm(term))
k::Int = 5
@inbounds while (norms[2]<norms[1] || norms[2]>1) && k ≤ 10000
S += out
ns = norm(S)
@inbounds ns ≈ norms[3] && break
term .= out
# term *= τ/(k*(k-1))
$(loop[2])
@inbounds norms .= (norms[2],norm(out),ns)
k += 2
end
return Spinor{V}(S)
end
end
exph(t) = Base.cosh(t)+Base.sinh(t)
for (logfast,expf) ∈ ((:log_fast,:exp),(:logh_fast,:exph))
@eval function $logfast(t::T) where T<:TensorAlgebra
V = Manifold(t)
term = Zero(V)
nrm = FixedVector{2}(0.,0.)
while true
en = $expf(term)
term -= 2(en-t)/(en+t)
@inbounds nrm .= (nrm[2],norm(term))
@inbounds nrm[1] ≈ nrm[2] && break
end
return term
end
end
#=function log(t::T) where T<:TensorAlgebra
V = Manifold(t)
norms::Tuple = (norm(t),0)
k::Int = 3
τ = t-1
if true #norms[1] ≤ 5/4
prods = τ^2
terms = TensorAlgebra{V}[τ,prods/2]
norms = (norms[1],norm(terms[2]))
while (norms[2]<norms[1] || norms[2]>1) && k ≤ 3000
prods = prods*t
push!(terms,prods/(k*(-1)^(k+1)))
norms = (norms[2],norm(terms[end]))
k += 1
end
else
s = inv(t*inv(τ))
prods = s^2
terms = TensorAlgebra{V}[s,2prods]
norms = (norm(terms[1]),norm(terms[2]))
while (norms[2]<norms[1] || norms[2]>1) && k ≤ 3000
prods = prods*s
push!(terms,k*prods)
norms = (norms[2],norm(terms[end]))
k += 1
end
end
return sum(terms[1:end-1])
end=#
function Base.angle(z::Couple{V,B}) where {V,B}
c = value(z)
if value(B^2) == -1
atan(imag(c),real(c))*B
elseif value(B^2) == 1
atanh(imag(c),real(c))*B
else
error("Unsupported trigonometric angle")
end
end
radius(z::Quaternion) = value(scalar(abs(z)))
function Base.angle(z::Quaternion,r=radius(z))
b = bivector(z)
(acos(value(scalar(z))/r)/value(abs(b)))*b
end
Base.atanh(y::Real, x::Real) = atanh(promote(float(y),float(x))...)
Base.atanh(y::T, x::T) where {T<:AbstractFloat} = Base.no_op_err("atanh", T)
function Base.atanh(y::T, x::T) where T<:Union{Float32, Float64}
# Method :
# M1) Reduce y to positive by atanh2(y,x)=-atanh2(-y,x).
# M2) Reduce x to positive by (if x and y are unexceptional):
# ARGH (x+iy) = arctanh(y/x) ... if x > 0,
# ARGH (x+iy) = arctanh[y/(-x)] ... if x < 0,
#
# Special cases:
#
# S1) ATANH2((anything), NaN ) is NaN;
# S2) ATANH2(NAN , (anything) ) is NaN;
# S3) ATANH2(+-0, +-(anything but NaN)) is +-0;
# S4) ATANH2(+-(anything but 0 and NaN), 0) is ERROR;
# S5) ATANH2(+-(anything but INF and NaN), +-INF) is +-0;
# S6) ATANH2(+-INF,+-INF ) is +-Inf;
# S7) ATANH2(+-INF, (anything but,0,NaN, and INF)) is ERROR;
if isnan(x) || isnan(y) # S1 or S2
return T(NaN)
end
if x == T(1.0) || x == T(-1.0) # then y/x = y, see M2
return atanh(y)
end
# generate an m ∈ {0, 1, 2, 3} to branch off of
m = 2*signbit(x) + 1*signbit(y)
if iszero(y)
return y # atanh(+-0, +-anything) = +-0
elseif iszero(x)
return atanh(copysign(T(Inf), y))
end
if isinf(x)
if isinf(y)
return y # atanh(+-Inf, +-Inf)
else
if m == 0 || m == 2
return zero(T) # atanh(+...,+-Inf) */
elseif m == 1 || m == 3
return -zero(T) # atanh(-...,+-Inf) */
end
end
end
# x wasn't Inf, but y is
isinf(y) && return atanh(y)
ypw = Base.Math.poshighword(y)
xpw = Base.Math.poshighword(x)
# compute y/x for Float32
k = reinterpret(Int32, ypw-xpw)>>Base.Math.ATAN2_RATIO_BIT_SHIFT(T)
if k > Base.Math.ATAN2_RATIO_THRESHOLD(T) # |y/x| > threshold
z=T(pi)/2+T(0.5)*Base.Math.ATAN2_PI_LO(T)
m&=1;
elseif x<0 && k < -Base.Math.ATAN2_RATIO_THRESHOLD(T) # 0 > |y|/x > threshold
z = zero(T)
else #safe to do y/x
z = atanh(abs(y/x))
end
if m == 0 || m == 2
return z # atanh(+,+-)
else # default case m == 1 || m == 3
return -z # atanh(-,+-)
end
end
function Cramer(N::Int,j=0)
t = j ≠ 0 ? :T : :t
x,y = Values{N}([Symbol(:x,i) for i ∈ list(1,N)]),Values{N}([Symbol(:y,i) for i ∈ list(1,N)])
xy = [:(($(x[1+i]),$(y[1+i])) = ($(x[i])∧$t[$(1+i-j)],$t[end-$i]∧$(y[i]))) for i ∈ list(1,N-1-j)]
return x,y,xy
end
DirectSum.Λ(x::Chain{V,1,<:Chain{V,1}},G) where V = compound(x,G)
compound(x,G::T) where T<:Integer = compound(x,Val(G))
@generated function compound(x::Chain{V,1,<:Chain{V,1}},::Val{G}) where {V,G}
Expr(:call,:(Chain{V,G}),Expr(:call,:Values,[Expr(:call,:∧,[:(@inbounds x[$i]) for i ∈ indices(j)]...) for j ∈ indexbasis(mdims(V),G)]...))
end
@generated function Base.:\(t::Values{M,<:Chain{V,1}},v::Chain{V,1}) where {M,V}
W = M≠mdims(V) ? Submanifold(M) : V; N = M-1
if M == 1 && (V === ℝ1 || V == 1)
return :(@inbounds Chain{V,1}(Values(v[1]/t[1][1])))
elseif M == 2 && (V === ℝ2 || V == 2)
return quote
@inbounds (a,A),(b,B),(c,C) = value(t[1]),value(t[2]),value(v)
x1 = (c-C*(b/B))/(a-A*(b/B))
return Chain{V,1}(x1,(C-A*x1)/B)
end
elseif M == 3 && (V === ℝ3 || V == 3)
return quote
dv = @inbounds v/∧(t)[1]; c1,c2,c3 = value(t)
return @inbounds Chain{V,1}(
(c2[2]*c3[3] - c3[2]*c2[3])*dv[1] +
(c3[1]*c2[3] - c2[1]*c3[3])*dv[2] +
(c2[1]*c3[2] - c3[1]*c2[2])*dv[3],
(c3[2]*c1[3] - c1[2]*c3[3])*dv[1] +
(c1[1]*c3[3] - c3[1]*c1[3])*dv[2] +
(c3[1]*c1[2] - c1[1]*c3[2])*dv[3],
(c1[2]*c2[3] - c2[2]*c1[3])*dv[1] +
(c2[1]*c1[3] - c1[1]*c2[3])*dv[2] +
(c1[1]*c2[2] - c2[1]*c1[2])*dv[3])
end
end
N<1 && (return :(inv(t)⋅v))
M > mdims(V) && (return :(tt=_transpose(t,$W); tt⋅(inv(Chain{$W,1}(t)⋅tt)⋅v)))
x,y,xy = Grassmann.Cramer(N) # paste this into the REPL for faster eval
mid = [:($(x[i])∧v∧$(y[end-i])) for i ∈ list(1,N-1)]
out = Expr(:call,:Values,:(v∧$(y[end])),mid...,:($(x[end])∧v))
detx = :(detx = @inbounds (t[1]∧$(y[end])))
return Expr(:block,:((x1,y1)=@inbounds (t[1],t[end])),xy...,detx,
:(Chain{$W,1}(column($(Expr(:call,:.⋅,out,:(Ref(detx))))./abs2(detx)))))
end
@generated function Base.in(v::Chain{V,1},t::Values{N,<:Chain{V,1}}) where {V,N}
if N == mdims(V)
x,y,xy = Grassmann.Cramer(N-1)
mid = [:(s==signbit(@inbounds ($(x[i])∧v∧$(y[end-i]))[1])) for i ∈ list(1,N-2)]
out = Values(:(s==signbit(@inbounds (v∧$(y[end]))[1])),mid...,:(s==signbit(@inbounds ($(x[end])∧v)[1])))
return Expr(:block,:((x1,y1)=@inbounds (t[1],t[end])),xy...,:(s=signbit(@inbounds (t[1]∧$(y[end]))[1])),ands(out))
else
x,y,xy = Grassmann.Cramer(N-1,1)
mid = [:(signscalar(($(x[i])∧(v-x1)∧$(y[end-i]))/d)) for i ∈ list(1,N-2)]
out = Values(:(signscalar((v∧∧(vectors(t,v)))/d)),mid...,:(signscalar(($(x[end])∧(v-x1))/d)))
return Expr(:block,:(T=vectors(t)),:((x1,y1)=@inbounds (t[1],T[end])),xy...,
:($(x[end])=$(x[end-1])∧T[end-1];d=$(x[end])∧T[end]),ands(out))
end
end
@generated function Base.inv(t::Values{M,<:Chain{V,1}}) where {M,V}
W = M≠mdims(V) ? Submanifold(M) : V; N = M-1
N<1 && (return :(_transpose(Values(inv(@inbounds t[1])),$W)))
M > mdims(V) && (return :(tt = _transpose(t,$W); tt⋅inv(Chain{$W,1}(t)⋅tt)))
x,y,xy = Grassmann.Cramer(N)
val = if iseven(N)
Expr(:call,:Values,y[end],[:($(y[end-i])∧$(x[i])) for i ∈ list(1,N-1)]...,x[end])
elseif M≠mdims(V)
Expr(:call,:Values,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ list(1,N-1)]...,:(-$(x[end])))
else
Expr(:call,:Values,:(-$(y[end])),[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ list(1,N-1)]...,x[end])
end
out = if M≠mdims(V)
:(vector.($(Expr(:call,:./,val,:(@inbounds (t[1]∧$(y[end])))))))
else
:(.⋆($(Expr(:call,:./,val,:(@inbounds (t[1]∧$(y[end]))[1])))))
end
return Expr(:block,:((x1,y1)=@inbounds (t[1],t[end])),xy...,:(_transpose($out,$W)))
end
@generated function grad(T::Values{M,<:Chain{V,1}}) where {M,V}
W = M≠mdims(V) ? Submanifold(M) : V; N = mdims(V)-1
M < mdims(V) && (return :(ct = Chain{$W,1}(T); map(↓(V),ct⋅inv(_transpose(T,$W)⋅ct))))
x,y,xy = Grassmann.Cramer(N)
val = if iseven(N)
Expr(:call,:Values,[:($(y[end-i])∧$(x[i])) for i ∈ list(1,N-1)]...,x[end])
elseif M≠mdims(V)
Expr(:call,:Values,y[end],[:($(iseven(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ list(1,N-1)]...,:(-$(x[end])))
else
Expr(:call,:Values,[:($(isodd(i) ? :+ : :-)($(y[end-i])∧$(x[i]))) for i ∈ list(1,N-1)]...,x[end])
end
out = if M≠mdims(V)
:(vector.($(Expr(:call,:./,val,:(@inbounds (t[1]∧$(y[end])))))))
else
:(.⋆($(Expr(:call,:./,val,:(@inbounds (t[1]∧$(y[end]))[1])))))
end
return Expr(:block,:(t=_transpose(T,$W)),:((x1,y1)=@inbounds (t[1],t[end])),xy...,:(_transpose($out,↓(V))))
end
@generated function Base.:\(t::Values{N,<:Chain{M,1}},v::Chain{V,1}) where {N,M,V}
W = M≠mdims(V) ? Submanifold(N) : V
if mdims(M) > mdims(V)
:(ct=Chain{$W,1}(t); ct⋅(inv(_transpose(t,$W)⋅ct)⋅v))
else # mdims(M) < mdims(V) ? inv(tt⋅t)⋅(tt⋅v) : tt⋅(inv(t⋅tt)⋅v)
:(_transpose(t,$W)\v)
end
end
function inv_approx(t::Chain{M,1,<:Chain{V,1}}) where {M,V}
tt = transpose(t)
mdims(M) < mdims(V) ? (inv(tt⋅t))⋅tt : tt⋅inv(t⋅tt)
end
Base.:\(t::Chain{M,1,<:Chain{W,1}},v::Chain{V,1}) where {M,W,V} = value(t)\v
Base.in(v::Chain{V,1},t::Chain{W,1,<:Chain{V,1}}) where {V,W} = v ∈ value(t)
Base.inv(t::Chain{V,1,<:Chain{W,1}}) where {W,V} = inv(value(t))
grad(t::Chain{V,1,<:Chain{W,1}}) where {V,W} = grad(value(t))
export vandermonde
@generated approx(x,y::Chain{V}) where V = :(polynom(x,$(Val(mdims(V))))⋅y)
approx(x,y::Values{N}) where N = value(polynom(x,Val(N)))⋅y
approx(x,y::AbstractVector) = [x^i for i ∈ 0:length(y)-1]⋅y
vandermonde(x::Array,y::Array,N::Int) = vandermonde(x,N)\y[:] # compute ((inv(X'*X))*X')*y
function vandermonde(x::Array,N)
V = zeros(length(x),N)
for d ∈ list(0,N-1)
V[:,d+1] = x.^d
end
return V # Vandermonde
end
vandermonde(x,y,V) = (length(x)≠mdims(V) ? _vandermonde(x,V) : vandermonde(x,V))\y
vandermonde(x,V) = transpose(_vandermonde(x,V))
_vandermonde(x::Chain,V) = _vandermonde(value(x),V)
@generated _vandermonde(x::Values{N},V) where N = :(Chain{$(Submanifold(N)),1}(polynom.(x,$(Val(mdims(V))))))
@generated polynom(x,::Val{N}) where N = Expr(:call,:(Chain{$(Submanifold(N)),1}),Expr(:call,:Values,[:(x^$i) for i ∈ list(0,N-1)]...))
function vandermondeinterp(x,y,V,grid) # grid=384
coef = vandermonde(x,y,V) # Vandermonde ((inv(X'*X))*X')*y
minx,maxx = minimum(x),maximum(x)
xp,yp = [minx:(maxx-minx)/grid:maxx...],coef[1]*ones(grid+1)
for d ∈ 1:length(coef)-1
yp += coef[d+1].*xp.^d
end # fill in polynomial terms
return coef,xp,yp # coefficients, interpolation
end
@generated function vectors(t,c=columns(t))
v = Expr(:tuple,[:(M.(p[c[$i]]-A)) for i ∈ list(2,mdims(t))]...)
quote
p = points(t)
M,A = ↓(Manifold(p)),p[c[1]]
Chain{M,1}.($(Expr(:.,:Values,v)))
end
end
@pure list(a::Int,b::Int) = Values{max(0,b-a+1),Int}(a:b...)
@pure evens(a::Int,b::Int) = Values{((b-a)÷2)+1,Int}(a:2:b...)
vectors(x::Values{N,<:Chain{V}},y=x[1]) where {N,V} = ↓(V).(x[list(2,N)].-y)
vectors(x::Chain{V,1},y=x[1]) where V = vectors(value(x),y)
#point(x,y=x[1]) = y∧∧(vectors(x))
signscalar(x::Submanifold{V,0} where V) = true
signscalar(x::Single{V,0} where V) = !signbit(value(x))
signscalar(x::Single) = false
signscalar(x::Chain) = false
signscalar(x::Chain{V,0} where V) = !signbit(@inbounds x[1])
signscalar(x::Multivector) = isscalar(x) && !signbit(value(scalar(x)))
ands(x,i=length(x)-1) = i ≠ 0 ? Expr(:&&,x[end-i],ands(x,i-1)) : x[end-i]
function Base.findfirst(P,t::Vector{<:Chain{V,1,<:Chain}} where V)
for i ∈ 1:length(t)
@inbounds P ∈ t[i] && (return i)
end
return 0
end
function Base.findfirst(P,t::ChainBundle)
p = points(t)
for i ∈ 1:length(t)
P ∈ p[t[i]] && (return i)
end
return 0
end
function Base.findlast(P,t::Vector{<:Chain{V,1,<:Chain}} where V)
for i ∈ length(t):-1:1
@inbounds P ∈ t[i] && (return i)
end
return 0
end
function Base.findlast(P,t::ChainBundle)
p = points(t)
for i ∈ length(t):-1:1
P ∈ p[t[i]] && (return i)
end
return 0
end
Base.findall(P,t) = findall(P .∈ getindex.(points(t),value(t)))
export volumes, detsimplex, initmesh, refinemesh, refinemesh!, select, submesh
edgelength(e) = (v=points(e)[value(e)]; value(abs(v[2]-v[1])))
volumes(m,dets) = value.(abs.(.⋆(dets)))
volumes(m) = mdims(Manifold(m))≠2 ? volumes(m,detsimplex(m)) : edgelength.(value(m))
detsimplex(m::Vector{<:Chain{V}}) where V = ∧(m)/factorial(mdims(V)-1)
detsimplex(m::ChainBundle) = detsimplex(value(m))
mean(m::T) where T<:AbstractVector{<:Chain} = sum(m)/length(m)
mean(m::T) where T<:Values = sum(m)/length(m)
mean(m::Chain{V,1,<:Chain} where V) = mean(value(m))
barycenter(m::Values{N,<:Chain}) where N = (s=sum(m);@inbounds s/s[1])
barycenter(m::Vector{<:Chain}) = (s=sum(m);@inbounds s/s[1])
barycenter(m::Chain{V,1,<:Chain} where V) = barycenter(value(m))
curl(m::FixedVector{N,<:Chain{V}} where N) where V = curl(Chain{V,1}(m))
curl(m::Values{N,<:Chain{V}} where N) where V = curl(Chain{V,1}(m))
curl(m::T) where T<:TensorAlgebra = Manifold(m)(∇)×m
LinearAlgebra.det(t::Chain{V,1,<:Chain} where V) = ∧(t)
LinearAlgebra.det(m::Vector{<:Chain{V}}) where V = ∧(m)
LinearAlgebra.det(m::ChainBundle) = ∧(m)
∧(m::ChainBundle) = ∧(value(m))
function ∧(m::Vector{<:Chain{V}}) where V
p = points(m); pm = p[m]
if mdims(p)>mdims(V)
.∧(vectors.(pm))
else
Chain{↓(Manifold(V)),mdims(V)-1}.(value.(.∧(pm)))
end
end
for op ∈ (:mean,:barycenter,:curl)
ops = Symbol(op,:s)
@eval begin
export $op, $ops
$ops(m::Vector{<:Chain{p}}) where p = $ops(m,p)
@pure $ops(m::ChainBundle{p}) where p = $ops(m,p)
@pure $ops(m,::Submanifold{p}) where p = $ops(m,p)
@pure $ops(m,p) = $op.(getindex.(Ref(p),value.(value(m))))
end
end
function area(m::Vector{<:Chain})
S = m[end]∧m[1]
for i ∈ 1:length(m)-1
S += m[i]∧m[i+1]
end
return value(abs(⋆(S))/2)
end
initedges(p::ChainBundle) = Chain{p,1}.(1:length(p)-1,2:length(p))
initedges(r::R) where R<:AbstractVector = initedges(ChainBundle(initpoints(r)))
function initmesh(r::R) where R<:AbstractVector
t = initedges(r); p = points(t)
p,ChainBundle(Chain{↓(p),1}.([1,length(p)])),t
end
select(η,ϵ=sqrt(norm(η)^2/length(η))) = sort!(findall(x->x>ϵ,η))
refinemesh(g::R,args...) where R<:AbstractRange = (g,initmesh(g,args...)...)
function refinemesh!(::R,p::ChainBundle{W},e,t,η,_=nothing) where {W,R<:AbstractRange}
p = points(t)
x,T,V = value(p),value(t),Manifold(p)
for i ∈ η
push!(x,Chain{V,1}(Values(1,(x[i+1][2]+x[i][2])/2)))
end
sort!(x,by=x->x[2]); submesh!(p)
e[end] = Chain{p(2),1}(Values(length(x)))
for i ∈ length(t)+2:length(x)
push!(T,Chain{p,1}(Values{2,Int}(i-1,i)))
end
end
const array_cache = (Array{T,2} where T)[]
array(m::Vector{<:Chain}) = [m[i][j] for i∈1:length(m),j∈1:mdims(Manifold(m))]
function array(m::ChainBundle{V,G,T,B} where {V,G,T}) where B
for k ∈ length(array_cache):B
push!(array_cache,Array{Any,2}(undef,0,0))
end
isempty(array_cache[B]) && (array_cache[B] = array(value(m)))
return array_cache[B]
end
function array!(m::ChainBundle{V,G,T,B} where {V,G,T}) where B
length(array_cache) ≥ B && (array_cache[B] = Array{Any,2}(undef,0,0))
end
const submesh_cache = (Array{T,2} where T)[]
submesh(m) = [m[i][j] for i∈1:length(m),j∈2:mdims(Manifold(m))]
function submesh(m::ChainBundle{V,G,T,B} where {V,G,T}) where B
for k ∈ length(submesh_cache):B
push!(submesh_cache,Array{Any,2}(undef,0,0))
end
isempty(submesh_cache[B]) && (submesh_cache[B] = submesh(value(m)))
return submesh_cache[B]
end
function submesh!(m::ChainBundle{V,G,T,B} where {V,G,T}) where B
length(submesh_cache) ≥ B && (submesh_cache[B] = Array{Any,2}(undef,0,0))
end
for op ∈ (:div,:rem,:mod,:mod1,:fld,:fld1,:cld,:ldexp)
@eval begin
Base.$op(a::Couple{V,B},m) where {V,B} = Couple{V,B}($op(value(a),m))
Base.$op(a::PseudoCouple{V,B},m) where {V,B} = PseudoCouple{V,B}($op(value(a),m))
Base.$op(a::Chain{V,G,T},m) where {V,G,T} = Chain{V,G}($op.(value(a),m))
Base.$op(a::Spinor{V,T},m) where {T,V} = Spinor{V}($op.(value(a),m))
Base.$op(a::AntiSpinor{V,T},m) where {T,V} = AntiSpinor{V}($op.(value(a),m))
Base.$op(a::Multivector{V,T},m) where {T,V} = Multivector{V}($op.(value(a),m))
end
end
for op ∈ (:mod2pi,:rem2pi,:rad2deg,:deg2rad,:round)
@eval begin
Base.$op(a::Couple{V,B};args...) where {V,B} = Couple{V,B}($op(value(a);args...))