Exported symbols from CliffordAlgebras.jl:

Base.:*Method
a * b

Calculates the geometric product of two MultiVectors a and b.

Base.:/Method
a / b
(/)(a::MultiVector{CA}, b::MuliVector{CA}) where CA

Calculates the MultiVector quotient a/b by evaluating a*inv(b).

Base.:\Method
b \ a
(\)(b::MultiVector{CA}, b::MuliVector{CA}) where CA

Calculates the MultiVector quotient a/b by evaluating inv(b)*a.

Base.:~Method
~a
(~)(::MultiVector)

Returns the reversed MultiVector reverse(a).

Base.conjMethod
conj(mv::MultiVector)

Return the conjugate of the MultiVector, i.e. reverse(grin(mv)).

Base.invMethod
inv(::MultiVector)

Finds the inverse of the MultiVector. If no inverse exists a SingularException is thrown.

Base.isapproxMethod
isapprox(mv1::MultiVector, mv2::MultiVector; kw...)

Check if mv1 and mv2 belong to the same algebra and their coefficients are close.

Base.reverseMethod
reverse(::MultiVector)

Returns the MultiVector that has all the basis vector products reversed.

CliffordAlgebras.algebraMethod
algebra(::MultiVector)
algebra(::Type{<:MultiVector})

Returns the CliffordAlgebra instance to which the MultiVector belongs.

CliffordAlgebras.basesymbolMethod
basesymbol(::CliffordAlgebra, n::Integer)
basesymbol(::Type{<:CliffordAlgebra}, n::Integer)

Returns the symbol used for the n-th basis multivector of the algebra.

CliffordAlgebras.basevectorMethod
basevector(::CliffordAlgebra, n::Integer)
basevector(::Type{<:CliffordAlgebra}, n::Integer)

Returns the n-th basis MultiVector of the given CliffordAlgebra.

CliffordAlgebras.basevectorMethod
basevector(::CliffordAlgebra, name::Symbol)
basevector(::Type{<:CliffordAlgebra}, name::Symbol)

Returns the basis MultiVector with the specified name from the given Clifford Algebra.

CliffordAlgebras.cayleytableMethod
caleytable(io::IO, ca::CliffordAlgebra)
caleytable(io::IO, CA::Type{<:CliffordAlgebra})

Generates a Cayley table view of the algebra.

CliffordAlgebras.characterMethod
character(::CliffordAlgebra)
character(::Type{<:CliffordAlgebra})

Returns the square of the pseudoscalar of the algebra.

CliffordAlgebras.coefficientMethod
coefficient(::MultiVector, n::Integer)

Returns the multivector coefficients for the n-th basis vector. Returns 0 if n is out of bounds.

CliffordAlgebras.coefficientMethod
coefficient(::MultiVector, s::Symbol)

Returns the multivector coefficients for the basis vector belonging to the symbol s. Returns 0 if the symbol is not a valid basis symbol.

CliffordAlgebras.dimensionMethod
dimension(::CliffordAlgebra)
dimension(::Type{<:CliffordAlgebra})

Returns the dimension of the algebra, i.e. the number of coefficients in a general multivector.

CliffordAlgebras.dualMethod
dual(mv::MultiVector)

Returns the Poincaré dual of the MultiVector, such that for all basis MultiVectors mv * dual(mv) = pseudoscalar. Dual is a linear map and the images of other MultiVectors follow from the images of the basis MultiVectors.

CliffordAlgebras.extendMethod
extend(::MultiVector)

Returns a new MultiVector with a non-sparse coefficient coding. This can be useful to manage type stability.

CliffordAlgebras.maxgradeMethod
maxgrade(::MultiVector ; rtol = 1e-8)

Projects the MultiVector onto the subspace of the largest grade with non-vanishing norm. Returns a tuple of the resulting multivector and its grade.

CliffordAlgebras.mingradeMethod
mingrade(::MultiVector ; rtol = 1e-8)

Projects the MultiVector onto the subspace of the largest grade with non-vanishing norm. Returns a tuple of the resulting multivector and its grade.

CliffordAlgebras.orderMethod
order(::CliffordAlgebra)
order(::Type{<:CliffordAlgebra})

Returns the order of the algebra. The order is the sum of the signature and the dimension of the underlying 1-vector space and the maximum grade for multivectors.

CliffordAlgebras.outermorphismMethod
outermorphism(A::AbstractMatrix, mv::MultiVector)

Calculates the outermorphism f of the MultiVector defined by f(v) = Av if v is in the grade-1 subspace of the algebra.

CliffordAlgebras.polarizeMethod
polarize(mv::MultiVector)
mv'

Calculates the polarization of the MultiVector, i.e. mv * pseudoscalar.

CliffordAlgebras.pruneMethod
prune(::MultiVector ; rtol = 1e-8 )

Returns a new MultiVector with all basis vectors removed from the sparse basis whose coefficients fall below the relative magnitude threshold. This function is not type stable, because the return type depends on the sparse basis.

CliffordAlgebras.scalarMethod
scalar(mv::MultiVector)

Returns the scalar component of the multivector. The result if of the internal storage type eltype(mv).

CliffordAlgebras.signaturetableMethod
signaturetable(io::IO, ca::CliffordAlgebra)
signaturetable(io::IO, CA::Type{<:CliffordAlgebra})

Prints the 1-vector basis symbols and their squares.

CliffordAlgebras.ΛᵏMethod
Λᵏ(::MultiVector, ::Val{k}) where k
Λᵏ(::MultiVector, k::Integer)

Projects the MultiVector onto k-vectors. Similar to grade(mv,k), but uses @generated code and compile time optimizations.

LinearAlgebra.normMethod
norm(::MultiVector)

Calculates the MultiVector norm defined as sqrt(scalar(mv*reverse(mv))).

CliffordAlgebras.CliffordAlgebraType
CliffordAlgebra(Npos::Integer, Nneg::Integer, Nzero::Integer, S::NTuple(N,Symbol))

Singleton instance of the type CliffordAlgebra that describes a geometric algebra with the signature (Npos,Nneg,Nzero), base symbols S. The base symbols are in order of the signature.

CliffordAlgebras.CliffordAlgebraMethod
CliffordAlgebra(a::Symbol)

Generates a predefined algebra from a identifier. Known algebras are - :Hyperbolic or :Hyper - :Complex or :ℂ - :Dual or :Grassmann - :Grassmann2D or :G2 - :Grassmann3D or :G3 - :Quaternions or :ℍ - :Cl2 and :Cl3 - :Spacetime - :PGA2D or :Projective2D or :Plane2D - :PGA3D or :Projective3D or :Plane3D - :CGA2D or :Conformal2D - :CGA3D or :Conformal3D - :DCGA3D or :DoubleConformal3D - :TCGA3D or :TripleConformal3D - :DCGSTA or :DoubleConformalSpacetime - :QCGA or :QuadricConformal

CliffordAlgebras.MultiVectorType
MultiVector{CA,T,BI}

Type for a multivector belonging to the algebra CA<:CliffordAlgbra with vector coefficients of type T. Coefficients are stored using a sparse coding, and only the coefficients of the basis indices stored in the tuple BI are considered.

CliffordAlgebras.MultiVectorMethod
MultiVector(::CliffordAlgebra, v::NTuple{N,T}) where {N,T<:Real}
MultiVector(::Type{<:CliffordAlgebra}, v::NTuple{N,T}) where {N,T<:Real}

Creates a MultiVector by converting the provided vector v to a 1-vector. The internal storage type of the MultiVector is T.

CliffordAlgebras.MultiVectorMethod
MultiVector(::CliffordAlgebra, a::Real)
MultiVector(::Type{<:CliffordAlgebra}, a::Real)

Creates a MultiVector from the real number a with only a scalar component. The internal storage type of the MultiVector is the type of a.