API

CheckerboardMatrix{T<:Union{AbstractFloat, Complex{<:AbstractFloat}}}

A type to represent a checkerboard decomposition matrix.

Fields

• transposed::Bool: If the checkerboard matrix is transposed.
• inverted::Bool: If the checkerboard matrix is inverted.
• Nsites::Int: Number of sites/orbitals in lattice.
• Nneighbors::Int: Number of neighbors.
• Ncolors::Int: Number of checkerboard colors/groups.
• neighbor_table::Matrix{Int}: Neighbor table represented by a (2,Nneighbors) dimension matrix, where each column contains a pair of neighboring sites.
• coshΔτt::Vector{T}: The $\cosh(\Delta\tau t)$ values.
• sinhΔτt::Vector{T}: The $\sinh(\Delta\tau t)$ values.
• perm::Vector{Int}: The checkerboard permutation order relative to the ordering of the original neighbor table.
• inv_perm::Vector{Int}: Inverse permuation of perm.
• colors::Matrix{Int}: The bounds of each checkerboard color/group in neighbor_table.

CheckerboardMatrix(
    # ARGUMENTS
    neighbor_table::Matrix{Int},
    t::AbstractVector{T},
    Δτ::E;
    # KEYWORD ARGUMENTS
    transposed::Bool=false,
    inverted::Bool=false
) where {T, E<:AbstractFloat}

Given a neighbor_table along with the corresponding hopping amplitudes t and discretzation in imaginary time Δτ, construct an instance of the type CheckerboardMatrix.

CheckerboardMatrix(
    # ARGUMENTS
    Γ::CheckerboardMatrix{T};
    # KEYWORD ARGUMENTS
    transposed::Bool = Γ.transposed,
    inverted::Bool   = Γ.inverted,
    new_matrix::Bool = false
) where {T}

Construct a new instance of CheckerboardMatrix based on a current instance Γ of CheckerboardMatrix. If new_matrix=true then allocate new coshΔτt and sinhΔτt arrays.

checkerboard_matrices(
    # ARGUMENTS
    neighbor_table::Matrix{Int},
    t::AbstractMatrix{T},
    # KEYWORD ARGUMENTS
    Δτ::E;
    transposed::Bool=false,
    inverted::Bool=false
) where {T<:Continuous, E<:AbstractFloat}

Return a vector of CheckerboardMatrix, one for each column of t, all sharing the same neighbor_table.

update!(
    Γ::CheckerboardMatrix{T}, t::AbstractVector{T}, Δτ::E
) where {T<:Continuous, E<:AbstractFloat}

Update the CheckerboardMatrix based on new hopping parameters t and discretezation in imaginary time Δτ.

update!(
    Γs::AbstractVector{CheckerboardMatrix{T}}, t::AbstractMatrix{T}, Δτ::E
) where {T<:Continuous, E<:AbstractFloat}

Update a vector of CheckerboardMatrix based on new hopping parameters t and discretezation in imaginary time Δτ.

update!(
    coshΔτt::AbstractVector{T}, sinhΔτt::AbstractVector{T}, t::AbstractVector{T},
    perm::AbstractVector{Int}, Δτ::E
) where {T<:Continuous, E<:AbstractFloat}

Update the coshΔτt and sinhΔτt associated with a checkerboard decomposition based on new hopping parameters t and discretezation in imaginary time Δτ.

eltype(Γ::CheckerboardMatrix{T}) where {T}

Return the matrix element type T of checkerboard decomposition Γ.

size(Γ::CheckerboardMatrix)

size(Γ::CheckerboardMatrix, dim::Int)

Return the dimensions of the checkerboard decomposition matrix Γ.

transpose(Γ::CheckerboardMatrix)

Return a transposed/adjoint version of the checkerboard matrix Γ.

adjoint(Γ::CheckerboardMatrix)

Return a transposed/adjoint version of the checkerboard matrix Γ.

inv(Γ::CheckerboardMatrix)

Return the inverse of the checkerboard matrix Γ.

mul!(u::AbstractVecOrMat, Γ::CheckerboardMatrix, v::AbstractVecOrMat)

Evaluate the matrix-vector or matrix-matrix product u=Γ⋅v.

mul!(u::AbstractVecOrMat, Γ::CheckerboardMatrix, v::AbstractVecOrMat, color::Int)

Evaluate the matrix-vector or matrix-matrix product u=Γ[c]⋅v where Γ[c] is the matrix associated with the color checkerboard color matrix.

mul!(u::AbstractVecOrMat, v::AbstractVecOrMat, Γ::CheckerboardMatrix)

Evaluate the matrix-vector or matrix-matrix product u=v⋅Γ.

mul!(u::AbstractVecOrMat, v::AbstractVecOrMat, Γ::CheckerboardMatrix, color::Int)

Evaluate the matrix-vector or matrix-matrix product u=v⋅Γ[c] where Γ[c] is the matrix associated with the color checkerboard color matrix.

lmul!(Γ::CheckerboardMatrix, u::AbstractVecOrMat)

Evaluate in-place the matrix-vector or matrix-matrix product u=Γ⋅u, where u gets over-written.

lmul!(Γ::CheckerboardMatrix, u::AbstractVecOrMat, color::Int)

Evaluate in-place the matrix-vector or matrix-matrix product u=Γ[c]⋅u, where Γ[c] is the matrix associated with the color checkerboard color matrix.

rmul!(u::AbstractVecOrMat, Γ::CheckerboardMatrix)

Evaluate in-place the matrix-vector or matrix-matrix product u=u⋅Γ, where u gets over-written.

rmul!(u::AbstractVecOrMat, Γ::CheckerboardMatrix, color::Int)

Evaluate in-place the matrix-vector or matrix-matrix product u=u⋅Γ[c], where Γ[c] is the matrix associated with the color checkerboard color matrix.

ldiv!(Γ::CheckerboardMatrix, v::AbstractVecOrMat)

Evaluate in-place the matrix-vector or matrix-matrix product v=Γ⁻¹⋅v.

ldiv!(u::AbstractVecOrMat, Γ::CheckerboardMatrix, v::AbstractVecOrMat)

Evaluate the matrix-vector or matrix-matrix product u=Γ⁻¹⋅v.

ldiv!(Γ::CheckerboardMatrix, v::AbstractVecOrMat, color::Int)

Evaluate in-place the matrix-vector or matrix-matrix product v=Γ⁻¹[c]⋅v where Γ⁻¹[c] is the inverse of the matrix associated with the color checkerboard color matrix.

ldiv!(u::AbstractVecOrMat, Γ::CheckerboardMatrix, v::AbstractVecOrMat, color::Int)

Evaluate the matrix-vector or matrix-matrix product u=Γ⁻¹[c]⋅v where Γ⁻¹[c] is the inverse of the matrix associated with the color checkerboard color matrix.

rdiv!(u::AbstractVecOrMat, Γ::CheckerboardMatrix)

Evaluate in-place the matrix-vector or matrix-matrix product u=u⋅Γ⁻¹, where u gets over-written.

rdiv!(u::AbstractVecOrMat, v::AbstractVecOrMat, Γ::CheckerboardMatrix)

Evaluate the matrix-vector or matrix-matrix product u=v⋅Γ⁻¹, where u gets over-written.

rdiv!(u::AbstractVecOrMat, Γ::CheckerboardMatrix, color::Int)

Evaluate in-place the matrix-vector or matrix-matrix product u=u⋅Γ⁻¹[c], where Γ[c] is the matrix associated with the color checkerboard color matrix.

rdiv!(u::AbstractVecOrMat, v::AbstractVecOrMat, Γ::CheckerboardMatrix, color::Int)

Evaluate the matrix-vector or matrix-matrix product u=v⋅Γ⁻¹[c], where Γ[c] is the matrix associated with the color checkerboard color matrix.

AbstractVecOrMat{T} = Union{AbstractVector{T}, AbstractMatrix{T}}

Abstract type defining union of AbstractVector and AbstractMatrix.

Continuous = Union{AbstractFloat, Complex{<:AbstractFloat}}

Abstract type to represent continuous real or complex numbers.

checkerboard_decomposition!(neighbor_table::Matrix{Int})

Given a neighbor_table, construct the checkerboard decomposition, which results in the columns of neighbor_table being re-ordered in-place. Two additional arrays are also returned:

• perm::Vector{Int}: The permutation used to re-order the columns of neighbor_table according to the checkerboard decomposition.
• colors::Matrix{Int}: Marks the column indice boundaries for each checkerboard color in neighbor_table.

checkerboard_lmul!(
    # ARGUMENTS
    B::AbstractMatrix{T},
    neighbor_table::Matrix{Int},
    coshΔτt::AbstractVector{E},
    sinhΔτt::AbstractVector{E},
    colors::Matrix{Int};
    # KEYWORD ARGUMENTS
    transposed::Bool=false,
    inverted::Bool=false
) where {T<:Continuous, E<:Continuous}

Evaluate the matrix-matrix product in-place B=Γ⋅B where Γ is the checkerboard matrix.

checkerboard_lmul!(
    # ARGUMENTS
    v::AbstractVector{T},
    neighbor_table::Matrix{Int},
    coshΔτt::AbstractVector{E},
    sinhΔτt::AbstractVector{E},
    colors::Matrix{Int};
    # KEYWORD ARGUMENTS
    transposed::Bool=false,
    inverted::Bool=false
) where {T<:Continuous, E<:Continuous}

Multiply in-place the vector v by the checkerboard matrix.

checkerboard_rmul!(
    # ARGUMENTS
    B::AbstractMatrix{T},
    neighbor_table::Matrix{Int},
    coshΔτt::AbstractVector{E},
    sinhΔτt::AbstractVector{E},
    colors::Matrix{Int};
    # KEYWORD ARGUMENTS
    transposed::Bool=false,
    inverted::Bool=false
) where {T<:Continuous, E<:Continuous}

Evaluate the matrix-matrix product in-place B=B⋅Γ where Γ is the checkerboard matrix.

checkerboard_color_lmul!(
    # ARGUMENTS
    B::AbstractMatrix{T},
    color::Int,
    neighbor_table::Matrix{Int},
    coshΔτt::AbstractVector{E},
    sinhΔτt::AbstractVector{E},
    colors::Matrix{Int};
    # KEYWORD ARGUMENTS
    inverted::Bool=false
) where {T<:Continuous, E<:Continuous}

Evaluate the matrix-matrix product in-place B=Γ[c]⋅B where Γ[c] is the color checkerboard color matrix.

checkerboard_color_lmul!(
    # ARGUMENTS
    v::AbstractVector{T},
    color::Int,
    neighbor_table::Matrix{Int},
    coshΔτt::AbstractVector{E},
    sinhΔτt::AbstractVector{E},
    colors::Matrix{Int};
    # KEYWORD ARGUMENTS
    inverted::Bool=false
) where {T<:Continuous, E<:Continuous}

Multiply in-place the vector v by the color checkerboard color matrix.

checkerboard_color_rmul!(
    # ARGUMENTS
    B::AbstractMatrix{T},
    color::Int,
    neighbor_table::Matrix{Int},
    coshΔτt::AbstractVector{E},
    sinhΔτt::AbstractVector{E},
    colors::Matrix{Int};
    # KEYWORD ARGUMENTS
    inverted::Bool=false
) where {T<:Continuous, E<:Continuous}

Evaluate the matrix-matrix product in-place B=B⋅Γ[c] where Γ[c] is the color checkerboard color matrix.