Public API

LDR{T<:Number, E<:Real} <: Factorization{T}

Represents the matrix factorization $A = L D R$ for a square matrix $A$, where $L$ is a unitary matrix, $D$ is a diagonal matrix of strictly positive real numbers, and $R$ is defined such that $|\det R| = 1$.

This factorization is based on a column-pivoted QR decomposition $A P = Q R',$ such that

\[\begin{align*} L &:= Q \\ D &:= \vert \textrm{diag}(R') \vert \\ R &:= \vert \textrm{diag}(R') \vert^{-1} R' P^T\\ \end{align*}\]

Fields

• L::Matrix{T}: The left unitary matrix $L$ in a LDR factorization.
• d::Vector{E}: A vector representing the diagonal matrix $D$ in a LDR facotorization.
• R::Matrix{T}: The right matrix $R$ in a LDR factorization.

LDRWorkspace{T<:Number}

A workspace to avoid dyanmic memory allocations when performing computations with a LDR factorization.

Fields

• qr_ws::QRWorkspace{T,E}: QRWorkspace for calculating column pivoted QR factorization without dynamic memory allocations.
• lu_ws::LUWorkspace{T}: LUWorkspace for calculating LU factorization without dynamic memory allocations.
• M::Matrix{T}: Temporary storage matrix for avoiding dynamic memory allocations. This matrix is used/modified when a LDR factorization is calculated.
• M′::Matrix{T}: Temporary storage matrix for avoiding dynamic memory allocations.
• M″::Matrix{T}: Temporary storage matrix for avoiding dynamic memory allocations.
• v::Vector{T}: Temporary storage vector for avoiding dynamic memory allocations.

ldr(A::AbstractMatrix{T}) where {T}

Allocate an LDR factorization based on A, but does not calculate its LDR factorization, instead initializing the factorization to the identity matrix.

ldr(A::AbstractMatrix{T}, ws::LDRWorkspace{T}) where {T}

Return the LDR factorization of the matrix A.

ldr(F::LDR{T}, ignore...) where {T}

Return a copy of the LDR factorization F.

ldr!(F::LDR{T}, A::AbstractMatrix{T}, ws::LDRWorkspace{T}) where {T}

Calculate the LDR factorization F for the matrix A.

ldr!(F::LDR{T}, I::UniformScaling, ignore...) where {T}

Set the LDR factorization equal to the identity matrix.

ldr!(Fout::LDR{T}, Fin::LDR{T}, ignore...) where {T}

Copy the LDR factorization Fin to Fout.

ldr!(F::LDR, ws::LDRWorkspace{T}) where {T}

Calculate the LDR factorization for the matrix F.L.

ldrs(A::AbstractMatrix{T}, N::Int) where {T}

Return a vector of LDR factorizations of length N, where each one represents the identity matrix of the same size as $A$.

ldrs(A::AbstractMatrix{T}, N::Int, ws::LDRWorkspace{T,E}) where {T,E}

Return a vector of LDR factorizations of length N, where each one represents the matrix A.

ldrs(A::AbstractArray{T,3}, ws::LDRWorkspace{T,E}) where {T,E}

Return a vector of LDR factorizations of length size(A, 3), where there is an LDR factorization for each matrix A[:,:,i].

ldrs!(Fs::AbstractVector{LDR{T,E}}, A::AbstractArray{T,3}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the LDR factorization Fs[i] for the matrix A[:,:,i].

ldr_workspace(A::AbstractMatrix)

ldr_workspace(F::LDR)

Return a LDRWorkspace that can be used to avoid dynamic memory allocations.

eltype(LDR{T}) where {T}

Return the matrix element type T of the LDR factorization F.

size(F::LDR, dim...)

Return the size of the LDR factorization F.

copyto!(ldrws_out::LDRWorkspace{T,E}, ldrws_in::LDRWorkspace{T,E}) where {T,E}

Copy the contents of ldrws_in into ldrws_out.

copyto!(U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Copy the matrix represented by the LDR factorization V into the matrix U.

copyto!(U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E} = ldr!(U,V,ws)

copyto!(U::LDR, I::UniformScaling, ignore...) = ldr!(U,I)

Copy the matrix V to the LDR factorization U, calculating the LDR factorization to represent V.

copyto!(U::LDR{T,E}, V::LDR{T,E}, ignore...) where {T,E}

Copy the LDR factorization V to U.

adjoint!(Aᵀ::AbstractMatrix{T}, A::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Given an LDR factorization $A$, construct the matrix representing its adjoint $A^{\dagger}.$

lmul!(U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate $V := U V$ where $U$ is a LDR factorization and $V$ is a matrix.

lmul!(U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $V := U V,$ where $U$ is a matrix and $V$ is an LDR factorization.

Algorithm

Calculate $V := U V$ using the procedure

\[\begin{align*} V:= & UV\\ = & \overset{L_{0}D_{0}R_{0}}{\overbrace{U[L_{v}D_{v}}}R_{v}]\\ = & \overset{L_{1}}{\overbrace{L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}}}\,\overset{R_{1}}{\overbrace{R_{0}R_{v}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

lmul!(U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $V := U V,$ where $U$ and $V$ are both LDR factorizations.

Algorithm

Calculate $V := U V$ using the procedure

\[\begin{align*} V:= & UV\\ = & [L_{u}D_{u}\overset{M}{\overbrace{R_{u}][L_{v}}}D_{v}R_{v}]\\ = & L_{u}\overset{L_{0}D_{0}R_{0}}{\overbrace{D_{u}MD_{v}}}R_{v}\\ = & \overset{L_{1}}{\overbrace{L_{u}L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}}}\,\overset{R_{1}}{\overbrace{R_{0}R_{v}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

rmul!(U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate $U := U V$ where $U$ is a matrix and $V$ is a LDR factorization.

rmul!(U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $U := U V,$ where $U$ is a LDR factorization and $V$ is a matrix.

Algorithm

Calculate $U := U V$ using the procedure

\[\begin{align*} U:= & UV\\ = & [L_{u}\overset{L_{0}D_{0}R_{0}}{\overbrace{D_{u}R_{u}]V}}\\ = & \overset{L_{1}}{\overbrace{L_{u}L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}}}\,\overset{R_{1}}{\overbrace{R_{0}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

rmul!(U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $U := U V,$ where both $U$ and $V$ are LDR factorizations.

Algorithm

Calculate $U := U V$ using the procedure

\[\begin{align*} U:= & UV\\ = & [L_{u}D_{u}\overset{M}{\overbrace{R_{u}][L_{v}}}D_{v}R_{v}]\\ = & L_{u}\overset{L_{0}D_{0}R_{0}}{\overbrace{D_{u}MD_{v}}}R_{v}\\ = & \overset{L_{1}}{\overbrace{L_{u}L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}}}\,\overset{R_{1}}{\overbrace{R_{0}R_{v}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

mul!(H::AbstractMatrix{T}, U::LDR{T,E}, V::AbstractMatrix{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the matrix product $H := U V$, where $H$ and $V$ are matrices and $U$ is a LDR factorization.

mul!(H::AbstractMatrix{T}, U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the matrix product $H := U V$, where $H$ and $U$ are matrices and $V$ is a LDR factorization.

mul!(H::LDR{T,E}, U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $H := U V$, where $U$ is matrix, and $H$ and $V$ are both LDR factorization. For the algorithm refer to documentation for lmul!.

mul!(H::LDR{T,E}, U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $H := U V$, where $V$ is matrix, and $H$ and $U$ are both LDR factorizations. For the algorithm refer to the documentation for rmul!.

mul!(H::LDR{T,E}, U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable matrix product $H := U V,$ where $H,$ $U$ and $V$ are all LDR factorizations. For the algorithm refer to the documentation for lmul!.

ldiv!(U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate $V := U^{-1} V,$ where $V$ is a matrix, and $U$ is an LDR factorization.

ldiv!(H::AbstractMatrix{T}, U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate $H := U^{-1} V,$ where $H$ and $V$ are matrices, and $U$ is an LDR factorization.

ldiv!(U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $V := U^{-1}V$ where both $U$ and $V$ are LDR factorizations. Note that an intermediate LU factorization is required to calucate the matrix inverse $R_u^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

Algorithm

Calculate $V := U^{-1}V$ using the procedure

\[\begin{align*} V:= & U^{-1}V\\ = & [L_{u}D_{u}R_{u}]^{-1}[L_{v}D_{v}R_{v}]\\ = & R_{u}^{-1}D_{u}^{-1}\overset{M}{\overbrace{L_{u}^{\dagger}L_{v}}}D_{v}R_{v}\\ = & \overset{L_{0}D_{0}R_{0}}{\overbrace{R_{u}^{-1}D_{u}^{-1}MD_{v}}}R_{v}\\ = & \overset{L_{1}}{\overbrace{L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}^{\phantom{1}}}}\,\overset{R_{1}}{\overbrace{R_{0}R_{v}^{\phantom{1}}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

ldiv!(H::LDR{T,E}, U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $H := U^{-1} V,$ where $H,$ $U$ and $V$ are all LDR factorizations. Note that an intermediate LU factorization is required to calucate the matrix inverse $R_u^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

ldiv!(U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $V := U^{-1} V,$ where $U$ is a matrix and $V$ is a LDR factorization. Note that an intermediate LU factorization is required as well to calucate the matrix inverse $U^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

Algorithm

The numerically stable procdure used to evaluate $V := U^{-1} V$ is

\[\begin{align*} V:= & U^{-1}V\\ = & \overset{L_{0}D_{0}R_{0}}{\overbrace{U^{-1}[L_{v}D_{v}}}R_{v}]\\ = & \overset{L_{1}}{\overbrace{L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}}}\,\overset{R_{1}}{\overbrace{R_{0}R_{v}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

ldiv!(H::LDR{T,E}, U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $H := U^{-1} V,$ where $H$ and $V$ are LDR factorizations and $U$ is a matrix. Note that an intermediate LU factorization is required to calculate $U^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

rdiv!(U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the matrix product $U := U V^{-1},$ where $V$ is an LDR factorization and $U$ is a matrix. Note that this requires two intermediate LU factorizations to calculate $L_v^{-1}$ and $R_v^{-1}$.

rdiv!(H::AbstractMatrix{T}, U::AbstractMatrix{T}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the matrix product $H := U V^{-1},$ where $H$ and $U$ are matrices and $V$ is a LDR factorization. Note that this requires two intermediate LU factorizations to calculate $L_v^{-1}$ and $R_v^{-1}$.

rdiv!(U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $U := U V^{-1}$ where both $U$ and $V$ are LDR factorizations. Note that an intermediate LU factorization is required to calucate the matrix inverse $L_v^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

Algorithm

Calculate $U := UV^{-1}$ using the procedure

\[\begin{align*} U:= & UV^{-1}\\ = & [L_{u}D_{u}R_{u}][L_{v}D_{v}R_{v}]^{-1}\\ = & L_{u}D_{u}\overset{M}{\overbrace{R_{u}R_{v}^{-1}}}D_{v}^{-1}L_{v}^{\dagger}\\ = & L_{u}\overset{L_{0}D_{0}R_{0}}{\overbrace{D_{u}MD_{v}^{-1}}}L_{v}^{\dagger}\\ = & \overset{L_{1}}{\overbrace{L_{u}L_{0}^{\phantom{1}}}}\,\overset{D_{1}}{\overbrace{D_{0}^{\phantom{1}}}}\,\overset{R_{1}}{\overbrace{R_{0}L_{v}^{\dagger}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

rdiv!(H::LDR{T,E}, U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $H := U V^{-1}$ where $H,$ $U$ and $V$ are all LDR factorizations. Note that an intermediate LU factorization is required to calucate the matrix inverse $L_v^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

rdiv!(U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $U := U V^{-1},$ where $V$ is a matrix and $U$ is an LDR factorization. Note that an intermediate LU factorization is required as well to calucate the matrix inverse $V^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

Algorithm

The numerically stable procdure used to evaluate $U := U V^{-1}$ is

\[\begin{align*} U:= & UV^{-1}\\ = & [L_{u}\overset{L_{0}D_{0}R_{0}}{\overbrace{D_{u}R_{u}]V^{-1}}}\\ = & \overset{L_{1}}{\overbrace{L_{u}L_{0}}}\,\overset{D_{1}}{\overbrace{D_{0}}}\,\overset{R_{1}}{\overbrace{R_{0}}}\\ = & L_{1}D_{1}R_{1}. \end{align*}\]

rdiv!(H::LDR{T,E}, U::LDR{T,E}, V::AbstractMatrix{T}, ws::LDRWorkspace{T,E}) where {T,E}

Calculate the numerically stable product $H := U V^{-1},$ where $V$ is a matrix and $H$ and $U$ is an LDR factorization. Note that an intermediate LU factorization is required as well to calucate the matrix inverse $V^{-1},$ in addition to the intermediate LDR factorization that needs to occur.

logabsdet(A::LDR{T}, ws::LDRWorkspace{T}) where {T}

Calculate $\log(\vert \det A \vert)$ and $\textrm{sign}(\det A)$ for the LDR factorization $A.$

inv_IpA!(G::AbstractMatrix{T}, A::LDR{T,E}, ws::LDRWorkspace{T,E})::Tuple{E,T} where {T,E}

Calculate the numerically stable inverse $G := [I + A]^{-1},$ where $G$ is a matrix, and $A$ is represented by a LDR factorization. This method also returns $\log( \vert \det G\vert )$ and $\textrm{sign}(\det G).$

Algorithm

The numerically stable inverse $G := [I + A]^{-1}$ is calculated using the procedure

\[\begin{align*} G:= & [I+A]^{-1}\\ = & [I+L_{a}D_{a}R_{a}]^{-1}\\ = & [I+L_{a}D_{a,\min}D_{a,\max}R_{a}]^{-1}\\ = & [(R_{a}^{-1}D_{a,\max}^{-1}+L_{a}D_{a,\min})D_{a,\max}R_{a}]^{-1}\\ = & R_{a}^{-1}D_{a,\max}^{-1}[\overset{M}{\overbrace{R_{a}^{-1}D_{a,\max}^{-1}+L_{a}D_{a,\min}}}]^{-1}\\ = & R_{a}^{-1}D_{a,\max}^{-1}M^{-1}, \end{align*}\]

where $D_{a,\min} = \min(D_a, 1)$ and $D_{a,\max} = \max(D_a, 1).$ Intermediate matrix inversions and relevant determinant calculations are performed via LU factorizations with partial pivoting.

inv_IpUV!(G::AbstractMatrix{T}, U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E})::Tuple{E,T} where {T,E}

Calculate the numerically stable inverse $G := [I + UV]^{-1},$ where $G$ is a matrix and $U$ and $V$ are represented by LDR factorizations. This method also returns $\log( \vert \det G\vert )$ and $\textrm{sign}(\det G).$

Algorithm

The numerically stable inverse $G := [I + UV]^{-1}$ is calculated using the procedure

\[\begin{align*} G:= & [I+UV]^{-1}\\ = & [I+L_{u}D_{u}R_{u}L_{v}D_{v}R_{v}]^{-1}\\ = & [I+L_{u}D_{u,\max}D_{u,\min}R_{u}L_{v}D_{v,\min}D_{v,\max}R_{v}]^{-1}\\ = & [L_{u}D_{u,\max}(D_{u,\max}^{-1}L_{u}^{\dagger}R_{v}^{-1}D_{v,\max}^{-1}+D_{u,\min}R_{u}L_{v}D_{v,\min})D_{v,\max}R_{v}]^{-1}\\ = & R_{v}^{-1}D_{v,\max}^{-1}[\overset{M}{\overbrace{D_{u,\max}^{-1}L_{u}^{\dagger}R_{v}^{-1}D_{v,\max}^{-1}+D_{u,\min}R_{u}L_{v}D_{v,\min}}}]^{-1}D_{u,\max}^{-1}L_{u}^{\dagger}\\ = & R_{v}^{-1}D_{v,\max}^{-1}M^{-1}D_{u,\max}^{-1}L_{u}^{\dagger} \end{align*}\]

Intermediate matrix inversions and relevant determinant calculations are performed via LU factorizations with partial pivoting.

inv_UpV!(G::AbstractMatrix{T}, U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E})::Tuple{E,T} where {T,E}

Calculate the numerically stable inverse $G := [U+V]^{-1},$ where $G$ is a matrix and $U$ and $V$ are represented by LDR factorizations. This method also returns $\log( \vert \det G\vert )$ and $\textrm{sign}(\det G).$

Algorithm

The numerically stable inverse $G := [U+V]^{-1}$ is calculated using the procedure

\[\begin{align*} G:= & [U+V]^{-1}\\ = & [\overset{D_{u,\max}D_{u,\min}}{L_{u}\overbrace{D_{u}}R_{u}}+\overset{D_{v,\min}D_{v,\max}}{L_{v}\overbrace{D_{v}}R_{v}}]^{-1}\\ = & [L_{u}D_{u,\max}D_{u,\min}R_{u}+L_{v}D_{v,\min}D_{v,\max}R_{v}]^{-1}\\ = & [L_{u}D_{u,\max}(D_{u,\min}R_{u}R_{v}^{-1}D_{v,\max}^{-1}+D_{u,\max}^{-1}L_{u}^{\dagger}L_{v}D_{v,\min})D_{v,\max}R_{v}]^{-1}\\ = & R_{v}^{-1}D_{v,\max}^{-1}[\overset{M}{\overbrace{D_{u,\min}R_{u}R_{v}^{-1}D_{v,\max}^{-1}+D_{u,\max}^{-1}L_{u}^{\dagger}L_{v}D_{v,\min}}}]^{-1}D_{u,\max}^{-1}L_{u}^{\dagger}\\ = & R_{v}^{-1}D_{v,\max}^{-1}M^{-1}D_{u,\max}^{-1}L_{u}^{\dagger}, \end{align*}\]

where

\[\begin{align*} D_{u,\min} = & \min(D_u,1)\\ D_{u,\max} = & \max(D_u,1)\\ D_{v,\min} = & \min(D_v,1)\\ D_{v,\max} = & \max(D_v,1), \end{align*}\]

and all intermediate matrix inversions and determinant calculations are performed via LU factorizations with partial pivoting.

inv_invUpV!(G::AbstractMatrix{T}, U::LDR{T,E}, V::LDR{T,E}, ws::LDRWorkspace{T,E})::Tuple{E,T} where {T,E}

Calculate the numerically stable inverse $G := [U^{-1}+V]^{-1},$ where $G$ is a matrix and $U$ and $V$ are represented by LDR factorizations. This method also returns $\log( \vert \det G\vert )$ and $\textrm{sign}(\det G).$

Algorithm

The numerically stable inverse $G := [U^{-1}+V]^{-1}$ is calculated using the procedure

\[\begin{align*} G:= & [U^{-1}+V]^{-1}\\ = & [\overset{D_{u,\max}D_{u,\min}}{(L_{u}\overbrace{D_{u}}R_{u})^{-1}}+\overset{D_{v,\min}D_{v,\max}}{L_{v}\overbrace{D_{v}}R_{v}}]^{-1}\\ = & [(L_{u}D_{u,\max}D_{u,\min}R_{u})^{-1}+L_{v}D_{v,\min}D_{v,\max}R_{v}]^{-1}\\ = & [R_{u}^{-1}D_{u,\min}^{-1}D_{u,\max}^{-1}L_{u}^{\dagger}+L_{v}D_{v,\min}D_{v,\max}R_{v}]^{-1}\\ = & [R_{u}^{-1}D_{u,\min}^{-1}(D_{u,\max}^{-1}L_{u}^{\dagger}R_{v}^{-1}D_{v,\max}^{-1}+D_{u,\min}R_{u}L_{v}D_{v,\min})D_{v,\max}R_{v}]^{-1}\\ = & R_{v}^{-1}D_{v,\max}^{-1}[\overset{M}{\overbrace{D_{u,\max}^{-1}L_{u}^{\dagger}R_{v}^{-1}D_{v,\max}^{-1}+D_{u,\min}R_{u}L_{v}D_{v,\min}}}]^{-1}D_{u,\min}R_{u}\\ = & R_{v}^{-1}D_{v,\max}^{-1}M^{-1}D_{u,\min}R_{u} \end{align*}\]

where

\[\begin{align*} D_{u,\min} = & \min(D_u,1)\\ D_{u,\max} = & \max(D_u,1)\\ D_{v,\min} = & \min(D_v,1)\\ D_{v,\max} = & \max(D_v,1), \end{align*}\]

and all intermediate matrix inversions and determinant calculations are performed via LU factorizations with partial pivoting.