Reference

A_scaled, D1, D2 = equilibrate(A::AbstractMatrix{T}; A_transposed = false, kwargs...)

Performs the equilibration algorithm on the matrix A and returns the scaled matrix matrix A_scaled with its diagonal scaling factors D1 and D2.

Q_scaled, D = equilibrate(Q::Symmetric{T}; kwargs...) where {T}

Performs the equilibration algorithm on the symmetric matrix Q and returns the scaled matrix matrix Q_scaled with its diagonal scaling factor D.

See equilibrate! for the keyword arguments.

equilibrate!(A::AbstractMatrix{T},
             D1::Diagonal{T, S}, D2::Diagonal{T, S},
             R_k::Diagonal{T, S}, C_k::Diagonal{T, S};
             ϵ::T = T(1.0e-2), max_iter::Int = 100,
             A_transposed::Bool = false) where {T, S <: AbstractVector{T}}

Performs the equilibration algorithm to scale A so that its rows and columns have a infinity norm of 1 with a tolerance ϵ. D1 and D2 are diagonal scaling factors of sizes size(A, 1) and size(A, 2) respectively. Once A is scaled, the identity inv(D1) * A * inv(D2) gives the unscaled matrix. R_k and C_k are diagonal matrices of sizes size(A, 1) and size(A, 2) respectively that should be pre-allocated. max_iter is the maximal number of iterations.

If A_transposed = true, then, once A is scaled, the identity inv(D2) * A * inv(D1) gives the unscaled matrix. When using A_transposed = true, D1 and D2 should have sizes size(A, 2) and size(A, 1).

equilibrate!(Q::Symmetric{T}, D::Diagonal{T, S}, C_k::Diagonal{T, S};
             ϵ::T = T(1.0e-2), max_iter::Int = 100) where {T, S <: AbstractVector{T}}

Performs the equilibration algorithm to scale the symmetric matrix Q so that its rows and columns have a infinity norm of 1 with a tolerance ϵ. D is a diagonal scaling factors of size size(Q, 1). Once Q is scaled, the identity inv(D) * Q * inv(D) gives the unscaled matrix. C_k is a diagonal matrix of size size(A, 1) that should be pre-allocated. max_iter is the maximal number of iterations.

Reference

• D. Ruiz, A Scaling Algorithm to Equilibrate Both Rows and Columns Norms in Matrices, RAL-TR-2001-034, 2001.