Reference

Exponentials <: AbstractExpFit

A type representing a sum of exponentials of the form

f(t) = ∑ cᵢ exp(-aᵢ(t-t₀))

where aᵢ and cᵢ are the exponents and coefficients of the sum, respectively.

Fields

• expon::AbstractVector{<:Number}: The exponents aᵢ of the sum of exponentials.
• coeff::AbstractVector{<:Number}: The coefficients cᵢ of the sum of exponentials.

balanced_truncation(a::AbstractVector{<:Number}, c::AbstractVector{<:Number}, eps::Float64) :: Exponentials

Given the exponents a and coefficients c of a sum of exponentials, compute a new sum of exponentials with a reduced number of terms for a given tolerance eps. The initial exponents must have positive real part.

balanced_truncation(a::AbstractVector{<:Number}, c::AbstractVector{<:Number}, M::Int) :: Exponentials

Given the exponents a and coefficients c of a sum of exponentials, compute a new sum of exponentials with a given number of terms M. The initial exponents must have positive real part.

espira1(f::Vector{<:Number}, dt::Real, M::Int) :: Exponentials

Perform the ESPIRA-I algorithm using discrete data f and the sampling interval dt for a given model order M.

espira1(f::Vector{<:Number}, dt::Real, eps::Real) :: Exponentials

Perform the ESPIRA-I algorithm using discrete data f and the sampling interval dt for a given tolerance eps.

espira1(func::Function, tmin::Real, tmax::Real, nsamples::Int, M::Int) :: Exponentials

Perform the ESPIRA-I algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given model order M.

espira1(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real) :: Exponentials

Perform the ESPIRA-I algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given tolerance eps.

espira1(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int) :: Exponentials

Perform the ESPIRA-I algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given model order M.

espira1(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real) :: Exponentials

Perform the ESPIRA-I algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given tolerance eps.

espira2(f::Vector{<:Number}, dt::Real, M::Int) :: Exponentials

Perform the ESPIRA-II algorithm using discrete data f and the sampling interval dt for a given model order M.

espira2(f::Vector{<:Number}, dt::Real, eps::Real)  :: Exponentials

Perform the ESPIRA-II algorithm using discrete data f and the sampling interval dt for a given tolerance eps.

espira2(func::Function, tmin::Real, tmax::Real, nsamples::Int, M::Int) :: Exponentials

Perform the ESPIRA-II algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given model order M.

espira2(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real) :: Exponentials

Perform the ESPIRA-II algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given tolerance eps.

espira2(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int) :: Exponentials

Perform the ESPIRA-II algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given model order M.

espira2(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real) :: Exponentials

Perform the ESPIRA-II algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given tolerance eps.

esprit(hk::Vector{<:Number}, dt::Real, M::Int; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the ESPRIT algorithm using discrete data hk and the sampling interval dt for a given model order M.

esprit(hk::AbstractVector{<:Number}, dt::Real, eps::Real; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the ESPRIT algorithm using discrete data hk and the sampling interval dt for a given tolerance eps.

esprit(func::Function, tmin::Real, tmax::Real, nsamples::Int, M::Int; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the ESPRIT algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given model order M.

esprit(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the ESPRIT algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given tolerance eps.

esprit(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the ESPRIT algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given model order M.

esprit(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the ESPRIT algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given tolerance eps.

expfit(f::AbstractVector{<:Number}, dt::Real, M::Int; alg::AbstractExpFit=ESPRIT()) :: Exponentials

Estimate the exponents and coefficients of the sum of exponentials for the function f using the algorithm alg.

Arguments

• f::AbstractVector{<:Number}: The samples of the function.
• dt::Real: The sampling interval.
• M::Int: The model order.
• alg::AbstractExpFit=ESPRIT(): The algorithm to use for the estimation.

expfit(f::AbstractVector{<:Number}, dt::Real, eps::Real; alg::AbstractExpFit=ESPRIT()) :: Exponentials

Estimate the exponents and coefficients of the sum of exponentials for the function f using the algorithm alg.

Arguments

• f::AbstractVector{<:Number}: The samples of the function.
• dt::Real: The sampling interval.
• eps::Real: The tolerance for the algorithm.
• alg::AbstractExpFit=ESPRIT(): The algorithm to use for the estimation.

expfit(f::AbstractVector{<:Number}, dt::Real, eps::Real; alg::AbstractExpFit=ESPRIT()) :: Exponentials

Estimate the exponents and coefficients of the sum of exponentials for the function f using the algorithm alg.

Arguments

• f::AbstractVector{<:Number}: The samples of the function.
• dt::Real: The sampling interval.
• eps::Real: The tolerance for the algorithm.
• alg::AbstractExpFit=ESPRIT(): The algorithm to use for the estimation.

expfit(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real; alg::AbstractExpFit=ESPRIT()) :: Exponentials

Estimate the exponents and coefficients of the sum of exponentials for the function func using the algorithm alg.

Arguments

• func::Function: The function to estimate the exponents and coefficients.
• tmin::Real: The initial time.
• tmax::Real: The final time.
• nsamples::Int: The number of samples to take in the interval [tmin, tmax].
• eps::Real: The tolerance for the algorithm.
• alg::AbstractExpFit=ESPRIT(): The algorithm to use for the estimation.

expfit(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int; alg::AbstractExpFit=ESPRIT()) :: Exponentials

Estimate the exponents and coefficients of the sum of exponentials for the function func using the algorithm alg.

Arguments

• func::Function: The function to estimate the exponents and coefficients.
• tmin::Real: The initial time.
• tmax::Real: The final time.
• dt::Real: The sampling interval.
• M::Int: The model order.
• alg::AbstractExpFit=ESPRIT(): The algorithm to use for the estimation.

expfit(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real; alg::AbstractExpFit=ESPRIT()) :: Exponentials

Estimate the exponents and coefficients of the sum of exponentials for the function func using the algorithm alg.

Arguments

• func::Function: The function to estimate the exponents and coefficients.
• tmin::Real: The initial time.
• tmax::Real: The final time.
• dt::Real: The sampling interval.
• eps::Real: The tolerance for the algorithm.
• alg::AbstractExpFit=ESPRIT(): The algorithm to use for the estimation.

expred(a::AbstractVector{<:Number}, c::AbstractVector{<:Number}, M::Int) :: Exponentials

Finding a new set of exponents and coefficients for the sum of exponentials using the balanced truncation method.

Arguments

• a::AbstractVector{<:Number}: The exponents aᵢ of the sum of exponentials.
• c::AbstractVector{<:Number}: The coefficients cᵢ of the sum of exponentials.
• M::Int: The model order.

expred(a::AbstractVector{<:Number}, c::AbstractVector{<:Number}, eps::Real) :: Exponentials

Finding a new set of exponents and coefficients for the sum of exponentials using the balanced truncation method.

Arguments

• a::AbstractVector{<:Number}: The exponents aᵢ of the sum of exponentials.
• c::AbstractVector{<:Number}: The coefficients cᵢ of the sum of exponentials.
• eps::Real: The tolerance for the balanced truncation method.

fast_esprit(hk::Vector{<:Number}, dt::Real, M::Int) :: Exponentials

Perform the Fast ESPRIT algorithm using discrete data hk and the sampling interval dt for a given model order M.

fast_esprit(hk::Vector{<:Number}, dt::Real, eps::Real) :: Exponentials

Perform the Fast ESPRIT algorithm using discrete data hk and the sampling interval dt for a given tolerance eps.

fast_esprit(func::Function, tmin::Real, tmax::Real, nsamples::Int, M::Int) :: Exponentials

Perform the Fast ESPRIT algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given model order M.

fast_esprit(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real) :: Exponentials

Perform the Fast ESPRIT algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given tolerance eps.

fast_esprit(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int) :: Exponentials

Perform the Fast ESPRIT algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given model order M.

fast_esprit(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real) :: Exponentials

Perform the Fast ESPRIT algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given tolerance eps.

matrix_pencil(hk::Vector{<:Number}, dt::Real, M::Int; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the matrix pencil algorithm using discrete data hk and the sampling interval dt for a given model order M.

matrix_pencil(hk::AbstractVector{<:Number}, dt::Real, eps::Real; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the matrix pencil algorithm using discrete data hk and the sampling interval dt for a given tolerance eps.

matrix_pencil(func::Function, tmin::Real, tmax::Real, nsamples::Int, M::Int; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the matrix pencil algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given model order M.

matrix_pencil(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real; ncols::Union{Int,Nothing}=nothing)

Perform the matrix pencil algorithm using a function func in the range [tmin,tmax] and nsamples sampling points for a given tolerance eps.

matrix_pencil(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the matrix pencil algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given model order M.

matrix_pencil(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real; ncols::Union{Int,Nothing}=nothing) :: Exponentials

Perform the matrix pencil algorithm using a function func in the range [tmin,tmax] and a sampling interval dt for a given tolerance eps.

prony(hk::Vector{<:Number}, dt::Real, M::Int) :: Exponentials

Perform the Prony method using discrete data hk and the sampling interval dt for a given model order M.

prony(hk::Vector{<:Number}, dt::Real, eps::Real) :: Exponentials

Perform the Prony method using discrete data hk and the sampling interval dt for a given tolerance eps.

prony(func::Function, tmin::Real, tmax::Real, nsamples::Int, eps::Real) :: Exponentials

Perform the Prony method using a function func in the range [tmin,tmax] and nsamples sampling points for a given tolerance eps.

prony(func::Function, tmin::Real, tmax::Real, nsamples::Int, M::Int) :: Exponentials

Perform the Prony method using a function func in the range [tmin,tmax] and nsamples sampling points for a given model order M.

prony(func::Function, tmin::Real, tmax::Real, dt::Real, eps::Real) :: Exponentials

Perform the Prony method using a function func in the range [tmin,tmax] and a sampling interval dt for a given tolerance eps.

prony(func::Function, tmin::Real, tmax::Real, dt::Real, M::Int) :: Exponentials

Perform the Prony method using a function func in the range [tmin,tmax] and a sampling interval dt for a given model order M.