Exponential Fitting

Exponentials type

First, we introduce ExpFit.Exponentials type. This type contains two fields: Exponents and coefficients of a sum of exponentials. Let us show you how to use it.

We define an instance using exponent and coefficient.

exponent = [1.0 + 2.0im, 3.0 + 4.0im]
coefficient = [5.0 + 6.0im, 7.0 + 8.0im]
ef = Exponentials(exponent, coefficient)

The instance ef contains the exponents and coefficients.

julia> ef.expon
2-element Vector{ComplexF64}:
 1.0 + 2.0im
 3.0 + 4.0im
julia> ef.coeff
2-element Vector{ComplexF64}:
 5.0 + 6.0im
 7.0 + 8.0im

ef also has a method and can be used as a function that returns sum(ef.coeff .* exp.(-ef.expon .* (t-t0)))

julia> f = t -> ef(t)
#1 (generic function with 1 method)

julia> g = t -> ef(t, 1.0)
#3 (generic function with 1 method)

expfit

As an example to demonstrate how to use the expfit function, consider approximating a Bessel function with a sum of exponentials. The following is the code.

sing LinearAlgebra
using ExpFit
using SpecialFunctions

tmin = 0.0
tmax  = 50.0      
tol = 1e-3     
N = 100
f = t -> besselj(0,t) + 1.0im*besselj(1,t)
ef = expfit(f, tmin, tmax, N, tol)
print("Approximation order = ", length(ef.coeff), "\n")

t = range(tmin, tmax, length=N*2)
err = abs.(ef.(t) .- f.(t))
println("Root mean square = ", norm(err)/sqrt(N))

Here, tmin and tmax specify the range used for the approximation, N is the number of sample points, and tol is the tolerance. ef is the instance of ExpFit.Exponentials.

When executed, the terminal will display the following.

Approximation order = 6
Root mean square = 0.0005937677782521255

The Bessel function was approximated with six terms for the tolerance of $10^{-2}$.

The results are also illustrated below.

We observe that the absolute error $\delta f(t)$ is within the tolerance.

As an alternative use case, equally spaced discrete data can be input into the expfit function. Here, dt represents the time interval.

dt = (tmax-tmin) / (N-1)
fv = f.(t)
ef = expfit(fv, dt, tol)

Additionally, the approximation order can be set instead of the tolerance.

order = 10
ef = expfit(f, tmin, tmax, N, order)
ef = expfit(fv, dt, order)

Algorithms

The default algorithm used in expfit is ESPRIT algorithm. If you want use a different algorithm, you can set the alg option as

ef = expfit(fv, dt, tol; alg=ESPIRA1())

Available algorithms and corresponding options are shown below. | Alogrithm | Option "alg" | | –––– | –––––– | | ESPRIT | ESPRIT() | | Matrix Pencil | Pencil() | | Prony | Prony() | | ESPIRA-I | ESPIRA1() | | ESPIRA-II | ESPIRA2() | | Fast ESPRIT | FastESPRIT() |

Please see Reference for all the available methods.