Discrete data

The original AAA algorithm (Nakatsukasa, Sète, Trefethen 2018) works with a fixed set of points on the domain of approximation. There is a legacy aaa function that can work with this type of data:

using RationalFunctionApproximation, ComplexRegions
x = -1:0.01:1
f = x -> tanh(5 * (x - 0.2))
r = aaa(x, f.(x))

Barycentric{Float64, Float64} rational of type (35, 35):
    -0.999988=>-1.0,  0.999329=>1.0,  -0.99186=>-0.35,  …  -0.999727=>-0.69

However, it's preferable to use the approximate function for this purpose, as the result type is more useful within the package. Simply pass the function and, in the form of a vector, the domain.

r = approximate(f, x)

Barycentric{Float64, Float64} rational of type (11, 11) constructed from 201 samples

As long as there are no singularities as close to the domain as the sample points are to one another, this fully discrete approach should be fine:

I = unit_interval
println("nearest pole is $(minimum(dist(z, I) for z in poles(r))) away")
_, err = check(r);
println("max error on the given domain: ", maximum(abs, err))

nearest pole is 0.31415926535638594 away
[ Info: Max error is 2.24e-14
max error on the given domain: 2.2426505097428162e-14

But if the distance to a singularity is comparable to the sample spacing, the quality of the approximation may suffer. Even worse, the method may not be aware that it has failed.

f = x -> tanh(400 * (x - 0.2))
r = approximate(f, x)
println("nearest pole is $(minimum(dist(z, I) for z in poles(r))) away")
_, err = check(r);
println("max error on the given domain: ", maximum(abs, err))
err = maximum(abs(f(x)- r(x)) for x in range(-1, 1, 3000))
println("max error on finer test points: ", err)

nearest pole is 0.004569419284272946 away
[ Info: Max error is 7.84e-14
max error on the given domain: 7.838174553853605e-14
max error on finer test points: 0.058457305710306495

In the continuous mode, the adaptive sampling of the domain attempts to ensure that the approximation is accurate everywhere.

r = approximate(f, I; tol=1e-12)
err = maximum(abs(f(x)- r(x)) for x in range(-1, 1, 3000))
println("max error on finer test points: ", err)

┌ Warning: Stopping with estimated error 1.698e-12 after 60 iterations
└ @ RationalFunctionApproximation ~/work/RationalFunctionApproximation.jl/RationalFunctionApproximation.jl/src/barycentric.jl:326
max error on finer test points: 1.938782467902911e-12