BaryRational

"You want poles with that?"

This small package contains both one dimensional barycentric rational approximation, using the AAA algorithm [1], and one dimensional barycentric rational interpolation with the Floater-Hormann weights [2]. It can also calculate the derivatives using the algorithm from [3].

The AAA approximation algorithm can model the poles of a function, if present. The FH interpolation is guaranteed to not contain any poles inside of the interpolation interval.

Usage

julia> using BaryRational
julia> x = [-3.0:0.1:3.0;];
julia> f = x -> sin(x) + 2exp(x)
julia> fh = FHInterp(x, f.(x), order=8, grid=true)
julia> fh(1.23)
7.78493669233287
julia> deriv(fh, 1.23)  # use deriv(fh, 1.23, m=2) for higher order derivatives
7.176696799673523

Note that the default order is 0. The best choice of the order parameter appears to be dependent on the number of points (see Table 2 of [1]) So for smaller data sets, order=3 or order=4 can be good choices. However, if you need more accurate derivatives, you may need to go to higher, as we did with order=8 above. This algorithm is not adaptive so you will have to try and see what works best for you.

If you know that the x points are on an even grid, use grid=true.

For approximation using aaa:

julia> a = aaa(x, f.(x))
julia> a(1.23)
7.784947874510929
julia> deriv(a, 1.23)
7.17669679970369
julia> deriv(a, 1.23, m=3)
6.508221345462802

and finally the exact results

julia> f(1.23)
7.784947874511044
julia> df = x -> cos(x) + 2exp(x)
julia> df(1.23)
7.1766967997038495
julia> df3 = x -> -cos(x) + 2exp(x)
julia> df3(1.23)
6.508221345454844

The AAA algorithm is adaptive in the subset of support points that it chooses to use.

NOTE: The aaa approximant is designed to take a scalar or vector input without broadcasting. It will still give the correct results if you inadvertently use a.(xx), but it will be much slower than using a(xx).

NOTE: ForwardDiff does not play well with BaryRational because when we interpolate at a support point, we just return the initial function value there. ForwardDiff recognizes this as a constant and returns derivative of a constant, which is zero. There is special handling in the algorithm of [3] for calculating the derivatives at support points and that is implemented here.

Examples

Here is an example of fitting f(x) = abs(x) with both FH and AAA. Note that because the first derivative is discontinuous at x = 0, we can achieve only linear convergence. (Note that systems like Chebfun and ApproxFun engineer around this by breaking up the interval at the points of discontinuity.) While the convergence order is the same for both algorithms, we see that the AAA has an error that is about a factor of 1.6 smaller than the Floater-Hormann scheme.

using PyPlot
using BaryRational
function plt_err_abs_x()
    pts = [40, 80, 160, 320, 640]
    fh_err = Float64[]
    aaa_err = Float64[]
    order = 3
    for p in pts
        xx = collect(range(-5.0, 5.0, length=2p - 1))
        xi = xx[1:2:end]
        xt = xx[2:2:end]
        yy = abs.(xi)
        fa = aaa(xi, yy)
        fh = FHInterp(xi, yy, order=order, grid=true)
        push!(aaa_err, maximum(abs.(fa(xt) .- abs.(xt))))
        push!(fh_err, maximum(abs.(fh.(xt) .- abs.(xt))))
    end
    plot(log.(pts), log.(fh_err), ".-", label="FH Error")
    plot(log.(pts), log.(aaa_err), ".-", label="AAA Error")
    xlabel("Log(Number of points)")
    ylabel("Log(Error)")
    legend()
    axis("equal")
    title("Error in approximating Abs(x)")
end
plt_err_abs_x()

Since both of these can approximate / interpolate on regular as well as irregular grid points, they can be used to create ApproxFun Fun's. ApproxFun needs to be able to evaluate, or have evaluated, a function on the Chebyshev points (1st kind here, 2nd kind for Chebfun), mostly if you have function values on a regular grid you are out of luck. Instead, use the AAA approximation algorithm to generate an approximation, use that to generate the values on the Chebyshev grid, use ApproxFun.transform to transform the function values to coefficients and then construct the Fun. The following shows how.

using LinearAlgebra
using ApproxFun
import BaryRational as br

# our function
f(x) = tanh(4x - 1)

# a regular grid
xx = [-1.0:0.01:1.0;];

# and evaluated on a regular grid
yy = f.(xx);

# and then approximated with AAA
faaa = br.aaa(xx, yy);

# but ApproxFun needs to be evaluated on the Chebyshev points
S = Chebyshev();
n = 129
pts = points(S, n);

# construct the Fun using the aaa approximation on the Chebyshev points
pn = Fun(S, ApproxFun.transform(S, faaa(pts)));

# now compare it to the "native" fun
x = Fun();
fapx = tanh(4x - 1);
println(norm(fapx - pn))

which yields an error norm of 3.0186087174306446e-14. Pretty nice.

As a final example, you can directly use the bary(), the barycentric interpolation formula, directly. In this case, it's really advised to use the Chebyshev points. Here is an example where we use the Float128 type from the Quadmath package:

using Quadmath
using BaryRational
using SpecialFunctions
T = Float128;
B = BigFloat;
num_points = 64;
# Test on the interval [-10.0, 0.0] where airyai is oscillatory
# and yet too small for asymptotic formulas to work.
# Create Chebyshev points and move to [-10.0, 0.0] interval
xx = T(5) * (chebpts(num_points, T) .- T(1));
# airyai does not work with Float128 but is OK with BigFloat
xb = B(5) * (chebpts(num_points, B) .- B(1));
fb = airyai.(xb);
f  = T.(fb);
# Random points for testing
xrat = rand(-10//1:1//100:0//1, 1000);
yb  = bary.(T.(xrat), (f,), (xx,));
ya  = airyai.(B.(xrat));
err = norm(yb - ya, Inf);
println("maximum error: ", T(err))

maximum error: 7.04169792675801867421523999050212504e-32

Which is also a nice result.

Poles

The function aaa, unlike in Chebfun, does not return the poles, zeros, and residues along with the approximant. To extract these, you need to use the prz function.

As an example, let us consider approximating $f(z) = 1/J_0(z)$ from random data in a rectangle in the complex plane, reproducing Figure 6.8 of [1].

using BaryRational
using StableRNGs
using CairoMakie
using Bessels

## Setup the data
seed = 7174444532485057091
rng = StableRNG(seed)
n = 2000
x = 10rand(rng, n)
y = 2rand(rng, n) .- 1
z = complex.(x, y)
f = z -> inv(besselj0(z))
fz = f.(z)

## Get the approximant and extract the poles and residues 
r = aaa(z, fz)
pol, res, zer = prz(r)

## Plot the error contours together with the data and computed poles
gridx = LinRange(-5, 15, 250)
gridy = LinRange(-3, 3, 250)
err = [abs(f(complex(x, y)) - r(complex(x, y))) for x in gridx, y in gridy]
fig = Figure()
ax = Axis(fig[1, 1], xlabel=L"x", ylabel=L"y", width=600, height=300)
contour!(ax, gridx, gridy, err, levels=10.0 .^ (-12:-1))
scatter!(ax, real(z), imag(z), color=:grey, markersize=4)
scatter!(ax, real(pol), imag(pol), color=:black, markersize=13)
xlims!(ax, -5, 15)
ylims!(ax, -3, 3)
resize_to_layout!(fig)
fig

We see that the computed poles of the solution, shown as black dots, are all real, even though we imposed no symmetry in our initial data - an impressive feature of the AAA algorithm. Moreover, these black dots are almost exactly equal to the true poles of $f$, i.e. the zeros of $J_0(z)$, again showing the power of the algorithm. Some of the poles do have some imaginary parts, though:

fig = Figure()
ax = Axis(fig[1, 1], xlabel=L"x", ylabel=L"y", width=600, height=300)
sc = scatter!(ax, real(pol), imag(pol), markersize=13, color=abs.(res))
lines!(ax, [0, 10, 10, 0, 0], [-1, -1, 1, 1, -1], linewidth=3, color=:black)
Colorbar(fig[1, 2], sc, label="|res|")
resize_to_layout!(fig)
fig

Those that are away from the real line are far from the original domain of the data, though, or have small residues, as shown.

Acknowledgements

• Some parts of Chebfun have been ported to Julia for this project
• A function from RationalFunctionApproximation has been borrowed (the continuum verion of the AAA algorithm on the interval [-1,1])

References

[1] The AAA algorithm for rational approximation

[2] Barycentric rational interpolation with no poles and high rates of approximation

[3] Some New Aspects of Rational Interpolation

[4] AAA rational approximation on a continuum