# Improving RationalInterpolation Result

The following code produces a {4,4} rational interpolation approximation accurate to 10^-6 or better over the range {0,Pi/2} for α in the range {.001,.999}:

Needs["FunctionApproximations"];

r[kz_, α_] := (Cos[(kz*α)/2]*Csc[kz/2]*Sin[(kz*α)/2])/α;
p[kz_, α_] :=
RationalInterpolation[
r[2 ArcSin[Sqrt[α]], α], {α, 4, 4},
Table[Max[N[Sin[i Pi/17]], .00001], {i, 0, 8}]] /. α -> Sin[kz/2]^2


However, applying this procedure to the alternative function

r[kz_, α_] := (Cos[(kz*α)/2]*Csc[kz/2]*Sin[(kz*α)/2])/(α*Sinc[kz/2])


produces much worse results at certain values of α, such as 0.653.

I am seeking advice on obtaining a more uniformly accurate rational approximation to the second function by a better choice of interpolation points or even by a better algorithm.

The code used to obtain the plot is

Manipulate[
pe = Evaluate[p[kz, α]];
re = Evaluate[r[kz, α]];
Plot[1 - pe/re, {kz, 0, Pi/2},
AxesLabel -> {"\!$$\*SubscriptBox[\(k$$, $$z$$]\) Δz",Error},
PlotRange -> {-.00001, .00001},
AxesStyle -> Directive[Bold, FontSize -> 16],
PlotStyle -> AbsoluteThickness[1.7]],
{{α, .653}, .01, .999, Appearance -> "Labeled"}]
`