How to get BigFloat accurate conversion operators?

[MarcoFasondini]

For OPs on y⁴ = 1 - x⁴ I need conversion operators between ChebyshevT() and certain semiclassical OPs, e.g.,

T = ChebyshevT{BigFloat}()
x = axes(T,1)
P₁ = LanczosPolynomial( (1 .+ x.^2).^(0.5) )
R₁ = P₁\T

This is accurate to only Float64 precision:

xv = big"0.1"
T[xv,3] - ( P₁[xv,1]*R₁[1,3] + P₁[xv,3]*R₁[3,3] )
-6.05859383536196727705447569248701020544252033186250536144991618812281214296214e-17

Another example:

P = Legendre{BigFloat}()
R = P\T

The entry R[3,3] is 4/3 but it's accurate to only Float64 precision:

R[3,3]
1.333333333333333250903485233170513237354072211061085640237818614115573313606597

[MikaelSlevinsky]

Legendre to Chebyshev to Legendre transforms are available in FastTransforms in BigFloat.

[dlfivefifty]

This one is actually subtle:

ClassicalOrthogonalPolynomials.jl/src/classical/chebyshev.jl

Line 223 in 459bdc6

function \(A::ChebyshevT, B::Legendre)

Here's a MWE:

julia> BroadcastArray{BigFloat}(/, 1:10, 2:11)
(10-element UnitRange{Int64}) ./ (10-element UnitRange{Int64}):
 0.5
 0.66666666666666662965923251249478198587894439697265625
 0.75
 0.8000000000000000444089209850062616169452667236328125
 0.83333333333333337034076748750521801412105560302734375
 0.85714285714285709527615608749329112470149993896484375
 0.875
 0.888888888888888839545643349993042647838592529296875
 0.90000000000000002220446049250313080847263336181640625
 0.90909090909090906063028114658663980662822723388671875

It doesn't know that integer division produces a Float64....

The workaround is to change it to do convert(AbstractVector{T}, 1:∞). Not sure if there's a more general solution.

Can you make this change and add a PR?

[TSGut]

@MarcoFasondini Did the PR associated with this (i.e. #89) fix this issue and can it be closed?

[dlfivefifty]

No its not fixed yet...