ApproxFun+IntervalArithmetic: make ODE solving really rigorous

The following code returns successfully for solving Airy's equation (the extra definitions are work arounds for bugs):

julia> Base.broadcast(::typeof(*), a::IntervalArithmetic.Interval, b::AbstractVector) = 
           map(x -> a*x, b)

julia> Base.Broadcast.broadcasted(::typeof(*), a::IntervalArithmetic.Interval, b::AbstractVector) = 
           map(x -> a*x, b)

julia> Base.length(d::UnitRange{<:IntervalArithmetic.Interval}) = Integer(maximum(d.stop.hi)-minimum(d.start.lo)+1)

julia> u = [Dirichlet(); D^2 - x] \ [[1, 0], 0]
Fun(Chebyshev([-1, -1]..[1, 1]),IntervalArithmetic.Interval{Float64}[[0.528647, 0.528648], [-0.522247, -0.522246], [-0.023162, -0.0231619], [0.0223854, 0.0223855], [-0.00557882, -0.00557881], [-0.000122104, -0.000122103], [9.37066e-05, 9.37067e-05], [-1.68122e-05, -1.68121e-05], [-2.4373e-07, -2.43729e-07], [1.63064e-07, 1.63065e-07]  …  [1.54639e-10, 1.5464e-10], [-1.89644e-11, -1.89643e-11], [-1.63813e-13, -1.63812e-13], [9.21385e-14, 9.21386e-14], [-9.91957e-15, -9.91956e-15], [-7.12466e-17, -7.12465e-17], [3.76658e-17, 3.76659e-17], [-3.63796e-18, -3.63795e-18], [-2.23433e-20, -2.23432e-20], [-1.12162e-20, -1.12161e-20]])

It happens to have worked:

julia> parse(BigFloat,"0.546136459064141770613483785351029091974067802242010192661484687977676021081218637004123196030567166414008915304513669838826570884623515686") in u(0) # true value from mathematica
true

However, this is mostly luck: the convergence criteria doesn't actually control the decay in the tail. To do this properly (and thereby have apriori guaranteed success) we would need to modify the convergence check at https://github.com/JuliaApproximation/ApproxFunBase.jl/blob/7dd9ef0a111ffd702f989c6f752a62538bc7e62b/src/Caching/matrix.jl#L122

to properly bound the tail.

Start in src/Caching/matrix.jl at the convergence check around line 122, then reproduce the Airy-equation example from the issue. Determine how the current check handles coefficient tails; done means the convergence criterion properly bounds the tail and provides an a priori guarantee rather than relying on observed success.