JuliaApproximation
arccos and arcsin
Firstly, they need a minor fix, since acos(x) has an infinite coefficient:
julia> acos(Fun(identity,-1.0..0.0))
Fun((1+x)^-0.5[Chebyshev(【-1.0,0.0】)]⊕ConstantSpace,[-1.07871, Inf, -1.10822, -0.0317578, -0.0024717, -0.000253871, -2.97623e-5, -3.7712e-6, -5.02982e-7, -6.957e-8 … -1.43517e-9, -2.11848e-10, -3.17037e-11, -4.79914e-12, -7.33506e-13, -1.13072e-13, -1.75372e-14, -2.72287e-15, -4.24105e-16, -9.27329e-17])
Secondly, something needs to be done when the interval is the entire real domain of acos:
julia> acos(Fun())
Fun(Chebyshev(【-1.0,1.0】)⊕(1-x^2)^-0.5[Chebyshev(【-1.0,1.0】)],[Inf, -9.11317e-16, -1.98721, 0.487335, 2.44702e-17, 6.0567e-16, 0.629283, -0.712968, 7.35972e-19, 8.33032e-17 … -3.59057e-17, 4.66159e-17, -1.62015e-18, -3.56283e-35, -3.50061e-17, 8.05625e-21, 0.0, -1.50605e-35, 0.0, 1.15429e-20])
The approximation space is wrong, since it should have a square-root singularity at the right endpoint. Perhaps it's best done via a PiecewiseSpace?
umm, acos does have square root singularities at both ±1 ….
On 18 Sep 2017, at 20:36, Richard Mikael Slevinsky [email protected] wrote:
Firstly, they need a minor fix, since acos(x) has an infinite coefficient:
julia> acos(Fun(identity,-1.0..0.0)) Fun((1+x)^-0.5[Chebyshev(【-1.0,0.0】)]⊕ConstantSpace,[-1.07871, Inf, -1.10822, -0.0317578, -0.0024717, -0.000253871, -2.97623e-5, -3.7712e-6, -5.02982e-7, -6.957e-8 … -1.43517e-9, -2.11848e-10, -3.17037e-11, -4.79914e-12, -7.33506e-13, -1.13072e-13, -1.75372e-14, -2.72287e-15, -4.24105e-16, -9.27329e-17]) Secondly, something needs to be done when the interval is the entire real domain of acos:
julia> acos(Fun()) Fun(Chebyshev(【-1.0,1.0】)⊕(1-x^2)^-0.5[Chebyshev(【-1.0,1.0】)],[Inf, -9.11317e-16, -1.98721, 0.487335, 2.44702e-17, 6.0567e-16, 0.629283, -0.712968, 7.35972e-19, 8.33032e-17 … -3.59057e-17, 4.66159e-17, -1.62015e-18, -3.56283e-35, -3.50061e-17, 8.05625e-21, 0.0, -1.50605e-35, 0.0, 1.15429e-20]) The approximation space is wrong, since it should have a square-root singularity at the right endpoint. Perhaps it's best done via a PiecewiseSpace?
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Then it's not working correctly, since it's returning inverse square-roots.
Probably the zero-endpoint conditions are currently being encoded in the coefficients à la Dirichlet spaces.