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JuliaApproximation / ApproxFun.jl

Julia package for function approximation
http://juliaapproximation.github.io/ApproxFun.jl/
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incorporate basis normalization in chop #185

Open dlfivefifty opened 9 years ago

dlfivefifty commented 9 years ago

Right now chop(::Fun) assumes all coefficients have equal weight. But many of the bases are not normalized. A routine like

basisapproxnorm(::FunctionSpace,k)

should be added to give an estimate "norm" of the k-th basis function.

dlfivefifty commented 8 years ago

We could also replicate chebfuns smart chop

MikaelSlevinsky commented 8 years ago

With a basis normalization, we could implement scale-invariant adaptive QR, hopefully dealing with this:

julia> using ApproxFun

julia> u = [dirichlet(Chebyshev());Derivative(Chebyshev(),2)-I]\[exp(1.0);exp(-1.0)]
Fun([1.2660658777520082,-1.1303182079849696,0.2714953395340765,-0.04433684984866379,0.005474240442093731,-0.0005429263119139437,4.4977322954295136e-5,-3.19843646240199e-6,1.9921248066727958e-7,-1.103677172551734e-8,5.505896079673748e-10,-2.497956616984982e-11,1.0391522306785702e-12,-3.991263356393109e-14,1.4237580108211739e-15,-4.7409164601184453e-17,1.4801777040733999e-18],Chebyshev(【-1.0,1.0】))

julia> u = [dirichlet(Chebyshev());Derivative(Chebyshev(),2)-I]\[1e-32exp(1.0);1e-32exp(-1.0)]
Fun([1.0287204232101624e-32,-1.1395890362606558e-32],Chebyshev(【-1.0,1.0】))

If someone needed scale-invariance right now, they could just supply linsolve an estimate for the appropriate tolerance, but generally u and f are in different spaces with different norms.