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‎README.md

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# ApproxFunOrthogonalPolynomials.jl
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Support for orthogonal polynomial-based spaces in ApproxFun
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[![](https://img.shields.io/badge/docs-stable-blue.svg)](https://JuliaApproximation.github.io/ApproxFun.jl/stable)
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[![](https://img.shields.io/badge/docs-latest-blue.svg)](https://JuliaApproximation.github.io/ApproxFun.jl/latest)
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[![Build Status](https://travis-ci.org/JuliaApproximation/ApproxFun.jl.svg?branch=master)](https://travis-ci.org/JuliaApproximation/ApproxFunOrthogonalPolynomials.jl)
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[![Build status](https://ci.appveyor.com/api/projects/status/q3e3ihfbxnrjn7ji?svg=true)](https://ci.appveyor.com/project/dlfivefifty/approxfunorthogonalpolynomials-jl)
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[![Coverage Status](https://img.shields.io/coveralls/JuliaApproximation/ApproxFunOrthogonalPolynomials.jl.svg)](https://coveralls.io/r/JuliaApproximation/ApproxFunOrthogonalPolynomials.jl?branch=master)
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[![Join the chat at https://gitter.im/JuliaApproximation/ApproxFun.jl](https://badges.gitter.im/JuliaApproximation/ApproxFun.jl.svg)](https://gitter.im/JuliaApproximation/ApproxFun.jl?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge)
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[ApproxFun.jl](https://github.com/JuliaApproximation/ApproxFun.jl) is a package for approximating functions. This package implements Fourier based spaces to ApproxFun.

‎src/ApproxFunOrthogonalPolynomials.jl

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module ApproxFunOrthogonalPolynomials
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using Base, LinearAlgebra, Reexport, BandedMatrices, AbstractFFTs, FFTW, InfiniteArrays, FillArrays, FastTransforms, IntervalSets,
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DomainSets, Statistics, SpecialFunctions
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using Base, LinearAlgebra, Reexport, BandedMatrices, BlockBandedMatrices, AbstractFFTs, FFTW, InfiniteArrays, BlockArrays, FillArrays, FastTransforms, IntervalSets,
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DomainSets, Statistics, SpecialFunctions, FastGaussQuadrature
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@reexport using ApproxFunBase
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@@ -17,30 +17,33 @@ import ApproxFunBase: normalize!, flipsign, FiniteRange, Fun, MatrixFun, UnsetSp
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ConcreteConversion, ConcreteMultiplication, ConcreteDerivative, ConcreteIntegral,
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ConcreteVolterra, Volterra, VolterraWrapper,
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MultiplicationWrapper, ConversionWrapper, DerivativeWrapper, Evaluation, EvaluationWrapper,
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Conversion, Multiplication, Derivative, Integral, bandwidths,
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Conversion, defaultConversion, defaultcoefficients, default_Fun, Multiplication, Derivative, Integral, bandwidths,
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ConcreteEvaluation, ConcreteDefiniteLineIntegral, ConcreteDefiniteIntegral, ConcreteIntegral,
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DefiniteLineIntegral, DefiniteIntegral, ConcreteDefiniteIntegral, ConcreteDefiniteLineIntegral,
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DefiniteLineIntegral, DefiniteIntegral, ConcreteDefiniteIntegral, ConcreteDefiniteLineIntegral, IntegralWrapper,
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ReverseOrientation, ReverseOrientationWrapper, ReverseWrapper, Reverse, NegateEven, Dirichlet, ConcreteDirichlet,
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TridiagonalOperator, SubOperator, Space, @containsconstants, spacescompatible,
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hasfasttransform, canonicalspace, domain, setdomain, prectype, domainscompatible,
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plan_transform, plan_itransform, plan_transform!, plan_itransform!, transform, itransform, hasfasttransform,
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CanonicalTransformPlan,
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CanonicalTransformPlan, ICanonicalTransformPlan,
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Integral,
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domainspace, rangespace, boundary,
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union_rule, conversion_rule, maxspace_rule, conversion_type, maxspace, hasconversion, points,
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rdirichlet, ldirichlet, lneumann, rneumann, ivp, bvp,
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linesum, differentiate, integrate, linebilinearform, bilinearform,
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UnsetNumber, coefficienttimes,
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UnsetNumber, coefficienttimes, subspace_coefficients, sumspacecoefficients, specialfunctionnormalizationpoint,
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Segment, IntervalOrSegmentDomain, PiecewiseSegment, isambiguous, Vec, eps, isperiodic,
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arclength, complexlength,
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invfromcanonicalD, fromcanonical, tocanonical, fromcanonicalD, tocanonicalD, canonicaldomain, setcanonicaldomain, mappoint,
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reverseorientation, checkpoints, evaluate, mul_coefficients, coefficients, isconvertible,
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clenshaw, ClenshawPlan, sineshaw,
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toeplitz_getindex, toeplitz_axpy!, sym_toeplitz_axpy!, hankel_axpy!, ToeplitzOperator, SymToeplitzOperator, hankel_getindex,
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SpaceOperator, ZeroOperator, InterlaceOperator,
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interlace!, reverseeven!, negateeven!, cfstype, pad!,
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extremal_args, hesseneigvals, chebyshev_clenshaw, recA, recB, recC, roots, chebmult_getindex, intpow, alternatingsum,
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domaintype, diagindshift, rangetype, weight, isapproxinteger, default_Dirichlet, scal!
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interlace!, reverseeven!, negateeven!, cfstype, pad!, alternatesign!, mobius,
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extremal_args, hesseneigvals, chebyshev_clenshaw, recA, recB, recC, roots,splitatroots,
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chebmult_getindex, intpow, alternatingsum,
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domaintype, diagindshift, rangetype, weight, isapproxinteger, default_Dirichlet, scal!, dotu,
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components, promoterangespace, promotedomainspace,
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block, blockstart, blockstop, blocklengths, isblockbanded, pointscompatible
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import DomainSets: Domain, indomain, UnionDomain, ProductDomain, FullSpace, Point, elements, DifferenceDomain,
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Interval, ChebyshevInterval, boundary, ∂, rightendpoint, leftendpoint,
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plan_chebyshevtransform!, plan_ichebyshevtransform, plan_ichebyshevtransform!,
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pochhammer
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import BlockBandedMatrices: blockbandwidths, subblockbandwidths
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# we need to import all special functions to use Calculus.symbolic_derivatives_1arg
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# we can't do importall Base as we replace some Base definitions
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import SpecialFunctions: sinpi, cospi, airy, besselh,

‎src/Domains/IntervalCurve.jl

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export IntervalCurve
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struct IntervalCurve{S<:Space,T,VT} <: SegmentDomain{T}
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curve::Fun{S,T,VT}

‎src/Spaces/Ultraspherical/ContinuousSpace.jl

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export ContinuousSpace
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struct ContinuousSpace{T,R} <: Space{PiecewiseSegment{T},R}
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domain::PiecewiseSegment{T}

‎src/roots.jl

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@@ -266,14 +266,6 @@ extrema(f::Fun{ContinuousSpace{T,R},T}) where {T<:Real,R<:Real} =
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mapreduce(extrema,(x,y)->extrema([x...;y...]),components(f))
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## Root finding for Laurent expansion
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function roots(f::Fun{P}) where P<:ContinuousSpace
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rts=mapreduce(roots,vcat,components(f))
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k=1

‎src/specialfunctions.jl

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locs[isjump]
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end
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#
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# These formulæ, appearing in Eq. (2.5) of:
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#
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# A.-K. Kassam and L. N. Trefethen, Fourth-order time-stepping for stiff PDEs, SIAM J. Sci. Comput., 26:1214--1233, 2005,
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#
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# are derived to implement ETDRK4 in double precision without numerical instability from cancellation.
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#
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expα_asy(x) = (exp(x)*(4-3x+x^2)-4-x)/x^3
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expβ_asy(x) = (exp(x)*(x-2)+x+2)/x^3
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expγ_asy(x) = (exp(x)*(4-x)-4-3x-x^2)/x^3
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# TODO: General types
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expα_taylor(x::Union{Float64,ComplexF64}) = @evalpoly(x,1/6,1/6,3/40,1/45,5/1008,1/1120,7/51840,1/56700,1/492800,1/4790016,11/566092800,1/605404800,13/100590336000,1/106748928000,1/1580833013760,1/25009272288000,17/7155594141696000,1/7508956815360000)
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expβ_taylor(x::Union{Float64,ComplexF64}) = @evalpoly(x,1/6,1/12,1/40,1/180,1/1008,1/6720,1/51840,1/453600,1/4435200,1/47900160,1/566092800,1/7264857600,1/100590336000,1/1494484992000,1/23712495206400,1/400148356608000,1/7155594141696000,1/135161222676480000)
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expγ_taylor(x::Union{Float64,ComplexF64}) = @evalpoly(x,1/6,0/1,-1/120,-1/360,-1/1680,-1/10080,-1/72576,-1/604800,-1/5702400,-1/59875200,-1/691891200,-1/8717829120,-1/118879488000,-1/1743565824000,-1/27360571392000,-1/457312407552000,-1/8109673360588800)
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expα(x::Float64) = abs(x) < 17/16 ? expα_taylor(x) : expα_asy(x)
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expβ(x::Float64) = abs(x) < 19/16 ? expβ_taylor(x) : expβ_asy(x)
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expγ(x::Float64) = abs(x) < 15/16 ? expγ_taylor(x) : expγ_asy(x)
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expα(x::ComplexF64) = abs2(x) < (17/16)^2 ? expα_taylor(x) : expα_asy(x)
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expβ(x::ComplexF64) = abs2(x) < (19/16)^2 ? expβ_taylor(x) : expβ_asy(x)
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expγ(x::ComplexF64) = abs2(x) < (15/16)^2 ? expγ_taylor(x) : expγ_asy(x)
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expα(x) = expα_asy(x)
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expβ(x) = expβ_asy(x)
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expγ(x) = expγ_asy(x)
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for f in (:(exp),:(expm1),:expα,:expβ,:expγ)
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@eval $f(op::Operator) = OperatorFunction(op,$f)
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end
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‎test/ETDRK4Test.jl

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‎test/ExtrasTest.jl

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‎test/FullPDETest.jl

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KO=Δ.op.ops[1].ops[1].op
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M=ApproxFun.BandedBlockBandedMatrix(view(KO,1:4,1:4))
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@test norm(ApproxFun.BandedBlockBandedMatrix(view(KO,1:4,2:4))-M[:,2:4]) < 10eps()
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@test norm(ApproxFun.BandedBlockBandedMatrix(view(KO,1:4,3:4))-M[:,3:4]) < 10eps()
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M=BandedBlockBandedMatrix(view(KO,1:4,1:4))
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@test norm(BandedBlockBandedMatrix(view(KO,1:4,2:4))-M[:,2:4]) < 10eps()
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@test norm(BandedBlockBandedMatrix(view(KO,1:4,3:4))-M[:,3:4]) < 10eps()
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end
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@testset "Check resizing" begin
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d=ChebyshevInterval()^2
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A=[Dirichlet(d);Laplacian()+100I]
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QRR = qr(A)
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@time ApproxFun.resizedata!(QRR.R_cache,:,2000)
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@time ApproxFunBase.resizedata!(QRR.R_cache,:,2000)
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@test norm(QRR.R_cache.data[1:200,1:200] - A[1:200,1:200]) ==0
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@time ApproxFun.resizedata!(QRR,:,200)
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@time ApproxFunBase.resizedata!(QRR,:,200)
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v=QRR.R_cache.op[1:100,j]
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@test norm(ldiv_coefficients(QRR.Q,v;maxlength=300)[j+1:end]) < 100eps()
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v=QRR.R_cache.op[1:ApproxFun.colstop(QRR.R_cache.op,j),j]
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v=QRR.R_cache.op[1:ApproxFunBase.colstop(QRR.R_cache.op,j),j]
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@test norm(ldiv_coefficients(QRR.Q,v;maxlength=1000)[j+1:end]) < 100eps()
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v=QRR.R_cache.op[1:ApproxFun.colstop(QRR.R_cache.op,j),j]
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v=QRR.R_cache.op[1:ApproxFunBase.colstop(QRR.R_cache.op,j),j]
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@test norm(ldiv_coefficients(QRR.Q,v;maxlength=1000)[j+1:end]) < j*20eps()
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@test ApproxFun.colstop(QRR.R_cache.op,195)-194 == ApproxFun.colstop(QRR.H,195)
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@test ApproxFunBase.colstop(QRR.R_cache.op,195)-194 == ApproxFunBase.colstop(QRR.H,195)
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QR1 = qr(A)
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@time ApproxFun.resizedata!(QR1.R_cache,:,1000)
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@time ApproxFunBase.resizedata!(QR1.R_cache,:,1000)
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QR2 = qr([Dirichlet(d);Laplacian()+100I])
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@time ApproxFun.resizedata!(QR2.R_cache,:,500)
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@time ApproxFunBase.resizedata!(QR2.R_cache,:,500)
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n=450;QR1.R_cache.data[1:n,1:n]-QR2.R_cache.data[1:n,1:n]|>norm
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@time ApproxFun.resizedata!(QR2.R_cache,:,1000)
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@time ApproxFunBase.resizedata!(QR2.R_cache,:,1000)
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N=450;QR1.R_cache.data[1:N,1:N]-QR2.R_cache.data[1:N,1:N]|>norm
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N=1000;QR1.R_cache.data[1:N,1:N]-QR2.R_cache.data[1:N,1:N]|>norm
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QR1 = qr(A)
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@time ApproxFun.resizedata!(QR1,:,1000)
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@time ApproxFunBase.resizedata!(QR1,:,1000)
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QR2 = qr([Dirichlet(d);Laplacian()+100I])
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@time ApproxFun.resizedata!(QR2,:,500)
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@time ApproxFun.resizedata!(QR2,:,1000)
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@time ApproxFunBase.resizedata!(QR2,:,500)
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@time ApproxFunBase.resizedata!(QR2,:,1000)
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@test norm(QR1.H[1:225,1:1000]-QR2.H[1:225,1:1000]) ≤ 10eps()
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QR1 = qr(A)
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@time ApproxFun.resizedata!(QR1,:,5000)
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@time ApproxFunBase.resizedata!(QR1,:,5000)
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@time u=\(QR1,[ones(∂(d));0.];tolerance=1E-7)
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@test norm((Dirichlet(d)*u-ones(∂(d))).coefficients) < 1E-7
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@test norm((A*u-Fun([ones(∂(d));0.])).coefficients) < 1E-7
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@test norm(((A*u)[2]-(Laplacian(space(u))+100I)*u).coefficients) < 1E-10
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@test eltype(ApproxFun.promotedomainspace(Laplacian(),space(u))) == Float64
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@test eltype(ApproxFun.promotedomainspace(Laplacian()+100I,space(u))) == Float64
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@test eltype(ApproxFunBase.promotedomainspace(Laplacian(),space(u))) == Float64
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@test eltype(ApproxFunBase.promotedomainspace(Laplacian()+100I,space(u))) == Float64
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@test norm(((A*u)[2]-(Laplacian()+100I)*u).coefficients) < 1E-10
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@test norm((Laplacian()*u+100*u - (A*u)[2]).coefficients) < 10E-10
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@time v=\(A,[ones(∂(d));0.];tolerance=1E-7)
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@test ApproxFun.rangetype(rangespace(continuity(s,0))) == Vector{Float64}
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@test ApproxFun.rangetype(rangespace(Bx)) == Vector{Any}
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@test ApproxFun.rangetype(rangespace(Bx⊗Operator(I,Chebyshev()))) == Vector{Any}
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@test ApproxFunBase.rangetype(rangespace(continuity(s,0))) == Vector{Float64}
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@test ApproxFunBase.rangetype(rangespace(Bx)) == Vector{Any}
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@test ApproxFunBase.rangetype(rangespace(Bx⊗Operator(I,Chebyshev()))) == Vector{Any}
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rhs = Fun([0,[0,[0,0]],0],rangespace([I⊗ldirichlet(dt);Bx⊗I;I⊗Dt+(a*Dx)⊗I]))
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@test rhs(-0.5,0.0) == [0,[0,[0,0]],0]

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