Support BigFloat in Hermite (#127)

* Support BigFloat in Hermite

* revert axes type

* allow powers of Hermite weight

Author: dlfivefifty

src/classical/hermite.jl

@@ -6,20 +6,27 @@ is a quasi-vector representing `exp(-x^2)` on ℝ.
 struct HermiteWeight{T} <: Weight{T} end
 
 HermiteWeight() = HermiteWeight{Float64}()
-axes(::HermiteWeight{T}) where T = (Inclusion(ℝ),)
+axes(::HermiteWeight{T}) where T = (Inclusion{T}(ℝ),)
 function getindex(w::HermiteWeight, x::Number)
     x ∈ axes(w,1) || throw(BoundsError())
     exp(-x^2)
 end
 
 sum(::HermiteWeight{T}) where T = sqrt(convert(T, π))
 
+function broadcasted(::typeof(^), H::HermiteWeight, k::Number)
+    x = axes(H,1)
+    exp.((-k) .* x .^2)
+end
+
+broadcasted(::typeof(sqrt), H::HermiteWeight{T}) where T = H .^ (one(T)/2)
+
 struct Hermite{T} <: OrthogonalPolynomial{T} end
 Hermite() = Hermite{Float64}()
 orthogonalityweight(::Hermite{T}) where T = HermiteWeight{T}()
 
 ==(::Hermite, ::Hermite) = true
-axes(::Hermite{T}) where T = (Inclusion(ℝ), oneto(∞))
+axes(::Hermite{T}) where T = (Inclusion{T}(ℝ), oneto(∞))
 
 """
      hermiteh(n, z)

test/test_hermite.jl

@@ -31,12 +31,15 @@ import DomainSets: ℝ
     end
 
     @testset "OrthonormalWeighted" begin
-        H = Hermite()
-        Q = OrthonormalWeighted(H)
-
-        @test Q'Q == Eye(∞)
+        @testset "orthogonality" begin
+            H = Hermite()
+            Q = OrthonormalWeighted(H)
+            @test Q'Q == Eye(∞)
+        end
 
         @testset "evaluation" begin
+            H = Hermite()
+            Q = OrthonormalWeighted(H)
             x = 0.1
             @test Q[x,1] ≈ exp(-x^2/2)/π^(1/4)
             @test Q[x,2] ≈ 2x*exp(-x^2/2)/(sqrt(2)π^(1/4))
@@ -46,6 +49,8 @@ import DomainSets: ℝ
         end
 
         @testset "Differentiation" begin
+            H = Hermite()
+            Q = OrthonormalWeighted(H)
             x = axes(Q,1)
             D = Derivative(x)
             D¹ = Q \ (D * Q)
@@ -54,5 +59,10 @@ import DomainSets: ℝ
             X = Q \ (x .* Q)
             @test (D² - X^2)[1:10,1:10] ≈ -Diagonal(1:2:19)
         end
+
+        @testset "BigFloat" begin
+            Q = OrthonormalWeighted(Hermite{BigFloat}())
+            @test Q[40.0, 1] ≈ exp(-big(40)^2/2) * big(π)^(-1/4)
+        end
     end
 end