Add normalize option to gausshermite (#122)

* added probabilist weights

* added tests

* removed extraneous file

src/gausshermite.jl

@@ -1,12 +1,18 @@
 @doc raw"""
-    gausshermite(n::Integer) -> x, w  # nodes, weights
+    gausshermite(n::Integer; normalize = false) -> x, w  # nodes, weights
 
 Return nodes `x` and weights `w` of [Gauss-Hermite quadrature](https://en.wikipedia.org/wiki/Gauss%E2%80%93Hermite_quadrature).
 
 ```math
 \int_{-\infty}^{+\infty} f(x) \exp(-x^2) dx \approx \sum_{i=1}^{n} w_i f(x_i)
 ```
 
+The option `normalize=true` instead computes
+```math
+\int_{-\infty}^{+\infty} f(x) \phi(x) dx \approx \sum_{i=1}^{n} w_i f(x_i),
+```
+where $\phi$ is the standard normal density function.
+
 # Examples
 ```jldoctest
 julia> x, w = gausshermite(3);
@@ -19,10 +25,10 @@ julia> I ≈ 3(√π)/4
 true
 ```
 """
-function gausshermite(n::Integer)
+function gausshermite(n::Integer; normalize = false)
     x,w = unweightedgausshermite(n)
     w .*= exp.(-x.^2)
-    x, w
+    normalize ? ( sqrt(2.0) * x, w / sqrt( π ) ) : ( x, w )
 end
 
 function unweightedgausshermite(n::Integer)

test/test_gausshermite.jl

@@ -73,6 +73,21 @@
         @test FastGaussQuadrature.hermpoly_rec(0:20^2,20)[end] ≈ FastGaussQuadrature.hermpoly_rec(20^2,20)[1]
     end
 
+    @testset "Normalize" begin
+        for t ∈ 1:6
+            x,w = gausshermite(t; normalize = true)
+            N = 2t - 1 
+            v = 1.0
+            for n ∈ 2:2:N
+                v *= (n-1)
+                @test v ≈ dot( x.^n, w )
+            end
+            for n ∈ 1:2:N
+                @test abs( 0.0 - dot( x.^n, w ) ) < 10 * tol
+            end
+        end
+    end
+
     @testset "All" begin
         for n in 2:220
             x,w = gausshermite(n)