From a51e8e7a41180ebd9cfc8a0fe95fd165956a93c2 Mon Sep 17 00:00:00 2001
From: James Cook <cookjws@gmail.com>
Date: Fri, 20 Oct 2023 07:28:10 +0100
Subject: [PATCH] Dualbesselix (#11)

* bugfix Dual of besselix
---
 src/Maths.jl        |  5 +++--
 test/maths/Maths.jl | 17 ++++++++++++++++-
 2 files changed, 19 insertions(+), 3 deletions(-)

diff --git a/src/Maths.jl b/src/Maths.jl
index 618bac5..1ef35bf 100644
--- a/src/Maths.jl
+++ b/src/Maths.jl
@@ -20,11 +20,12 @@ end
 
 
 import SpecialFunctions.besselix
-function besselix(n::Integer, x::DualNumbers.Dual)
+function besselix(n::Integer, x::DualNumbers.Dual{T}) where T
   r, d = realpart(x), dualpart(x)
   bix = besselix(n, r)
+  didr = (besseli(n - 1, r) + besseli(n + 1, r)) / 2
   return Dual(bix,
-    d * ((besselix(n - 1, r) + besselix(n + 1, r)) / 2 - abs(real(r)) * bix))
+    d * (didr * exp(-abs(real(r))) - sign(real(r)) * bix))
 end
 
 
diff --git a/test/maths/Maths.jl b/test/maths/Maths.jl
index 31b68a7..326142e 100644
--- a/test/maths/Maths.jl
+++ b/test/maths/Maths.jl
@@ -3,7 +3,7 @@ println("$(now()) $(@__FILE__)")
 
 using LinearMaxwellVlasov
 using Test, SpecialFunctions, QuadGK, HCubature, StaticArrays, Random
-using DualNumbers
+using DualNumbers, ForwardDiff
 
 const LMV = LinearMaxwellVlasov
 
@@ -128,4 +128,19 @@ const LMV = LinearMaxwellVlasov
       (besselj(0+im, 1.0) - besselj(2.0+im, 1.0)) / 2
   end
 
+  @testset "Duals vs ForwardDiff" begin
+    for n in (3, 4, -3, -4), x in (-5.0, 5.0)
+      @test DualNumbers.dualpart(besselix(n, Dual(x, 1))) ≈
+        ForwardDiff.derivative(z->besselix(n, z), x)
+      @test DualNumbers.dualpart(besselj(n, Dual(x, 1))) ≈
+        ForwardDiff.derivative(z->besselj(n, z), x)
+    end
+    for n in (3, 4), x in (-5.0, 5.0)
+      @assert ForwardDiff.derivative(z->besseli(n, z), x) ≈
+        (besseli(n-1, x) + besseli(n+1, x)) / 2
+      @test DualNumbers.dualpart(n^Dual(x, 1)) ≈
+        ForwardDiff.derivative(z->n^z, x)
+    end
+  end
+
 end