Update README for test data generation

Author: Chris Geoga

test/data/besselk/enzyme/README.md

@@ -1,20 +1,19 @@
 ```julia
-using FiniteDifferences, ArbNumerics, DelimitedFiles, SpecialFunctions
-
-if !(@isdefined vx)
-  vgrid = sort(vcat(range(0.05, 15.0, length=20), [0.5, 1.5, 2.5, 3.5]))
-  xgrid = range(15.0, 30.0, length=20)
-  const vx = vec(collect(Iterators.product(vgrid, xgrid)))
-end
+using ArbNumerics, DelimitedFiles, SpecialFunctions
 
+# Because you can't just use FiniteDifferences due to the "numerical noise".
 simplefd(f,x,h=ArbReal(1e-40)) = (f(x+h)-f(x))/h
 
 ArbNumerics.setprecision(ArbReal, digits=100)
+arb_besselk(v,x)  = ArbNumerics.besselk(ArbReal(v), ArbReal(x))
+arb_besselkx(v,x) = arb_besselk(v,x)*ArbNumerics.exp(ArbReal(x))
 
-function arb_besselkx(v,x) 
-  (av, ax) = ArbReal.((v,x))
-  ArbNumerics.besselk(av, ax)*ArbNumerics.exp(ax)
-end
+#
+# besselkx_levin test:
+#
+vgrid = sort(vcat(range(0.05, 15.0, length=20), [0.5, 1.5, 2.5, 3.5]))
+xgrid = range(15.0, 30.0, length=20)
+vx = vec(collect(Iterators.product(vgrid, xgrid)))
 
 ref_values = map(vx) do vxj
   (v,x) = vxj
@@ -29,4 +28,25 @@ out_matrix = hcat(getindex.(vx, 1),          # test v argument
                   getindex.(ref_values, 2))  # test d/dx value
 
 writedlm("besselkx_levin_enzyme_tests.csv", out_matrix)
+
+#
+# besselk_power_series test:
+#
+vgrid = [-1e-8, 1e-8, 1-1e-8, 1.0, 1+1e-8, 2-1e-8, 2.0, 2+1e-8, 3-1e-8, 3.0, 3+1e-8] 
+xgrid = range(1e-5, 1.5, length=20)
+vx = vec(collect(Iterators.product(vgrid, xgrid)))
+
+ref_values = map(vx) do vxj
+  (v,x) = vxj
+  dx = simplefd(_x->arb_besselk(v, _x), x) # NOT besselkx!
+  dv = simplefd(_v->arb_besselk(_v, x), v) # NOT besselkx!
+  Float64.((dv, dx))
+end
+
+out_matrix = hcat(getindex.(vx, 1),          # test v argument
+                  getindex.(vx, 2),          # test x argument
+                  getindex.(ref_values, 1),  # test d/dv value
+                  getindex.(ref_values, 2))  # test d/dx value
+
+writedlm("besselk_power_series_enzyme_tests.csv", out_matrix)
 ```