Switch convergence tests to Math.check_convergence, add second rule to

`Enzyme` extension for two-argument `Math.check_convergence`.

Author: Chris Geoga

ext/BesselsEnzymeCoreExt.jl

@@ -24,6 +24,13 @@ module BesselsEnzymeCoreExt
     (t.val <= eps(T)) && (t.dval <= eps(T))
   end
 
+  function EnzymeRules.forward(func::Const{typeof(check_convergence)},
+                               ::Type{Const{Bool}},
+                               t::Duplicated{T},
+                               s::Duplicated{T}) where{T}
+    (t.val <= eps(T)*s.val) && (t.dval <= eps(T)*s.dval)
+  end
+
   # This will be fixed upstream: see #861 for Enzyme.jl whenever the next
   # release occurs.
   function EnzymeRules.forward(func::Const{typeof(sinpi)}, 

src/BesselFunctions/besseli.jl

@@ -624,7 +624,7 @@ function besseli_power_series(v, x::ComplexOrReal{T}) where T
     xx = x * x * T(0.25)
     for i in 0:MaxIter
         s += t
-        abs(t) < eps(T) * abs(s) && break
+        Math.check_convergence(t, s) && break
         t *= xx / ((v + i + 1) * (i + 1))
     end
     return s * ((x/2)^v / gamma(v + 1))

src/BesselFunctions/besselk.jl

@@ -458,7 +458,7 @@ function besselkx_large_args(v, x::ComplexOrReal{T}) where T
     for i in 1:MaxIter
         t *= invx * ((4*v^2 - (2i - 1)^2) / i)
         s += t
-        abs(t) <= eps(T) && break
+        Math.check_convergence(t) && break
     end
     return s * sqrt(π / (2 * x))
 end
@@ -552,7 +552,7 @@ function besselk_power_series(v, x::ComplexOrReal{T}) where T
         s2 += t2
         t1 *= x^2 / (4k * (k - v))
         t2 *= x^2 / (4k * (k + v))
-        abs(t1) < eps(T) && break
+        Math.check_convergence(t1) && break
     end
 
     xpv = (x/2)^v
@@ -590,8 +590,7 @@ function besselk_temme_series(v::T, x::T) where T <: Float64
         term_vp1 = ck * (pk - (k-1) * fk)
         out_v += term_v
         out_vp1 += term_vp1
-        ((abs(term_v) < eps(T)) && (abs(term_vp1) < eps(T))) && break
-
+        (Math.check_convergence(term_v) && Math.check_convergence(term_vp1)) && break
         fk = (k * fk + pk + qk) / (k^2 - v^2)
         pk /= k - v
         qk /= k + v