Fix negative order case for sphericalbesselk in special integer routine and add tests.

Author: cgeoga

src/modifiedsphericalbessel.jl

@@ -8,7 +8,14 @@ sphericalbesselk(nu, x) = _sphericalbesselk(nu, float(x))
 
 function _sphericalbesselk(nu, x::T) where T
     if isinteger(nu) && nu < 41.5
-        return sphericalbesselk_int(nu, x)
+        # using ifelse here to hopefully cut out a branch on nu < 0 or not. The
+        # symmetry here is that 
+        # k_{-n} = (...)*K_{-n     + 1/2}
+        #        = (...)*K_{|n|    - 1/2}
+        #        = (...)*K_{|n|-1  + 1/2}
+        #        = k_{|n|-1}  
+        _nu = ifelse(nu<zero(nu), abs(nu)-one(nu), nu)
+        return sphericalbesselk_int(Int(_nu), x)
     else
         return inv(SQRT_PID2(T)*sqrt(x))*besselk(nu+1/2, x)
     end

test/sphericalbessel_test.jl

@@ -58,3 +58,9 @@ v, x = -4.0, 5.6
 v, x = -6.8, 15.6
 @test isapprox(Bessels.sphericalbesselj(v, x), 0.04386355397884301866595, rtol=3e-12)
 @test isapprox(Bessels.sphericalbessely(v, x), 0.05061013363904335437354, rtol=3e-12)
+
+# test for negative order of spherical modified besselk in the special integer
+# routine:
+for v in 1:10
+  @test Bessels.sphericalbesselk(-v, 1.1) ≈ Bessels.sphericalbesselk(v-1, 1.1)
+end