Power series Temme-like method for near-integer/integer orders that

gives valid derivatives. Tests TODO/to debug still.

Author: Chris Geoga

src/BesselFunctions/besselk.jl

@@ -499,38 +499,25 @@ besselk_power_series(v, x::Float32) = Float32(besselk_power_series(v, Float64(x)
 besselk_power_series(v, x::ComplexF32) = ComplexF32(besselk_power_series(v, ComplexF64(x)))
 
 function besselk_power_series(v, x::ComplexOrReal{T}) where T
-    MaxIter = 1000
-    S = eltype(x)
-    v, x = S(v), S(x)
-
-    z  = x / 2
-    zz = z * z
-    logz = log(z)
-    xd2_v = exp(v*logz)
-    xd2_nv = inv(xd2_v)
-
-    # use the reflection identify to calculate gamma(-v)
-    # use relation gamma(v)*v = gamma(v+1) to avoid two gamma calls
-    gam_v = gamma(v)
-    gam_nv = π / (sinpi(-abs(v)) * gam_v * v)
-    gam_1mv = -gam_nv * v
-    gam_1mnv = gam_v * v
-
-    _t1 = gam_v * xd2_nv * gam_1mv
-    _t2 = gam_nv * xd2_v * gam_1mnv
-    (xd2_pow, fact_k, out) = (one(S), one(S), zero(S))
-    for k in 0:MaxIter
-        t1 = xd2_pow * T(0.5)
-        tmp = muladd(_t1, gam_1mnv, _t2 * gam_1mv)
-        tmp *= inv(gam_1mv * gam_1mnv * fact_k)
-        term = t1 * tmp
-        out += term
-        abs(term / out) < eps(T) && break
-        (gam_1mnv, gam_1mv) = (gam_1mnv*(one(S) + v + k), gam_1mv*(one(S) - v + k)) 
-        xd2_pow *= zz
-        fact_k *= k + one(S)
+    Math.isnearint(v) && return besselk_power_series_int(v, x)
+    MaxIter = 5000
+    gam = gamma(v)
+    ngam = π / (sinpi(-abs(v)) * gam * v)
+
+    s1, s2 = zero(T), zero(T)
+    t1, t2 = one(T), one(T)
+
+    for k in 1:MaxIter
+        s1 += t1
+        s2 += t2
+        t1 *= x^2 / (4k * (k - v))
+        t2 *= x^2 / (4k * (k + v))
+        abs(t1) < eps(T) && break
     end
-    return out
+
+    xpv = (x/2)^v
+    s = gam * s1 + xpv^2 * ngam * s2
+    return s / (2*xpv)
 end
 besselk_power_series_cutoff(nu, x::Float64) = x < 2.0 || nu > 1.6x - 1.0
 besselk_power_series_cutoff(nu, x::Float32) = x < 10.0f0 || nu > 1.65f0*x - 8.0f0
@@ -615,3 +602,68 @@ end
         end
     )
 end
+
+# This is an expansion of the function
+#
+# f_0(v, x) = (x^v)*gamma(-v)  + (x^(-v))*gamma(v)
+#           = (x^v)*(gamma(-v) + (x^(-2*v))*gamma(v))
+#
+# around v ∼ 0. As you can see by plugging that second form into Wolfram Alpha
+# and getting an expansion back, this is actually a bivariate polynomial in
+# (v^2, log(x)). So that's how this is structured.
+@inline function f0_local_expansion_v0(v, x)
+    lx = log(x)
+    c0 = evalpoly(lx, (-1.1544313298030657,  -2.0))
+    c2 = evalpoly(lx, ( 1.4336878944573288,  -1.978111990655945, -0.5772156649015329,  -0.3333333333333333))
+    c4 = evalpoly(lx, (-0.6290784463642211,  -1.4584260788225176,   -0.23263776388631713, -0.32968533177599085, -0.048101305408461074, -0.016666666666666666))
+    evalpoly(v*v, (c0,c2,c4))/2
+end
+
+# This function assumes |v| < 1e-6 or 1e-7!
+# 
+# TODO (cg 2023/05/16 18:07): lots of micro-optimizations.
+function besselk_power_series_temme_basal(v::V, x::Float64) where{V}
+    max_iter = 50
+    T   = promote_type(V,Float64)
+    z   = x/2
+    zz  = z*z
+    fk  = f0_local_expansion_v0(v, x/2)
+    zv  = z^v
+    znv = inv(zv)
+    gam_1pv = GammaFunctions.gamma_near_1(1+v)
+    gam_1nv = GammaFunctions.gamma_near_1(1-v)
+    (pk, qk, _ck, factk, vv) = (znv*gam_1pv/2, zv*gam_1nv/2, one(T), one(T), v*v)
+    (out_v, out_vp1) = (zero(T), zero(T))
+    for k in 1:max_iter
+        # add to the series:
+        ck       = _ck/factk
+        term_v   = ck*fk
+        term_vp1 = ck*(pk - (k-1)*fk)
+        out_v   += term_v
+        out_vp1 += term_vp1
+        # check for convergence:
+        ((abs(term_v) < eps(T)) && (abs(term_vp1) < eps(T))) && break
+        # otherwise, increment new quantities:
+        fk     = (k*fk + pk + qk)/(k^2 - vv)
+        pk    /= (k-v)
+        qk    /= (k+v)
+        _ck   *= zz
+        factk *= k
+    end
+    (out_v, out_vp1/z)
+end
+
+function besselk_power_series_int(v, x::Float64)
+    v < zero(v) && return besselk_power_series_int(-v, x)
+    flv = Int(floor(v))
+    _v  = v - flv
+    (kv, kvp1) = besselk_power_series_temme_basal(_v, x)
+    abs(v) < 1/2 && return kv
+    twodx = 2/x
+    for _ in 1:(flv-1)
+        _v += 1
+        (kv, kvp1) = (kvp1, muladd(twodx*_v, kvp1, kv))
+    end
+    kvp1
+end
+

src/GammaFunctions/gamma.jl

@@ -113,3 +113,5 @@ function gamma(n::Integer)
     n > 20 && return gamma(float(n))
     @inbounds return Float64(factorial(n-1))
 end
+
+gamma_near_1(x) = evalpoly(x-one(x), (1.0, -0.5772156649015329, 0.9890559953279725, -0.23263776388631713))

src/Math/Math.jl

@@ -154,4 +154,7 @@ end
     )
 end
 
+# TODO (cg 2023/05/16 18:09): dispute this cutoff.
+isnearint(x) = abs(x-round(x)) < 1e-7
+
 end