update docs

Author: heltonmc

src/Bessels.jl

@@ -35,6 +35,7 @@ include("Float128/constants.jl")
 
 include("math_constants.jl")
 include("U_polynomials.jl")
+include("recurrence.jl")
 #include("parse.jl")
 #include("hankel.jl")
 

src/U_polynomials.jl

@@ -50,38 +50,8 @@ function Uk_poly_Kn(p, v, p2, ::Type{T}) where T <: BigFloat
     u10 = 1 / 88580102706155225088000 * evalpoly(p2, (9745329584487361980740625, -1230031256571145165088463750, 27299183373230345667273718125, -246750339886026017414509498824, 1177120360439828012193658602930, -3327704366990695147540934069220, 5876803711285273203043452095250, -6564241639632418015173104205000, 4513386761946134740461797128125, -1745632061522350031610173343750, 290938676920391671935028890625))
     return evalpoly(-p/v, (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10))
 end
-function besselk_large_orders(v, x::T) where T <: AbstractFloat
-    x = Float64(x)
-    z = x / v
-    zs = sqrt(1 + z^2)
-    n = zs + log(z / (1 + zs))
-    coef = exp(-v * n) * sqrt(pi / (2*v)) / sqrt(zs)
-    p = inv(zs)
-    p2  = p*p
-    u0 = one(Float64)
-    u1 = 1 / 24 * evalpoly(p2, (3, -5))
-    u2 = 1 / 1152 * evalpoly(p2, (81, -462, 385))
-    u3 = 1 / 414720 * evalpoly(p2, (30375, -369603, 765765, -425425))
-    u4 = 1 / 39813120 * evalpoly(p2, (4465125, -94121676, 349922430, -446185740, 185910725))
-    u5 = 1 / 6688604160 * evalpoly(p2, (1519035525, -49286948607, 284499769554, -614135872350, 566098157625, -188699385875))
-    u6 = 1 / 4815794995200 * evalpoly(p2, (2757049477875, -127577298354750, 1050760774457901, -3369032068261860,5104696716244125, -3685299006138750, 1023694168371875))
-    u7 = 1 / 115579079884800 * evalpoly(p2, (199689155040375, -12493049053044375, 138799253740521843, -613221795981706275, 1347119637570231525, -1570320948552481125, 931766432052080625,  -221849150488590625))
-    u8 = 1 / 22191183337881600 * evalpoly(p2, (134790179652253125, -10960565081605263000, 157768535329832893644, -914113758588905038248, 2711772922412520971550, -4513690624987320777000, 4272845805510421639500, -2152114239059719935000, 448357133137441653125))
-    u9 = 1 / 263631258054033408000 * evalpoly(p2, (6427469716717690265625, -659033454841709672064375, 11921080954211358275362500, -87432034049652400520788332, 334380732677827878090447630, -741743213039573443221773250, 992115946599792610768672500, -790370708270219620781737500, 345821892003106984030190625, -64041091111686478524109375))
-    u10 = 1 / 88580102706155225088000 * evalpoly(p2, (9745329584487361980740625, -1230031256571145165088463750, 27299183373230345667273718125, -246750339886026017414509498824, 1177120360439828012193658602930, -3327704366990695147540934069220, 5876803711285273203043452095250, -6564241639632418015173104205000, 4513386761946134740461797128125, -1745632061522350031610173343750, 290938676920391671935028890625))
 
-    return coef*evalpoly(-p/v, (u0, u1, u2, u3, u4, u5, u6, u7, u8, u9, u10))
-end
 
-function besselk_large_orders(v, x::T) where T <: AbstractFloat
-    z = x / v
-    zs = sqrt(1 + z^2)
- 
-    n = zs + log(z / (1 + zs))
-    coef = exp(-v * n) * sqrt(pi / (2*v)) / sqrt(zs)
- 
-    p = inv(zs)
- 
     u0 = one(x)
     u1 = p / 24 * (3 - 5*p^2) * -1 / v
     u2 = p^2 / 1152 * (81 - 462*p^2 + 385*p^4) / v^2
@@ -109,27 +79,4 @@ function besselk_large_orders(v, x::T) where T <: AbstractFloat
         + 5136561256208409671660362298619778869859994724706875*p^16 - 2284251621937242886581917667066122422330060024456125*p^14 + 730367145705123976114617970888707594104468177381925*p^12 - 163359140754958502896104062604202934925448173291477*p^10
         + 24403480234538299231733883413666768614198435948125*p^8 - 2254933495791765108580529087615802250458013685625*p^6 + 112597271053778779753048514469995937998172890625*p^4 - 2303431987527333128955769182299845911305390625*p^2 + 8178936810213560828419581728001773291015625) * -1 / v^15
 
-
-    out1 = ((u15 + u14 + u13 + u12 + u11 + u10 + u9) + u8) + u7
-    out2 = ((out1 + u6 + u5) + u4) + u3
-    out = ((out2 + u2) + u1) + u0
- 
-    return coef*out
- end
-
-    u11 = 1 / 27636992044320430227456000 * (-1363312191078326248171914924296875*p^22 + 8997860461116953237934638500359375*p^20 - 25963913760458280822131901737878125*p^18 + 42936745153513007436411401865860625*p^16 - 44801790321820682384740638703320750*p^14 + 30589806122850866110941936529264950*p^12 - 13704902022868787041097596217578170*p^10
-        + 3926191452593448964331218647028194*p^8 - 676389476843440422173605288596087*p^6 + 62011003282542082472252466220875*p^4 - 2321657500166464779660536015625*p^2 + 15237265774872558064250728125) * -1 / v^11
-    u12 = 1 / 39797268543821419527536640000 * (32426380464797989812768996474401171875*p^24 - 233469939346545526651936774615688437500*p^22 + 743739612850105971846901081692858843750*p^20 - 1378260730939829908037976894025628887500*p^18 + 1642838631056253395899341314188134178125*p^16 - 1314368459332124683504418467809129387000*p^14
-        + 714528665351965363868467348116538170900*p^12 - 261201165596865827608687437905740780920*p^10 + 62055079517573388459132793029571876461*p^8 - 8958590476947726766450604043798559500*p^6 + 692277766674325563109617997687563750*p^4 - 21882222767154197351962677311437500*p^2 + 120907703923613748239829527671875) * 1 / v^12
-    u13 = 1 / 955134445051714068660879360000 * (-14020668045586884672121117629431460546875*p^26 + 109361210755577700442544717509565392265625*p^24 - 381190503845282445953419057314665534156250*p^22 + 782463969315283937856703223178540650343750*p^20
-        - 1049095945162229046324321461816274931715625*p^18 + 962926533925253374359704288384340809260875*p^16 - 616410216242554698436702353237166008656700*p^14 + 274983827478138958623041508409195988431140*p^12 - 83924867223075156862785921508524155665245*p^10
-        + 16843538631795795357786827345307534156063*p^8 - 2069933923586966756183324291232117362250*p^6 + 136735019134677724428035696765082813750*p^4 - 3698121486504259988094897605296209375*p^2 + 17438611142828905996129258798828125) * -1 / v^13
-    u14 = 1 / 45846453362482275295722209280000 * (13133360053559028970728309756597440972265625*p^28 - 110320224449895843354117801955418504167031250*p^26 + 417630985812040040477569028872405152769921875*p^24
-        - 940627071986145750405920450097257805227812500*p^22 + 1401302601668131482630653233972052729323190625*p^20 - 1451823699927947602004297385351260623500372750*p^18 + 1070439683260179398514645952824098811025619475*p^16
-        - 564850830044980230366140582063618983657685400*p^14 + 211477117385619365164298957821904115470535555*p^12 - 54857705817658080981995319669299096598482382*p^10 + 9440449669103391509091075981237243128469201*p^8
-        - 1000503839668383458999731491914516564625300*p^6 + 57170953417612444837142230812990944671875*p^4 - 1338184074771428116079233795614103631250*p^2 + 5448320367052402487647812713291015625) * 1 / v^14
-    u15 = 1 / 9820310310243703368343697227776000000 * (- 59115551939078562227317999668652486368337724609375*p^30 + 532039967451707060045861997017872377315039521484375*p^28 - 2173722139119126857509156976742601742357422740234375*p^26
-        + 5329871927856528282113994744458999865006055974609375*p^24 - 8735135969643867540297524795790262235822823374296875*p^22 + 10085018700249896522602267572484630409640764997271875*p^20 - 8420533422834140468835467666391400380550043688871875*p^18
-        + 5136561256208409671660362298619778869859994724706875*p^16 - 2284251621937242886581917667066122422330060024456125*p^14 + 730367145705123976114617970888707594104468177381925*p^12 - 163359140754958502896104062604202934925448173291477*p^10
-        + 24403480234538299231733883413666768614198435948125*p^8 - 2254933495791765108580529087615802250458013685625*p^6 + 112597271053778779753048514469995937998172890625*p^4 - 2303431987527333128955769182299845911305390625*p^2 + 8178936810213560828419581728001773291015625) * -1 / v^15
 =#

src/besseli.jl

@@ -4,6 +4,9 @@
 #    Scaled modified Bessel functions of the first kind of order zero and one
 #                       besseli0x, besselix
 #
+#    (Scaled) Modified Bessel functions of the first kind of order nu
+#                       besseli, besselix
+#
 #    Calculation of besseli0 is done in two branches using polynomial approximations [1]
 #
 #    Branch 1: x < 7.75 
@@ -35,11 +38,18 @@
 #
 #    Horner's scheme is then used to evaluate all polynomials.
 #    ArbNumerics.jl is used as the reference bessel implementations with 75 digits.
+#
+#    Calculation of besseli and besselkx can be done with downward recursion starting with
+#    besseli_{nu+1} and besseli_{nu}. Higher orders are determined by a uniform asymptotic
+#    expansion similar to besselk (see notes there) using Equation 10.41.3 [3].
+#
 # 
 # [1] "Rational Approximations for the Modified Bessel Function of the First Kind 
 #     - I0(x) for Computations with Double Precision" by Pavel Holoborodko     
 # [2] "Rational Approximations for the Modified Bessel Function of the First Kind 
-#     - I1(x) for Computations with Double Precision" by Pavel Holoborodko        
+#     - I1(x) for Computations with Double Precision" by Pavel Holoborodko
+# [3] https://dlmf.nist.gov/10.41
+
 """
     besseli0(x::T) where T <: Union{Float32, Float64}
 
@@ -56,6 +66,7 @@ function besseli0(x::T) where T <: Union{Float32, Float64}
         return a * s
     end
 end
+
 """
     besseli0x(x::T) where T <: Union{Float32, Float64}
 
@@ -73,6 +84,7 @@ function besseli0x(x::T) where T <: Union{Float32, Float64}
         return evalpoly(inv(x), besseli0_large_coefs(T)) / sqrt(x)
     end
 end
+
 """
     besseli1(x::T) where T <: Union{Float32, Float64}
 
@@ -94,6 +106,7 @@ function besseli1(x::T) where T <: Union{Float32, Float64}
     end
     return z
 end
+
 """
     besseli1x(x::T) where T <: Union{Float32, Float64}
 
@@ -136,6 +149,7 @@ function besseli(nu, x::T) where T <: Union{Float32, Float64, BigFloat}
         return besseli_large_orders(nu, x)
     end
 end
+
 """
     besselix(nu, x::T) where T <: Union{Float32, Float64}
 
@@ -155,24 +169,6 @@ function besselix(nu, x::T) where T <: Union{Float32, Float64, BigFloat}
         return besseli_large_orders_scaled(nu, x)
     end
 end
-
-
-@inline function down_recurrence(x, in, inp1, nu, branch)
-    # this prevents us from looping through large values of nu when the loop will always return zero
-    (iszero(in) || iszero(inp1)) && return zero(x)
-    (isinf(inp1) || isinf(inp1)) && return in
-
-    inm1 = in
-    x2 = 2 / x
-    for n in branch:-1:nu+1
-        a = x2 * n
-        inm1 = muladd(a, in, inp1)
-        inp1 = in
-        in = inm1
-    end
-    return inm1
-end
-
 function besseli_large_orders(v, x::T) where T <: Union{Float32, Float64, BigFloat}
     S = promote_type(T, Float64)
     x = S(x)
@@ -196,4 +192,4 @@ function besseli_large_orders_scaled(v, x::T) where T <: Union{Float32, Float64,
     p2  = v^2/fma(max(v,x), max(v,x), min(v,x)^2)
 
     return T(coef*Uk_poly_In(p, v, p2, T))
-end
+end

src/besselk.jl

@@ -55,34 +55,7 @@
 #
 #    However, calculation of these higher order U polynomials are tedious. These have been hand
 #    calculated and somewhat crosschecked with symbolic math. There could be errors. They are listed
-#    here as a reference as higher orders are impossible to find while being needed for any meaningfully accurate calculation.
-
-#    u0 = one(x)
-#    u1 = p / 24 * (3 - 5*p^2) * -1 / v
-#    u2 = p^2 / 1152 * (81 - 462*p^2 + 385*p^4) / v^2
-#    u3 = p^3 / 414720 * (30375 - 369603 * p^2 + 765765*p^4 - 425425*p^6) * -1 / v^3
-#    u4 = p^4 / 39813120 * (4465125 - 94121676*p^2 + 349922430*p^4 - 446185740*p^6 + 185910725*p^8) / v^4
-#    u5 = p^5 / 6688604160 * (-188699385875*p^10 + 566098157625*p^8 - 614135872350*p^6 + 284499769554*p^4 - 49286948607*p^2 + 1519035525) * -1 / v^5
-#    u6 = p^6 / 4815794995200 * (1023694168371875*p^12 - 3685299006138750*p^10 + 5104696716244125*p^8 - 3369032068261860*p^6 
-#         + 1050760774457901*p^4 - 127577298354750*p^2 + 2757049477875) * 1 / v^6
-#    u7 = p^7 / 115579079884800 * (-221849150488590625*p^14 + 931766432052080625*p^12 - 1570320948552481125*p^10 + 1347119637570231525*p^8 
-#         - 613221795981706275*p^6 + 138799253740521843*p^4 - 12493049053044375*p^2 + 199689155040375) * -1 / v^7
-#    u8 = p^8 / 22191183337881600 * (448357133137441653125*p^16 - 2152114239059719935000*p^14 + 4272845805510421639500*p^12 - 4513690624987320777000*p^10 
-#         + 2711772922412520971550*p^8 - 914113758588905038248*p^6 + 157768535329832893644*p^4 - 10960565081605263000*p^2 + 134790179652253125) * 1 / v^8
-#    u9 = p^9 / 263631258054033408000 * (-64041091111686478524109375*p^18 + 345821892003106984030190625*p^16 - 790370708270219620781737500*p^14 
-#         + 992115946599792610768672500*p^12 - 741743213039573443221773250*p^10 + 334380732677827878090447630*p^8 - 87432034049652400520788332*p^6 
-#         + 11921080954211358275362500*p^4 - 659033454841709672064375*p^2 + 6427469716717690265625) * -1 / v^9
-#    u10 = p^10 / 88580102706155225088000 * (290938676920391671935028890625*p^20 - 1745632061522350031610173343750*p^18 + 4513386761946134740461797128125*p^16 
-#         - 6564241639632418015173104205000*p^14 + 5876803711285273203043452095250*p^12 - 3327704366990695147540934069220*p^10 + 1177120360439828012193658602930*p^8 
-#         - 246750339886026017414509498824*p^6 + 27299183373230345667273718125*p^4 - 1230031256571145165088463750*p^2 + 9745329584487361980740625) * 1 / v^10
-#    u11 = p^11 / 27636992044320430227456000 * (-1363312191078326248171914924296875*p^22 + 8997860461116953237934638500359375*p^20 - 25963913760458280822131901737878125*p^18 + 42936745153513007436411401865860625*p^16 - 44801790321820682384740638703320750*p^14 + 30589806122850866110941936529264950*p^12 - 13704902022868787041097596217578170*p^10
-#        + 3926191452593448964331218647028194*p^8 - 676389476843440422173605288596087*p^6 + 62011003282542082472252466220875*p^4 - 2321657500166464779660536015625*p^2 + 15237265774872558064250728125) * -1 / v^11
-#    u12 = p^12 / 39797268543821419527536640000 * (32426380464797989812768996474401171875*p^24 - 233469939346545526651936774615688437500*p^22 + 743739612850105971846901081692858843750*p^20 - 1378260730939829908037976894025628887500*p^18 + 1642838631056253395899341314188134178125*p^16 - 1314368459332124683504418467809129387000*p^14
-#        + 714528665351965363868467348116538170900*p^12 - 261201165596865827608687437905740780920*p^10 + 62055079517573388459132793029571876461*p^8 - 8958590476947726766450604043798559500*p^6 + 692277766674325563109617997687563750*p^4 - 21882222767154197351962677311437500*p^2 + 120907703923613748239829527671875) * 1 / v^12
-#    u13 = p^13 / 955134445051714068660879360000 * (-14020668045586884672121117629431460546875*p^26 + 109361210755577700442544717509565392265625*p^24 - 381190503845282445953419057314665534156250*p^22 + 782463969315283937856703223178540650343750*p^20
-#        - 1049095945162229046324321461816274931715625*p^18 + 962926533925253374359704288384340809260875*p^16 - 616410216242554698436702353237166008656700*p^14 + 274983827478138958623041508409195988431140*p^12 - 83924867223075156862785921508524155665245*p^10
-#        + 16843538631795795357786827345307534156063*p^8 - 2069933923586966756183324291232117362250*p^6 + 136735019134677724428035696765082813750*p^4 - 3698121486504259988094897605296209375*p^2 + 17438611142828905996129258798828125) * -1 / v^13
-#        
+#    in src/U_polynomials.jl as a reference as higher orders are impossible to find while being needed for any meaningfully accurate calculation.
 #    For large orders these formulas will converge much faster than using upward recurrence.
 #
 #    
@@ -97,6 +70,7 @@
 # [6] Abramowitz, Milton, and Irene A. Stegun, eds. Handbook of mathematical functions with formulas, graphs, and mathematical tables. 
 #     Vol. 55. US Government printing office, 1964.
 #
+
 """
     besselk0(x::T) where T <: Union{Float32, Float64}
 
@@ -206,25 +180,6 @@ function besselkx(nu::Int, x::T) where T <: Union{Float32, Float64}
         return besselk_large_orders_scaled(nu, x)
     end
 end
-
-@inline function up_recurrence(x, k0, k1, nu)
-    nu == 0 && return k0
-    nu == 1 && return k1
-
-    # this prevents us from looping through large values of nu when the loop will always return zero
-    (iszero(k0) || iszero(k1)) && return zero(x) 
-
-    k2 = k0
-    x2 = 2 / x
-    for n in 1:nu-1
-        a = x2 * n
-        k2 = muladd(a, k1, k0)
-        k0 = k1
-        k1 = k2
-    end
-    return k2
-end
-
 function besselk_large_orders(v, x::T) where T <: Union{Float32, Float64, BigFloat}
     S = promote_type(T, Float64)
     x = S(x)
@@ -237,7 +192,6 @@ function besselk_large_orders(v, x::T) where T <: Union{Float32, Float64, BigFlo
 
     return T(coef*Uk_poly_Kn(p, v, p2, T))
 end
-
 function besselk_large_orders_scaled(v, x::T) where T <: Union{Float32, Float64, BigFloat}
     S = promote_type(T, Float64)
     x = S(x)

src/recurrence.jl

@@ -0,0 +1,32 @@
+@inline function down_recurrence(x, in, inp1, nu, branch)
+    # this prevents us from looping through large values of nu when the loop will always return zero
+    (iszero(in) || iszero(inp1)) && return zero(x)
+    (isinf(inp1) || isinf(inp1)) && return in
+
+    inm1 = in
+    x2 = 2 / x
+    for n in branch:-1:nu+1
+        a = x2 * n
+        inm1 = muladd(a, in, inp1)
+        inp1 = in
+        in = inm1
+    end
+    return inm1
+end
+@inline function up_recurrence(x, k0, k1, nu)
+    nu == 0 && return k0
+    nu == 1 && return k1
+
+    # this prevents us from looping through large values of nu when the loop will always return zero
+    (iszero(k0) || iszero(k1)) && return zero(x) 
+
+    k2 = k0
+    x2 = 2 / x
+    for n in 1:nu-1
+        a = x2 * n
+        k2 = muladd(a, k1, k0)
+        k0 = k1
+        k1 = k2
+    end
+    return k2
+end