Merge pull request #15 from oscardssmith/faster-besselj0s

faster besselj0 for x<25

Author: heltonmc

src/Bessels.jl

@@ -36,6 +36,7 @@ include("math_constants.jl")
 include("U_polynomials.jl")
 include("recurrence.jl")
 include("misc.jl")
+include("Polynomials/besselj_polys.jl")
 #include("hankel.jl")
 
 end

src/Polynomials/besselj_polys.jl

@@ -0,0 +1,87 @@
+const J0_ROOTS(::Type{Float64}) = (
+    (2.4048255576957730, -1.176691651530894e-16),
+    (3.8317059702075125, -1.5269184090088067e-16),
+    (5.5200781102863110, 8.088597146146722e-17),
+    (7.0155866698156190, -9.414165653410389e-17),
+    (8.6537279129110130, -2.92812607320779e-16),
+    (10.173468135062722, 4.482162274768888e-16),
+    (11.791534439014281, 2.812956912778735e-16),
+    (13.323691936314223, 2.600408064718813e-16),
+    (14.930917708487787, -7.070514505983074e-16),
+    (16.470630050877634, -1.619019544798128e-15),
+    (18.071063967910924, -9.658048089426209e-16),
+    (19.615858510468243, -1.004445634526616e-15),
+    (21.211636629879260, 4.947077428784068e-16),
+    (22.760084380592772, -4.925749373614922e-16),
+    (24.352471530749302, 9.169067133951066e-16),
+    (25.903672087618382, 4.894530726419825e-16)
+)
+const J0_POLYS(::Type{Float64}) = (
+    (0.0000000000000000000, -0.5191474972894669, 0.10793870175491979, 0.05660177443794914, -0.008657669593292222, -0.0021942003590739974, 0.0002643770365942964, 4.37291931443113e-5, -4.338825419759815e-6, -5.304927679598406e-7, 4.469819175606667e-8, 4.3284827345621115e-9, -3.135111000732148e-10, -2.628876489517534e-11),
+    (-0.402759395702553, 2.476919088072758e-16, 0.20137969785127532, -0.017518715285670765, -0.013352611033152278, 0.0010359438492839443, 0.00037218755624680334, -2.4952042421113972e-5, -5.776086353844158e-6, 3.374317129436161e-7, 5.727482259215452e-8, -2.9561880489355444e-9, -3.905845672635605e-10, 1.971332566705736e-11),
+    (0.000000000000000000, 0.34026480655836816, -0.030820651425593214, -0.05298855286760721, 0.004631042145890388, 0.002257440229081131, -0.00017518572879518415, -4.6521091062878115e-5, 3.199785909661533e-6, 5.716500268232186e-7, -3.5112898510841466e-8, -4.684643389757727e-9, 2.562685034682206e-10, 2.7958958795750104e-11),
+    (0.30011575252613254, -1.6640272822046001e-16, -0.15005787626306408, 0.007129737603121546, 0.011742619737383848, -0.0006260583453094324, -0.00035093119008693753, 1.7929701912295164e-5, 5.6239324892485754e-6, -2.668437501970219e-7, -5.6648488273749086e-8, 2.48117399780498e-9, 3.8876537586241154e-10, -1.6657136713437192e-11),
+    (0.000000000000000000, -0.27145229992838193, 0.015684124960953488, 0.044033774963413, -0.0025093022271948434, -0.0020603351551475315, 0.00011243486771159352, 4.482303558813413e-5, -2.288390108003442e-6, -5.679383588459768e-7, 2.693939234375692e-8, 4.737285529934781e-9, -2.0612709555352797e-10, -2.8163166483726606e-11),
+    (-0.2497048770578432, 1.1807897766765572e-16, 0.12485243852891914, -0.0040907858517059345, -0.010102792347641438, 0.00038536375952213334, 0.00031859711440332953, -1.2373899600646271e-5, -5.3013932979548665e-6, 2.001098153528186e-7, 5.4711629662471434e-8, -1.9724572531751518e-9, -3.8121398193699247e-10, 1.3667679743782715e-11),
+    (0.000000000000000000, 0.23245983136472478, -0.009857064513825458, -0.03818600911162367, 0.0016073972920762946, 0.0018420433388794816, -7.581358465623415e-5, -4.159284549011371e-5, 1.650645590334915e-6, 5.425453494592871e-7, -2.0556207467977526e-8, -4.620018928884712e-9, 1.642028058414746e-10, 2.7701605444102412e-11),
+    (0.21835940724787295, -8.89726402965429e-17, -0.10917970362393398, 0.0027314677279632535, 0.008944552393700088, -0.00026391472261453583, -0.00028847875053074687, 8.858193371737123e-6, 4.9233776180403375e-6, -1.5077786827161215e-7, -5.190218733666561e-8, 1.5539413886301204e-9, 3.674809363354973e-10, -1.1113645791216594e-11),
+    (0.00000000000000000, -0.20654643307799603, 0.006916736034268416, 0.034115572697347704, -0.001137276252948717, -0.0016680057255530109, 5.4841792064182565e-5, 3.837965853474541e-5, -1.2335804050962046e-6, -5.106259295634553e-7, 1.592333632709497e-8, 4.423517565793139e-9, -1.3138837384184105e-10, -2.6809397212536384e-11),
+    (-0.19646537146865717, 6.979167865106427e-17, 0.09823268573432613, -0.001988037402152532, -0.008095530671166083, 0.00019440675128712672, 0.0002640383898036336, -6.666777683303928e-6, -4.5715696772304925e-6, 1.1666296153560847e-7, 4.8913639696764225e-8, -1.2379867207945651e-9, -3.508930968415813e-10, 9.07632091591013e-12),
+    (0.00000000000000000, 0.18772880304043943, -0.005194182350684612, -0.031096513233785917, 0.0008577442641273341, 0.0015312251534677639, -4.184307585284775e-5, -3.5603170534217916e-5, 9.58002109234601e-7, 4.795250964600283e-7, -1.263434500625308e-8, -4.20550167685701e-9, 1.065791122326672e-10, 2.5727417675684577e-11),
+    (0.18006337534431555, -5.638484737644332e-17, -0.09003168767215539, 0.0015299132863046024, 0.0074441453680234426, -0.00015060569680378768, -0.0002443398416353605, 5.227001519013193e-6, 4.267152607972633e-6, -9.295966007495808e-8, -4.610438011417262e-8, 1.0049092632275165e-9, 3.339442682325105e-10, -7.4991079301099e-12),
+    (0.00000000000000000, -0.17326589422922986, 0.004084217951979124, 0.028749284970146657, -0.0006761643016121907, -0.0014215899173758441, 3.320978125391802e-5, 3.3264379323815026e-5, -7.684444100376941e-7, -4.515479818833484e-7, 1.0271599456634145e-8, 3.993903280527723e-9, -8.793975528824604e-11, -2.4610374225652004e-11),
+    (-0.16718460047381803, 4.662597138876655e-17, 0.08359230023690671, -0.0012242529339116864, -0.006925682915280748, 0.00012100729852042988, 0.00022821854128498174, -4.230799709959437e-6, -4.007618360337058e-6, 7.601217108010105e-8, 4.358483157250522e-8, -8.317621452517008e-10, -3.178805429572483e-10, 6.284538399593043e-12),
+    (0.000000000000000000, 0.16170155068925002, -0.0033200234006036146, -0.02685937038656159, 0.0005505380905909195, 0.0013316994659124243, -2.715683205388875e-5, -3.1289544794362e-5, 6.326900360777323e-7, 4.2697141781883964e-7, -8.532447325590315e-9, -3.799009445641295e-9, 7.379552811590934e-11, 2.353654960834717e-11),
+    (0.15672498625285222, 1.1464880342445208e-19, -0.07836249312642612, 0.0010083833270351796, 0.006501011610557426, -9.993664895655577e-5, -0.00021478298462967253, 3.511086890959515e-6, 3.7857022791122945e-6, -6.350944888510364e-8, -4.135588012575766e-8, 7.135988717747828e-10, 3.2075281131621564e-10, 2.208506585533542e-12)
+)
+J0_POLY_PIO2(::Type{Float64}) = (
+1.000000000000000, -0.25, 0.01562499999999994, -0.00043402777777725544, 6.781684026082576e-6,
+-6.781683757550061e-8, 4.709479394601058e-10, -2.4016837144506874e-12, 9.104258208703104e-15
+)
+
+J1_ROOTS(::Type{Float64}) = (
+        (1.8411837813406593, 4.7898393919093694e-18),
+        (3.8317059702075125, -1.5269184090088067e-16),
+        (5.3314427735250325, 1.5109105349471405e-16),
+        (7.015586669815619, -9.414165653410389e-17),
+        (8.536316366346286, -1.5433871213307537e-16),
+        (10.173468135062722, 4.482162274768888e-16),
+        (11.706004902592063, 7.121366942298246e-16),
+        (13.323691936314223, 2.600408064718813e-16),
+        (14.863588633909034, -6.265788988781879e-16),
+        (16.470630050877634, -1.619019544798128e-15),
+        (18.015527862681804, -1.1196999448424267e-16),
+        (19.615858510468243, -1.004445634526616e-15),
+        (21.16436985918879, 1.7024131380423588e-15),
+        (22.760084380592772, -4.925749373614922e-16),
+        (24.311326857210776, -2.614798558537172e-16),
+        (25.903672087618382, 4.894530726419825e-16)
+        )
+
+@inline J1_POLYS(::Type{Float64}) = (
+         (0.5818652242815964, 8.973834293036876e-17, -0.20511071214777496, 0.006058948324597746, 0.013801769808047723, -0.0003723170971916689, -0.0003949590750416506, 9.202953658715798e-6, 6.267292697927347e-6, -1.2681571675674473e-7, -6.319326736018798e-8, 1.057852721352679e-9, 4.839653511789343e-10, -2.1534683937074676e-11),
+         (0.0, -0.402759395702553, 0.05255614585697725, 0.053410444132722765, -0.005179719245639115, -0.00223312533910116, 0.00017466429071755533, 4.6208700909637e-5, -3.0368632747366735e-6, -5.727790851165856e-7, 3.248220586398651e-8, 4.732693354840469e-9, -2.346792988667587e-10, -2.607612190048509e-11),
+         (-0.34612620185379156, 2.6002760076991505e-17, 0.16697453550109428, -0.009678268542879972, -0.012099225779175138, 0.0006654009006841076, 0.0003541389011182841, -1.742720356958137e-5, -5.655294960447268e-6, 2.484316221112215e-7, 5.7101369594782676e-8, -2.2587104440771904e-9, -3.9854619701676605e-10, 1.854508641982197e-11),
+         (0.0, 0.30011575252613254, -0.02138921280934146, -0.04697047894974007, 0.0031302917260424344, 0.0021055871432169443, -0.00012550790947431806, -4.499147505171648e-5, 2.401580334889493e-6, 5.665260872870135e-7, -2.7272058756924113e-8, -4.718921752439826e-9, 2.045523137027535e-10, 2.6923755198436e-11),
+         (0.27329994163319987, -1.3511933704624036e-16, -0.13477468037992396, 0.005116340346495344, 0.010631861751990702, -0.000448743683849691, -0.0003268000185531749, 1.3382556752353113e-5, 5.363177082527157e-6, -2.0647461181851255e-7, -5.499892476485602e-8, 1.9780335410646812e-9, 3.8456897851545063e-10, -1.5543946040865833e-11),
+         (0.0, -0.2497048770578432, 0.012272357555101388, 0.04041116939079181, -0.0019268187972545589, -0.0019115826893618788, 8.661729445925766e-5, 4.241116264477775e-5, -1.800978985430059e-6, -5.47159694982648e-7, 2.1682608288127194e-8, 4.629875997238198e-9, -1.6946465290961774e-10, -2.7001263147077087e-11),
+         (-0.23330441717143408, 1.3347222236161232e-16, 0.11580092244607755, -0.0032489977328297747, -0.009372527206041913, 0.0003036138212749204, 0.00029804555521007126, -9.813819271763006e-6, -5.0242293237237345e-6, 1.6136482548346424e-7, 5.251755806094281e-8, -1.6213723779041528e-9, -3.7130918416720704e-10, 1.2840797195120732e-11),
+         (0.0, 0.21835940724787295, -0.008194403183877394, -0.03577820957502993, 0.001319573612804675, 0.0017308725061602815, -6.200735153951766e-5, -3.938703730297954e-5, 1.3569938030086903e-6, 5.190652330962483e-7, -1.7082573736581586e-8, -4.464580500726025e-9, 1.3835552663370753e-10, 2.6508049073606443e-11),
+         (0.20701265272531905, -1.1572365385200866e-16, -0.10303781563402706, 0.002289729546293356, 0.008432435058846974, -0.00021965941971569214, -0.000272721543802475, 7.36909520166486e-6, 4.682127415369e-6, -1.2611406291992532e-7, -4.9733737497256667e-8, 1.313333637862963e-9, 3.5584811981981077e-10, -1.0581432980865016e-11),
+         (0.0, -0.19646537146865717, 0.005964112206447895, 0.032382122684890324, -0.0009720337562254532, -0.0015842303417456071, 4.6667442166602986e-5, 3.6572573450276185e-5, -1.0499611641790313e-6, -4.891787571238003e-7, 1.3609422867092438e-8, 4.2640592578446675e-9, -1.1321195999108497e-10, -2.5685037344174917e-11),
+         (-0.18801748852581776, 9.806595982257605e-17, 0.09371909377243136, -0.0017233243363111925, -0.007714266760106843, 0.00016755862384546863, 0.000251827174474777, -5.7335177376628445e-6, -4.372425030172481e-6, 1.0045839852031556e-7, 4.698184959516747e-8, -1.0717704296017814e-9, -3.3952344362582674e-10, 8.773851724781324e-12),
+         (0.0, 0.18006337534431555, -0.004589739858905968, -0.029776581472313136, 0.0007530284838491313, 0.0014660390526475475, -3.658900932751018e-5, -3.4137236390211275e-5, 8.366325091910564e-7, 4.6108475352995054e-7, -1.1047200388967438e-8, -4.058793425595929e-9, 9.363513647344476e-11, 2.4720051360744732e-11),
+         (0.1734590492857464, -8.319859584484218e-17, -0.08653590193876039, 0.0013568189856187354, 0.007147216520714834, -0.00013295776598760658, -0.00023462959289705953, 4.6031437800947915e-6, 4.103196271673978e-6, -8.18373301294018e-8, -4.44403211204062e-8, 8.872617322304839e-10, 3.2359730989751983e-10, -7.354373969368619e-12),
+         (0.0, -0.16718460047381806, 0.0036727588017285762, 0.02770273166133465, -0.0006050364924617241, -0.0013693112504440895, 2.9615596890234806e-5, 3.206096184277923e-5, -6.841058767433935e-7, -4.358877331737043e-7, 9.143763763164119e-9, 3.864037532317281e-9, -7.851667555762691e-11, -2.3728757802646805e-11),
+         (-0.1618382095526585, 7.115722920799028e-17, 0.08078219522574913, -0.0011038527651152986, -0.006686444726790514, 0.00010870535083026479, 0.00022030197347228358, -3.791805814546456e-6, -3.871147931227952e-6, 6.80607764766372e-8, 4.2160129307780366e-8, -7.460427055677985e-10, -3.0872139357605097e-10, 6.24158293622982e-12),
+         (0.0, 0.15672498625285222, -0.0030251499811055965, -0.026004046442225856, 0.0004996832448037306, 0.001288697907658241, -2.457760943647847e-5, -3.02856199504406e-5, 5.715983781675481e-7, 4.136275513873783e-7, -7.703846567878601e-9, -3.6853569596797437e-9, 6.676439988396374e-11, 2.277101500922319e-11))#for i in 2:2:18
+
+J1_POLY_PIO2(::Type{Float64}) = (0.5, -0.0624999999999989, 0.002604166666657291, -5.42534721917933e-5, 6.781683542660179e-7, -5.651361336587487e-9, 3.36191211106159e-11, -1.4511302591871352e-13)
+
+#for i in 1:2:18
+#    display(((i+1)*pi/2-pts[i+1][1], (i+2)*pi/2-pts[i+1][1]))
+#    poly = Float64.(Tuple(ratfn_minimax(x->r2(x,pts[i]), (i*pi/2-pts[i][1], (i+1)*pi/2-pts[i][1]), 13, 0)[1]))
+#    push!(polys, poly)
+#    poly = Float64.(Tuple(ratfn_minimax(x->r(x,pts[i+1]), ((i+1)*pi/2-pts[i+1][1], (i+2)*pi/2-pts[i+1][1]), 12, 0)[1]))
+#    push!(polys, (0.0, poly...))
+#end