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| import math | |
| import pytest | |
| from mpmath import * | |
| def test_bessel(): | |
| mp.dps = 15 | |
| assert j0(1).ae(0.765197686557966551) | |
| assert j0(pi).ae(-0.304242177644093864) | |
| assert j0(1000).ae(0.0247866861524201746) | |
| assert j0(-25).ae(0.0962667832759581162) | |
| assert j1(1).ae(0.440050585744933516) | |
| assert j1(pi).ae(0.284615343179752757) | |
| assert j1(1000).ae(0.00472831190708952392) | |
| assert j1(-25).ae(0.125350249580289905) | |
| assert besselj(5,1).ae(0.000249757730211234431) | |
| assert besselj(5+0j,1).ae(0.000249757730211234431) | |
| assert besselj(5,pi).ae(0.0521411843671184747) | |
| assert besselj(5,1000).ae(0.00502540694523318607) | |
| assert besselj(5,-25).ae(0.0660079953984229934) | |
| assert besselj(-3,2).ae(-0.128943249474402051) | |
| assert besselj(-4,2).ae(0.0339957198075684341) | |
| assert besselj(3,3+2j).ae(0.424718794929639595942 + 0.625665327745785804812j) | |
| assert besselj(0.25,4).ae(-0.374760630804249715) | |
| assert besselj(1+2j,3+4j).ae(0.319247428741872131 - 0.669557748880365678j) | |
| assert (besselj(3, 10**10) * 10**5).ae(0.76765081748139204023) | |
| assert bessely(-0.5, 0) == 0 | |
| assert bessely(0.5, 0) == -inf | |
| assert bessely(1.5, 0) == -inf | |
| assert bessely(0,0) == -inf | |
| assert bessely(-0.4, 0) == -inf | |
| assert bessely(-0.6, 0) == inf | |
| assert bessely(-1, 0) == inf | |
| assert bessely(-1.4, 0) == inf | |
| assert bessely(-1.6, 0) == -inf | |
| assert bessely(-1, 0) == inf | |
| assert bessely(-2, 0) == -inf | |
| assert bessely(-3, 0) == inf | |
| assert bessely(0.5, 0) == -inf | |
| assert bessely(1, 0) == -inf | |
| assert bessely(1.5, 0) == -inf | |
| assert bessely(2, 0) == -inf | |
| assert bessely(2.5, 0) == -inf | |
| assert bessely(3, 0) == -inf | |
| assert bessely(0,0.5).ae(-0.44451873350670655715) | |
| assert bessely(1,0.5).ae(-1.4714723926702430692) | |
| assert bessely(-1,0.5).ae(1.4714723926702430692) | |
| assert bessely(3.5,0.5).ae(-138.86400867242488443) | |
| assert bessely(0,3+4j).ae(4.6047596915010138655-8.8110771408232264208j) | |
| assert bessely(0,j).ae(-0.26803248203398854876+1.26606587775200833560j) | |
| assert (bessely(3, 10**10) * 10**5).ae(0.21755917537013204058) | |
| assert besseli(0,0) == 1 | |
| assert besseli(1,0) == 0 | |
| assert besseli(2,0) == 0 | |
| assert besseli(-1,0) == 0 | |
| assert besseli(-2,0) == 0 | |
| assert besseli(0,0.5).ae(1.0634833707413235193) | |
| assert besseli(1,0.5).ae(0.25789430539089631636) | |
| assert besseli(-1,0.5).ae(0.25789430539089631636) | |
| assert besseli(3.5,0.5).ae(0.00068103597085793815863) | |
| assert besseli(0,3+4j).ae(-3.3924877882755196097-1.3239458916287264815j) | |
| assert besseli(0,j).ae(besselj(0,1)) | |
| assert (besseli(3, 10**10) * mpf(10)**(-4342944813)).ae(4.2996028505491271875) | |
| assert besselk(0,0) == inf | |
| assert besselk(1,0) == inf | |
| assert besselk(2,0) == inf | |
| assert besselk(-1,0) == inf | |
| assert besselk(-2,0) == inf | |
| assert besselk(0,0.5).ae(0.92441907122766586178) | |
| assert besselk(1,0.5).ae(1.6564411200033008937) | |
| assert besselk(-1,0.5).ae(1.6564411200033008937) | |
| assert besselk(3.5,0.5).ae(207.48418747548460607) | |
| assert besselk(0,3+4j).ae(-0.007239051213570155013+0.026510418350267677215j) | |
| assert besselk(0,j).ae(-0.13863371520405399968-1.20196971531720649914j) | |
| assert (besselk(3, 10**10) * mpf(10)**4342944824).ae(1.1628981033356187851) | |
| # test for issue 331, bug reported by Michael Hartmann | |
| for n in range(10,100,10): | |
| mp.dps = n | |
| assert besseli(91.5,24.7708).ae("4.00830632138673963619656140653537080438462342928377020695738635559218797348548092636896796324190271316137982810144874264e-41") | |
| def test_bessel_zeros(): | |
| mp.dps = 15 | |
| assert besseljzero(0,1).ae(2.40482555769577276869) | |
| assert besseljzero(2,1).ae(5.1356223018406825563) | |
| assert besseljzero(1,50).ae(157.86265540193029781) | |
| assert besseljzero(10,1).ae(14.475500686554541220) | |
| assert besseljzero(0.5,3).ae(9.4247779607693797153) | |
| assert besseljzero(2,1,1).ae(3.0542369282271403228) | |
| assert besselyzero(0,1).ae(0.89357696627916752158) | |
| assert besselyzero(2,1).ae(3.3842417671495934727) | |
| assert besselyzero(1,50).ae(156.29183520147840108) | |
| assert besselyzero(10,1).ae(12.128927704415439387) | |
| assert besselyzero(0.5,3).ae(7.8539816339744830962) | |
| assert besselyzero(2,1,1).ae(5.0025829314460639452) | |
| def test_hankel(): | |
| mp.dps = 15 | |
| assert hankel1(0,0.5).ae(0.93846980724081290423-0.44451873350670655715j) | |
| assert hankel1(1,0.5).ae(0.2422684576748738864-1.4714723926702430692j) | |
| assert hankel1(-1,0.5).ae(-0.2422684576748738864+1.4714723926702430692j) | |
| assert hankel1(1.5,0.5).ae(0.0917016996256513026-2.5214655504213378514j) | |
| assert hankel1(1.5,3+4j).ae(0.0066806866476728165382-0.0036684231610839127106j) | |
| assert hankel2(0,0.5).ae(0.93846980724081290423+0.44451873350670655715j) | |
| assert hankel2(1,0.5).ae(0.2422684576748738864+1.4714723926702430692j) | |
| assert hankel2(-1,0.5).ae(-0.2422684576748738864-1.4714723926702430692j) | |
| assert hankel2(1.5,0.5).ae(0.0917016996256513026+2.5214655504213378514j) | |
| assert hankel2(1.5,3+4j).ae(14.783528526098567526-7.397390270853446512j) | |
| def test_struve(): | |
| mp.dps = 15 | |
| assert struveh(2,3).ae(0.74238666967748318564) | |
| assert struveh(-2.5,3).ae(0.41271003220971599344) | |
| assert struvel(2,3).ae(1.7476573277362782744) | |
| assert struvel(-2.5,3).ae(1.5153394466819651377) | |
| def test_whittaker(): | |
| mp.dps = 15 | |
| assert whitm(2,3,4).ae(49.753745589025246591) | |
| assert whitw(2,3,4).ae(14.111656223052932215) | |
| def test_kelvin(): | |
| mp.dps = 15 | |
| assert ber(2,3).ae(0.80836846563726819091) | |
| assert ber(3,4).ae(-0.28262680167242600233) | |
| assert ber(-3,2).ae(-0.085611448496796363669) | |
| assert bei(2,3).ae(-0.89102236377977331571) | |
| assert bei(-3,2).ae(-0.14420994155731828415) | |
| assert ker(2,3).ae(0.12839126695733458928) | |
| assert ker(-3,2).ae(-0.29802153400559142783) | |
| assert ker(0.5,3).ae(-0.085662378535217097524) | |
| assert kei(2,3).ae(0.036804426134164634000) | |
| assert kei(-3,2).ae(0.88682069845786731114) | |
| assert kei(0.5,3).ae(0.013633041571314302948) | |
| def test_hyper_misc(): | |
| mp.dps = 15 | |
| assert hyp0f1(1,0) == 1 | |
| assert hyp1f1(1,2,0) == 1 | |
| assert hyp1f2(1,2,3,0) == 1 | |
| assert hyp2f1(1,2,3,0) == 1 | |
| assert hyp2f2(1,2,3,4,0) == 1 | |
| assert hyp2f3(1,2,3,4,5,0) == 1 | |
| # Degenerate case: 0F0 | |
| assert hyper([],[],0) == 1 | |
| assert hyper([],[],-2).ae(exp(-2)) | |
| # Degenerate case: 1F0 | |
| assert hyper([2],[],1.5) == 4 | |
| # | |
| assert hyp2f1((1,3),(2,3),(5,6),mpf(27)/32).ae(1.6) | |
| assert hyp2f1((1,4),(1,2),(3,4),mpf(80)/81).ae(1.8) | |
| assert hyp2f1((2,3),(1,1),(3,2),(2+j)/3).ae(1.327531603558679093+0.439585080092769253j) | |
| mp.dps = 25 | |
| v = mpc('1.2282306665029814734863026', '-0.1225033830118305184672133') | |
| assert hyper([(3,4),2+j,1],[1,5,j/3],mpf(1)/5+j/8).ae(v) | |
| mp.dps = 15 | |
| def test_elliptic_integrals(): | |
| mp.dps = 15 | |
| assert ellipk(0).ae(pi/2) | |
| assert ellipk(0.5).ae(gamma(0.25)**2/(4*sqrt(pi))) | |
| assert ellipk(1) == inf | |
| assert ellipk(1+0j) == inf | |
| assert ellipk(-1).ae('1.3110287771460599052') | |
| assert ellipk(-2).ae('1.1714200841467698589') | |
| assert isinstance(ellipk(-2), mpf) | |
| assert isinstance(ellipe(-2), mpf) | |
| assert ellipk(-50).ae('0.47103424540873331679') | |
| mp.dps = 30 | |
| n1 = +fraction(99999,100000) | |
| n2 = +fraction(100001,100000) | |
| mp.dps = 15 | |
| assert ellipk(n1).ae('7.1427724505817781901') | |
| assert ellipk(n2).ae(mpc('7.1427417367963090109', '-1.5707923998261688019')) | |
| assert ellipe(n1).ae('1.0000332138990829170') | |
| v = ellipe(n2) | |
| assert v.real.ae('0.999966786328145474069137') | |
| assert (v.imag*10**6).ae('7.853952181727432') | |
| assert ellipk(2).ae(mpc('1.3110287771460599052', '-1.3110287771460599052')) | |
| assert ellipk(50).ae(mpc('0.22326753950210985451', '-0.47434723226254522087')) | |
| assert ellipk(3+4j).ae(mpc('0.91119556380496500866', '0.63133428324134524388')) | |
| assert ellipk(3-4j).ae(mpc('0.91119556380496500866', '-0.63133428324134524388')) | |
| assert ellipk(-3+4j).ae(mpc('0.95357894880405122483', '0.23093044503746114444')) | |
| assert ellipk(-3-4j).ae(mpc('0.95357894880405122483', '-0.23093044503746114444')) | |
| assert isnan(ellipk(nan)) | |
| assert isnan(ellipe(nan)) | |
| assert ellipk(inf) == 0 | |
| assert isinstance(ellipk(inf), mpc) | |
| assert ellipk(-inf) == 0 | |
| assert ellipk(1+0j) == inf | |
| assert ellipe(0).ae(pi/2) | |
| assert ellipe(0.5).ae(pi**(mpf(3)/2)/gamma(0.25)**2 +gamma(0.25)**2/(8*sqrt(pi))) | |
| assert ellipe(1) == 1 | |
| assert ellipe(1+0j) == 1 | |
| assert ellipe(inf) == mpc(0,inf) | |
| assert ellipe(-inf) == inf | |
| assert ellipe(3+4j).ae(1.4995535209333469543-1.5778790079127582745j) | |
| assert ellipe(3-4j).ae(1.4995535209333469543+1.5778790079127582745j) | |
| assert ellipe(-3+4j).ae(2.5804237855343377803-0.8306096791000413778j) | |
| assert ellipe(-3-4j).ae(2.5804237855343377803+0.8306096791000413778j) | |
| assert ellipe(2).ae(0.59907011736779610372+0.59907011736779610372j) | |
| assert ellipe('1e-1000000000').ae(pi/2) | |
| assert ellipk('1e-1000000000').ae(pi/2) | |
| assert ellipe(-pi).ae(2.4535865983838923) | |
| mp.dps = 50 | |
| assert ellipk(1/pi).ae('1.724756270009501831744438120951614673874904182624739673') | |
| assert ellipe(1/pi).ae('1.437129808135123030101542922290970050337425479058225712') | |
| assert ellipk(-10*pi).ae('0.5519067523886233967683646782286965823151896970015484512') | |
| assert ellipe(-10*pi).ae('5.926192483740483797854383268707108012328213431657645509') | |
| v = ellipk(pi) | |
| assert v.real.ae('0.973089521698042334840454592642137667227167622330325225') | |
| assert v.imag.ae('-1.156151296372835303836814390793087600271609993858798016') | |
| v = ellipe(pi) | |
| assert v.real.ae('0.4632848917264710404078033487934663562998345622611263332') | |
| assert v.imag.ae('1.0637961621753130852473300451583414489944099504180510966') | |
| mp.dps = 15 | |
| def test_exp_integrals(): | |
| mp.dps = 15 | |
| x = +e | |
| z = e + sqrt(3)*j | |
| assert ei(x).ae(8.21168165538361560) | |
| assert li(x).ae(1.89511781635593676) | |
| assert si(x).ae(1.82104026914756705) | |
| assert ci(x).ae(0.213958001340379779) | |
| assert shi(x).ae(4.11520706247846193) | |
| assert chi(x).ae(4.09647459290515367) | |
| assert fresnels(x).ae(0.437189718149787643) | |
| assert fresnelc(x).ae(0.401777759590243012) | |
| assert airyai(x).ae(0.0108502401568586681) | |
| assert airybi(x).ae(8.98245748585468627) | |
| assert ei(z).ae(3.72597969491314951 + 7.34213212314224421j) | |
| assert li(z).ae(2.28662658112562502 + 1.50427225297269364j) | |
| assert si(z).ae(2.48122029237669054 + 0.12684703275254834j) | |
| assert ci(z).ae(0.169255590269456633 - 0.892020751420780353j) | |
| assert shi(z).ae(1.85810366559344468 + 3.66435842914920263j) | |
| assert chi(z).ae(1.86787602931970484 + 3.67777369399304159j) | |
| assert fresnels(z/3).ae(0.034534397197008182 + 0.754859844188218737j) | |
| assert fresnelc(z/3).ae(1.261581645990027372 + 0.417949198775061893j) | |
| assert airyai(z).ae(-0.0162552579839056062 - 0.0018045715700210556j) | |
| assert airybi(z).ae(-4.98856113282883371 + 2.08558537872180623j) | |
| assert li(0) == 0.0 | |
| assert li(1) == -inf | |
| assert li(inf) == inf | |
| assert isinstance(li(0.7), mpf) | |
| assert si(inf).ae(pi/2) | |
| assert si(-inf).ae(-pi/2) | |
| assert ci(inf) == 0 | |
| assert ci(0) == -inf | |
| assert isinstance(ei(-0.7), mpf) | |
| assert airyai(inf) == 0 | |
| assert airybi(inf) == inf | |
| assert airyai(-inf) == 0 | |
| assert airybi(-inf) == 0 | |
| assert fresnels(inf) == 0.5 | |
| assert fresnelc(inf) == 0.5 | |
| assert fresnels(-inf) == -0.5 | |
| assert fresnelc(-inf) == -0.5 | |
| assert shi(0) == 0 | |
| assert shi(inf) == inf | |
| assert shi(-inf) == -inf | |
| assert chi(0) == -inf | |
| assert chi(inf) == inf | |
| def test_ei(): | |
| mp.dps = 15 | |
| assert ei(0) == -inf | |
| assert ei(inf) == inf | |
| assert ei(-inf) == -0.0 | |
| assert ei(20+70j).ae(6.1041351911152984397e6 - 2.7324109310519928872e6j) | |
| # tests for the asymptotic expansion | |
| # values checked with Mathematica ExpIntegralEi | |
| mp.dps = 50 | |
| r = ei(20000) | |
| s = '3.8781962825045010930273870085501819470698476975019e+8681' | |
| assert str(r) == s | |
| r = ei(-200) | |
| s = '-6.8852261063076355977108174824557929738368086933303e-90' | |
| assert str(r) == s | |
| r =ei(20000 + 10*j) | |
| sre = '-3.255138234032069402493850638874410725961401274106e+8681' | |
| sim = '-2.1081929993474403520785942429469187647767369645423e+8681' | |
| assert str(r.real) == sre and str(r.imag) == sim | |
| mp.dps = 15 | |
| # More asymptotic expansions | |
| assert chi(-10**6+100j).ae('1.3077239389562548386e+434288 + 7.6808956999707408158e+434287j') | |
| assert shi(-10**6+100j).ae('-1.3077239389562548386e+434288 - 7.6808956999707408158e+434287j') | |
| mp.dps = 15 | |
| assert ei(10j).ae(-0.0454564330044553726+3.2291439210137706686j) | |
| assert ei(100j).ae(-0.0051488251426104921+3.1330217936839529126j) | |
| u = ei(fmul(10**20, j, exact=True)) | |
| assert u.real.ae(-6.4525128526578084421345e-21, abs_eps=0, rel_eps=8*eps) | |
| assert u.imag.ae(pi) | |
| assert ei(-10j).ae(-0.0454564330044553726-3.2291439210137706686j) | |
| assert ei(-100j).ae(-0.0051488251426104921-3.1330217936839529126j) | |
| u = ei(fmul(-10**20, j, exact=True)) | |
| assert u.real.ae(-6.4525128526578084421345e-21, abs_eps=0, rel_eps=8*eps) | |
| assert u.imag.ae(-pi) | |
| assert ei(10+10j).ae(-1576.1504265768517448+436.9192317011328140j) | |
| u = ei(-10+10j) | |
| assert u.real.ae(7.6698978415553488362543e-7, abs_eps=0, rel_eps=8*eps) | |
| assert u.imag.ae(3.141595611735621062025) | |
| def test_e1(): | |
| mp.dps = 15 | |
| assert e1(0) == inf | |
| assert e1(inf) == 0 | |
| assert e1(-inf) == mpc(-inf, -pi) | |
| assert e1(10j).ae(0.045456433004455372635 + 0.087551267423977430100j) | |
| assert e1(100j).ae(0.0051488251426104921444 - 0.0085708599058403258790j) | |
| assert e1(fmul(10**20, j, exact=True)).ae(6.4525128526578084421e-21 - 7.6397040444172830039e-21j, abs_eps=0, rel_eps=8*eps) | |
| assert e1(-10j).ae(0.045456433004455372635 - 0.087551267423977430100j) | |
| assert e1(-100j).ae(0.0051488251426104921444 + 0.0085708599058403258790j) | |
| assert e1(fmul(-10**20, j, exact=True)).ae(6.4525128526578084421e-21 + 7.6397040444172830039e-21j, abs_eps=0, rel_eps=8*eps) | |
| def test_expint(): | |
| mp.dps = 15 | |
| assert expint(0,0) == inf | |
| assert expint(0,1).ae(1/e) | |
| assert expint(0,1.5).ae(2/exp(1.5)/3) | |
| assert expint(1,1).ae(-ei(-1)) | |
| assert expint(2,0).ae(1) | |
| assert expint(3,0).ae(1/2.) | |
| assert expint(4,0).ae(1/3.) | |
| assert expint(-2, 0.5).ae(26/sqrt(e)) | |
| assert expint(-1,-1) == 0 | |
| assert expint(-2,-1).ae(-e) | |
| assert expint(5.5, 0).ae(2/9.) | |
| assert expint(2.00000001,0).ae(100000000./100000001) | |
| assert expint(2+3j,4-j).ae(0.0023461179581675065414+0.0020395540604713669262j) | |
| assert expint('1.01', '1e-1000').ae(99.9999999899412802) | |
| assert expint('1.000000000001', 3.5).ae(0.00697013985754701819446) | |
| assert expint(2,3).ae(3*ei(-3)+exp(-3)) | |
| assert (expint(10,20)*10**10).ae(0.694439055541231353) | |
| assert expint(3,inf) == 0 | |
| assert expint(3.2,inf) == 0 | |
| assert expint(3.2+2j,inf) == 0 | |
| assert expint(1,3j).ae(-0.11962978600800032763 + 0.27785620120457163717j) | |
| assert expint(1,3).ae(0.013048381094197037413) | |
| assert expint(1,-3).ae(-ei(3)-pi*j) | |
| #assert expint(3) == expint(1,3) | |
| assert expint(1,-20).ae(-25615652.66405658882 - 3.1415926535897932385j) | |
| assert expint(1000000,0).ae(1./999999) | |
| assert expint(0,2+3j).ae(-0.025019798357114678171 + 0.027980439405104419040j) | |
| assert expint(-1,2+3j).ae(-0.022411973626262070419 + 0.038058922011377716932j) | |
| assert expint(-1.5,0) == inf | |
| def test_trig_integrals(): | |
| mp.dps = 30 | |
| assert si(mpf(1)/1000000).ae('0.000000999999999999944444444444446111') | |
| assert ci(mpf(1)/1000000).ae('-13.2382948930629912435014366276') | |
| assert si(10**10).ae('1.5707963267075846569685111517747537') | |
| assert ci(10**10).ae('-4.87506025174822653785729773959e-11') | |
| assert si(10**100).ae(pi/2) | |
| assert (ci(10**100)*10**100).ae('-0.372376123661276688262086695553') | |
| assert si(-3) == -si(3) | |
| assert ci(-3).ae(ci(3) + pi*j) | |
| # Test complex structure | |
| mp.dps = 15 | |
| assert mp.ci(50).ae(-0.0056283863241163054402) | |
| assert mp.ci(50+2j).ae(-0.018378282946133067149+0.070352808023688336193j) | |
| assert mp.ci(20j).ae(1.28078263320282943611e7+1.5707963267949j) | |
| assert mp.ci(-2+20j).ae(-4.050116856873293505e6+1.207476188206989909e7j) | |
| assert mp.ci(-50+2j).ae(-0.0183782829461330671+3.0712398455661049023j) | |
| assert mp.ci(-50).ae(-0.0056283863241163054+3.1415926535897932385j) | |
| assert mp.ci(-50-2j).ae(-0.0183782829461330671-3.0712398455661049023j) | |
| assert mp.ci(-2-20j).ae(-4.050116856873293505e6-1.207476188206989909e7j) | |
| assert mp.ci(-20j).ae(1.28078263320282943611e7-1.5707963267949j) | |
| assert mp.ci(50-2j).ae(-0.018378282946133067149-0.070352808023688336193j) | |
| assert mp.si(50).ae(1.5516170724859358947) | |
| assert mp.si(50+2j).ae(1.497884414277228461-0.017515007378437448j) | |
| assert mp.si(20j).ae(1.2807826332028294459e7j) | |
| assert mp.si(-2+20j).ae(-1.20747603112735722103e7-4.050116856873293554e6j) | |
| assert mp.si(-50+2j).ae(-1.497884414277228461-0.017515007378437448j) | |
| assert mp.si(-50).ae(-1.5516170724859358947) | |
| assert mp.si(-50-2j).ae(-1.497884414277228461+0.017515007378437448j) | |
| assert mp.si(-2-20j).ae(-1.20747603112735722103e7+4.050116856873293554e6j) | |
| assert mp.si(-20j).ae(-1.2807826332028294459e7j) | |
| assert mp.si(50-2j).ae(1.497884414277228461+0.017515007378437448j) | |
| assert mp.chi(50j).ae(-0.0056283863241163054+1.5707963267948966192j) | |
| assert mp.chi(-2+50j).ae(-0.0183782829461330671+1.6411491348185849554j) | |
| assert mp.chi(-20).ae(1.28078263320282943611e7+3.1415926535898j) | |
| assert mp.chi(-20-2j).ae(-4.050116856873293505e6+1.20747571696809187053e7j) | |
| assert mp.chi(-2-50j).ae(-0.0183782829461330671-1.6411491348185849554j) | |
| assert mp.chi(-50j).ae(-0.0056283863241163054-1.5707963267948966192j) | |
| assert mp.chi(2-50j).ae(-0.0183782829461330671-1.500443518771208283j) | |
| assert mp.chi(20-2j).ae(-4.050116856873293505e6-1.20747603112735722951e7j) | |
| assert mp.chi(20).ae(1.2807826332028294361e7) | |
| assert mp.chi(2+50j).ae(-0.0183782829461330671+1.500443518771208283j) | |
| assert mp.shi(50j).ae(1.5516170724859358947j) | |
| assert mp.shi(-2+50j).ae(0.017515007378437448+1.497884414277228461j) | |
| assert mp.shi(-20).ae(-1.2807826332028294459e7) | |
| assert mp.shi(-20-2j).ae(4.050116856873293554e6-1.20747603112735722103e7j) | |
| assert mp.shi(-2-50j).ae(0.017515007378437448-1.497884414277228461j) | |
| assert mp.shi(-50j).ae(-1.5516170724859358947j) | |
| assert mp.shi(2-50j).ae(-0.017515007378437448-1.497884414277228461j) | |
| assert mp.shi(20-2j).ae(-4.050116856873293554e6-1.20747603112735722103e7j) | |
| assert mp.shi(20).ae(1.2807826332028294459e7) | |
| assert mp.shi(2+50j).ae(-0.017515007378437448+1.497884414277228461j) | |
| def ae(x,y,tol=1e-12): | |
| return abs(x-y) <= abs(y)*tol | |
| assert fp.ci(fp.inf) == 0 | |
| assert ae(fp.ci(fp.ninf), fp.pi*1j) | |
| assert ae(fp.si(fp.inf), fp.pi/2) | |
| assert ae(fp.si(fp.ninf), -fp.pi/2) | |
| assert fp.si(0) == 0 | |
| assert ae(fp.ci(50), -0.0056283863241163054402) | |
| assert ae(fp.ci(50+2j), -0.018378282946133067149+0.070352808023688336193j) | |
| assert ae(fp.ci(20j), 1.28078263320282943611e7+1.5707963267949j) | |
| assert ae(fp.ci(-2+20j), -4.050116856873293505e6+1.207476188206989909e7j) | |
| assert ae(fp.ci(-50+2j), -0.0183782829461330671+3.0712398455661049023j) | |
| assert ae(fp.ci(-50), -0.0056283863241163054+3.1415926535897932385j) | |
| assert ae(fp.ci(-50-2j), -0.0183782829461330671-3.0712398455661049023j) | |
| assert ae(fp.ci(-2-20j), -4.050116856873293505e6-1.207476188206989909e7j) | |
| assert ae(fp.ci(-20j), 1.28078263320282943611e7-1.5707963267949j) | |
| assert ae(fp.ci(50-2j), -0.018378282946133067149-0.070352808023688336193j) | |
| assert ae(fp.si(50), 1.5516170724859358947) | |
| assert ae(fp.si(50+2j), 1.497884414277228461-0.017515007378437448j) | |
| assert ae(fp.si(20j), 1.2807826332028294459e7j) | |
| assert ae(fp.si(-2+20j), -1.20747603112735722103e7-4.050116856873293554e6j) | |
| assert ae(fp.si(-50+2j), -1.497884414277228461-0.017515007378437448j) | |
| assert ae(fp.si(-50), -1.5516170724859358947) | |
| assert ae(fp.si(-50-2j), -1.497884414277228461+0.017515007378437448j) | |
| assert ae(fp.si(-2-20j), -1.20747603112735722103e7+4.050116856873293554e6j) | |
| assert ae(fp.si(-20j), -1.2807826332028294459e7j) | |
| assert ae(fp.si(50-2j), 1.497884414277228461+0.017515007378437448j) | |
| assert ae(fp.chi(50j), -0.0056283863241163054+1.5707963267948966192j) | |
| assert ae(fp.chi(-2+50j), -0.0183782829461330671+1.6411491348185849554j) | |
| assert ae(fp.chi(-20), 1.28078263320282943611e7+3.1415926535898j) | |
| assert ae(fp.chi(-20-2j), -4.050116856873293505e6+1.20747571696809187053e7j) | |
| assert ae(fp.chi(-2-50j), -0.0183782829461330671-1.6411491348185849554j) | |
| assert ae(fp.chi(-50j), -0.0056283863241163054-1.5707963267948966192j) | |
| assert ae(fp.chi(2-50j), -0.0183782829461330671-1.500443518771208283j) | |
| assert ae(fp.chi(20-2j), -4.050116856873293505e6-1.20747603112735722951e7j) | |
| assert ae(fp.chi(20), 1.2807826332028294361e7) | |
| assert ae(fp.chi(2+50j), -0.0183782829461330671+1.500443518771208283j) | |
| assert ae(fp.shi(50j), 1.5516170724859358947j) | |
| assert ae(fp.shi(-2+50j), 0.017515007378437448+1.497884414277228461j) | |
| assert ae(fp.shi(-20), -1.2807826332028294459e7) | |
| assert ae(fp.shi(-20-2j), 4.050116856873293554e6-1.20747603112735722103e7j) | |
| assert ae(fp.shi(-2-50j), 0.017515007378437448-1.497884414277228461j) | |
| assert ae(fp.shi(-50j), -1.5516170724859358947j) | |
| assert ae(fp.shi(2-50j), -0.017515007378437448-1.497884414277228461j) | |
| assert ae(fp.shi(20-2j), -4.050116856873293554e6-1.20747603112735722103e7j) | |
| assert ae(fp.shi(20), 1.2807826332028294459e7) | |
| assert ae(fp.shi(2+50j), -0.017515007378437448+1.497884414277228461j) | |
| def test_airy(): | |
| mp.dps = 15 | |
| assert (airyai(10)*10**10).ae(1.1047532552898687) | |
| assert (airybi(10)/10**9).ae(0.45564115354822515) | |
| assert (airyai(1000)*10**9158).ae(9.306933063179556004) | |
| assert (airybi(1000)/10**9154).ae(5.4077118391949465477) | |
| assert airyai(-1000).ae(0.055971895773019918842) | |
| assert airybi(-1000).ae(-0.083264574117080633012) | |
| assert (airyai(100+100j)*10**188).ae(2.9099582462207032076 + 2.353013591706178756j) | |
| assert (airybi(100+100j)/10**185).ae(1.7086751714463652039 - 3.1416590020830804578j) | |
| def test_hyper_0f1(): | |
| mp.dps = 15 | |
| v = 8.63911136507950465 | |
| assert hyper([],[(1,3)],1.5).ae(v) | |
| assert hyper([],[1/3.],1.5).ae(v) | |
| assert hyp0f1(1/3.,1.5).ae(v) | |
| assert hyp0f1((1,3),1.5).ae(v) | |
| # Asymptotic expansion | |
| assert hyp0f1(3,1e9).ae('4.9679055380347771271e+27455') | |
| assert hyp0f1(3,1e9j).ae('-2.1222788784457702157e+19410 + 5.0840597555401854116e+19410j') | |
| def test_hyper_1f1(): | |
| mp.dps = 15 | |
| v = 1.2917526488617656673 | |
| assert hyper([(1,2)],[(3,2)],0.7).ae(v) | |
| assert hyper([(1,2)],[(3,2)],0.7+0j).ae(v) | |
| assert hyper([0.5],[(3,2)],0.7).ae(v) | |
| assert hyper([0.5],[1.5],0.7).ae(v) | |
| assert hyper([0.5],[(3,2)],0.7+0j).ae(v) | |
| assert hyper([0.5],[1.5],0.7+0j).ae(v) | |
| assert hyper([(1,2)],[1.5+0j],0.7).ae(v) | |
| assert hyper([0.5+0j],[1.5],0.7).ae(v) | |
| assert hyper([0.5+0j],[1.5+0j],0.7+0j).ae(v) | |
| assert hyp1f1(0.5,1.5,0.7).ae(v) | |
| assert hyp1f1((1,2),1.5,0.7).ae(v) | |
| # Asymptotic expansion | |
| assert hyp1f1(2,3,1e10).ae('2.1555012157015796988e+4342944809') | |
| assert (hyp1f1(2,3,1e10j)*10**10).ae(-0.97501205020039745852 - 1.7462392454512132074j) | |
| # Shouldn't use asymptotic expansion | |
| assert hyp1f1(-2, 1, 10000).ae(49980001) | |
| # Bug | |
| assert hyp1f1(1j,fraction(1,3),0.415-69.739j).ae(25.857588206024346592 + 15.738060264515292063j) | |
| def test_hyper_2f1(): | |
| mp.dps = 15 | |
| v = 1.0652207633823291032 | |
| assert hyper([(1,2), (3,4)], [2], 0.3).ae(v) | |
| assert hyper([(1,2), 0.75], [2], 0.3).ae(v) | |
| assert hyper([0.5, 0.75], [2.0], 0.3).ae(v) | |
| assert hyper([0.5, 0.75], [2.0], 0.3+0j).ae(v) | |
| assert hyper([0.5+0j, (3,4)], [2.0], 0.3+0j).ae(v) | |
| assert hyper([0.5+0j, (3,4)], [2.0], 0.3).ae(v) | |
| assert hyper([0.5, (3,4)], [2.0+0j], 0.3).ae(v) | |
| assert hyper([0.5+0j, 0.75+0j], [2.0+0j], 0.3+0j).ae(v) | |
| v = 1.09234681096223231717 + 0.18104859169479360380j | |
| assert hyper([(1,2),0.75+j], [2], 0.5).ae(v) | |
| assert hyper([0.5,0.75+j], [2.0], 0.5).ae(v) | |
| assert hyper([0.5,0.75+j], [2.0], 0.5+0j).ae(v) | |
| assert hyper([0.5,0.75+j], [2.0+0j], 0.5+0j).ae(v) | |
| v = 0.9625 - 0.125j | |
| assert hyper([(3,2),-1],[4], 0.1+j/3).ae(v) | |
| assert hyper([1.5,-1.0],[4], 0.1+j/3).ae(v) | |
| assert hyper([1.5,-1.0],[4+0j], 0.1+j/3).ae(v) | |
| assert hyper([1.5+0j,-1.0+0j],[4+0j], 0.1+j/3).ae(v) | |
| v = 1.02111069501693445001 - 0.50402252613466859521j | |
| assert hyper([(2,10),(3,10)],[(4,10)],1.5).ae(v) | |
| assert hyper([0.2,(3,10)],[0.4+0j],1.5).ae(v) | |
| assert hyper([0.2,(3,10)],[0.4+0j],1.5+0j).ae(v) | |
| v = 0.76922501362865848528 + 0.32640579593235886194j | |
| assert hyper([(2,10),(3,10)],[(4,10)],4+2j).ae(v) | |
| assert hyper([0.2,(3,10)],[0.4+0j],4+2j).ae(v) | |
| assert hyper([0.2,(3,10)],[(4,10)],4+2j).ae(v) | |
| def test_hyper_2f1_hard(): | |
| mp.dps = 15 | |
| # Singular cases | |
| assert hyp2f1(2,-1,-1,3).ae(7) | |
| assert hyp2f1(2,-1,-1,3,eliminate_all=True).ae(0.25) | |
| assert hyp2f1(2,-2,-2,3).ae(34) | |
| assert hyp2f1(2,-2,-2,3,eliminate_all=True).ae(0.25) | |
| assert hyp2f1(2,-2,-3,3) == 14 | |
| assert hyp2f1(2,-3,-2,3) == inf | |
| assert hyp2f1(2,-1.5,-1.5,3) == 0.25 | |
| assert hyp2f1(1,2,3,0) == 1 | |
| assert hyp2f1(0,1,0,0) == 1 | |
| assert hyp2f1(0,0,0,0) == 1 | |
| assert isnan(hyp2f1(1,1,0,0)) | |
| assert hyp2f1(2,-1,-5, 0.25+0.25j).ae(1.1+0.1j) | |
| assert hyp2f1(2,-5,-5, 0.25+0.25j, eliminate=False).ae(163./128 + 125./128*j) | |
| assert hyp2f1(0.7235, -1, -5, 0.3).ae(1.04341) | |
| assert hyp2f1(0.7235, -5, -5, 0.3, eliminate=False).ae(1.2939225017815903812) | |
| assert hyp2f1(-1,-2,4,1) == 1.5 | |
| assert hyp2f1(1,2,-3,1) == inf | |
| assert hyp2f1(-2,-2,1,1) == 6 | |
| assert hyp2f1(1,-2,-4,1).ae(5./3) | |
| assert hyp2f1(0,-6,-4,1) == 1 | |
| assert hyp2f1(0,-3,-4,1) == 1 | |
| assert hyp2f1(0,0,0,1) == 1 | |
| assert hyp2f1(1,0,0,1,eliminate=False) == 1 | |
| assert hyp2f1(1,1,0,1) == inf | |
| assert hyp2f1(1,-6,-4,1) == inf | |
| assert hyp2f1(-7.2,-0.5,-4.5,1) == 0 | |
| assert hyp2f1(-7.2,-1,-2,1).ae(-2.6) | |
| assert hyp2f1(1,-0.5,-4.5, 1) == inf | |
| assert hyp2f1(1,0.5,-4.5, 1) == -inf | |
| # Check evaluation on / close to unit circle | |
| z = exp(j*pi/3) | |
| w = (nthroot(2,3)+1)*exp(j*pi/12)/nthroot(3,4)**3 | |
| assert hyp2f1('1/2','1/6','1/3', z).ae(w) | |
| assert hyp2f1('1/2','1/6','1/3', z.conjugate()).ae(w.conjugate()) | |
| assert hyp2f1(0.25, (1,3), 2, '0.999').ae(1.06826449496030635) | |
| assert hyp2f1(0.25, (1,3), 2, '1.001').ae(1.06867299254830309446-0.00001446586793975874j) | |
| assert hyp2f1(0.25, (1,3), 2, -1).ae(0.96656584492524351673) | |
| assert hyp2f1(0.25, (1,3), 2, j).ae(0.99041766248982072266+0.03777135604180735522j) | |
| assert hyp2f1(2,3,5,'0.99').ae(27.699347904322690602) | |
| assert hyp2f1((3,2),-0.5,3,'0.99').ae(0.68403036843911661388) | |
| assert hyp2f1(2,3,5,1j).ae(0.37290667145974386127+0.59210004902748285917j) | |
| assert fsum([hyp2f1((7,10),(2,3),(-1,2), 0.95*exp(j*k)) for k in range(1,15)]).ae(52.851400204289452922+6.244285013912953225j) | |
| assert fsum([hyp2f1((7,10),(2,3),(-1,2), 1.05*exp(j*k)) for k in range(1,15)]).ae(54.506013786220655330-3.000118813413217097j) | |
| assert fsum([hyp2f1((7,10),(2,3),(-1,2), exp(j*k)) for k in range(1,15)]).ae(55.792077935955314887+1.731986485778500241j) | |
| assert hyp2f1(2,2.5,-3.25,0.999).ae(218373932801217082543180041.33) | |
| # Branches | |
| assert hyp2f1(1,1,2,1.01).ae(4.5595744415723676911-3.1104877758314784539j) | |
| assert hyp2f1(1,1,2,1.01+0.1j).ae(2.4149427480552782484+1.4148224796836938829j) | |
| assert hyp2f1(1,1,2,3+4j).ae(0.14576709331407297807+0.48379185417980360773j) | |
| assert hyp2f1(1,1,2,4).ae(-0.27465307216702742285 - 0.78539816339744830962j) | |
| assert hyp2f1(1,1,2,-4).ae(0.40235947810852509365) | |
| # Other: | |
| # Cancellation with a large parameter involved (bug reported on sage-devel) | |
| assert hyp2f1(112, (51,10), (-9,10), -0.99999).ae(-1.6241361047970862961e-24, abs_eps=0, rel_eps=eps*16) | |
| def test_hyper_3f2_etc(): | |
| assert hyper([1,2,3],[1.5,8],-1).ae(0.67108992351533333030) | |
| assert hyper([1,2,3,4],[5,6,7], -1).ae(0.90232988035425506008) | |
| assert hyper([1,2,3],[1.25,5], 1).ae(28.924181329701905701) | |
| assert hyper([1,2,3,4],[5,6,7],5).ae(1.5192307344006649499-1.1529845225075537461j) | |
| assert hyper([1,2,3,4,5],[6,7,8,9],-1).ae(0.96288759462882357253) | |
| assert hyper([1,2,3,4,5],[6,7,8,9],1).ae(1.0428697385885855841) | |
| assert hyper([1,2,3,4,5],[6,7,8,9],5).ae(1.33980653631074769423-0.07143405251029226699j) | |
| assert hyper([1,2.79,3.08,4.37],[5.2,6.1,7.3],5).ae(1.0996321464692607231-1.7748052293979985001j) | |
| assert hyper([1,1,1],[1,2],1) == inf | |
| assert hyper([1,1,1],[2,(101,100)],1).ae(100.01621213528313220) | |
| # slow -- covered by doctests | |
| #assert hyper([1,1,1],[2,3],0.9999).ae(1.2897972005319693905) | |
| def test_hyper_u(): | |
| mp.dps = 15 | |
| assert hyperu(2,-3,0).ae(0.05) | |
| assert hyperu(2,-3.5,0).ae(4./99) | |
| assert hyperu(2,0,0) == 0.5 | |
| assert hyperu(-5,1,0) == -120 | |
| assert hyperu(-5,2,0) == inf | |
| assert hyperu(-5,-2,0) == 0 | |
| assert hyperu(7,7,3).ae(0.00014681269365593503986) #exp(3)*gammainc(-6,3) | |
| assert hyperu(2,-3,4).ae(0.011836478100271995559) | |
| assert hyperu(3,4,5).ae(1./125) | |
| assert hyperu(2,3,0.0625) == 256 | |
| assert hyperu(-1,2,0.25+0.5j) == -1.75+0.5j | |
| assert hyperu(0.5,1.5,7.25).ae(2/sqrt(29)) | |
| assert hyperu(2,6,pi).ae(0.55804439825913399130) | |
| assert (hyperu((3,2),8,100+201j)*10**4).ae(-0.3797318333856738798 - 2.9974928453561707782j) | |
| assert (hyperu((5,2),(-1,2),-5000)*10**10).ae(-5.6681877926881664678j) | |
| # XXX: fails because of undetected cancellation in low level series code | |
| # Alternatively: could use asymptotic series here, if convergence test | |
| # tweaked back to recognize this one | |
| #assert (hyperu((5,2),(-1,2),-500)*10**7).ae(-1.82526906001593252847j) | |
| def test_hyper_2f0(): | |
| mp.dps = 15 | |
| assert hyper([1,2],[],3) == hyp2f0(1,2,3) | |
| assert hyp2f0(2,3,7).ae(0.0116108068639728714668 - 0.0073727413865865802130j) | |
| assert hyp2f0(2,3,0) == 1 | |
| assert hyp2f0(0,0,0) == 1 | |
| assert hyp2f0(-1,-1,1).ae(2) | |
| assert hyp2f0(-4,1,1.5).ae(62.5) | |
| assert hyp2f0(-4,1,50).ae(147029801) | |
| assert hyp2f0(-4,1,0.0001).ae(0.99960011997600240000) | |
| assert hyp2f0(0.5,0.25,0.001).ae(1.0001251174078538115) | |
| assert hyp2f0(0.5,0.25,3+4j).ae(0.85548875824755163518 + 0.21636041283392292973j) | |
| # Important: cancellation check | |
| assert hyp2f0((1,6),(5,6),-0.02371708245126284498).ae(0.996785723120804309) | |
| # Should be exact; polynomial case | |
| assert hyp2f0(-2,1,0.5+0.5j,zeroprec=200) == 0 | |
| assert hyp2f0(1,-2,0.5+0.5j,zeroprec=200) == 0 | |
| # There used to be a bug in thresholds that made one of the following hang | |
| for d in [15, 50, 80]: | |
| mp.dps = d | |
| assert hyp2f0(1.5, 0.5, 0.009).ae('1.006867007239309717945323585695344927904000945829843527398772456281301440034218290443367270629519483 + 1.238277162240704919639384945859073461954721356062919829456053965502443570466701567100438048602352623e-46j') | |
| def test_hyper_1f2(): | |
| mp.dps = 15 | |
| assert hyper([1],[2,3],4) == hyp1f2(1,2,3,4) | |
| a1,b1,b2 = (1,10),(2,3),1./16 | |
| assert hyp1f2(a1,b1,b2,10).ae(298.7482725554557568) | |
| assert hyp1f2(a1,b1,b2,100).ae(224128961.48602947604) | |
| assert hyp1f2(a1,b1,b2,1000).ae(1.1669528298622675109e+27) | |
| assert hyp1f2(a1,b1,b2,10000).ae(2.4780514622487212192e+86) | |
| assert hyp1f2(a1,b1,b2,100000).ae(1.3885391458871523997e+274) | |
| assert hyp1f2(a1,b1,b2,1000000).ae('9.8851796978960318255e+867') | |
| assert hyp1f2(a1,b1,b2,10**7).ae('1.1505659189516303646e+2746') | |
| assert hyp1f2(a1,b1,b2,10**8).ae('1.4672005404314334081e+8685') | |
| assert hyp1f2(a1,b1,b2,10**20).ae('3.6888217332150976493e+8685889636') | |
| assert hyp1f2(a1,b1,b2,10*j).ae(-16.163252524618572878 - 44.321567896480184312j) | |
| assert hyp1f2(a1,b1,b2,100*j).ae(61938.155294517848171 + 637349.45215942348739j) | |
| assert hyp1f2(a1,b1,b2,1000*j).ae(8455057657257695958.7 + 6261969266997571510.6j) | |
| assert hyp1f2(a1,b1,b2,10000*j).ae(-8.9771211184008593089e+60 + 4.6550528111731631456e+59j) | |
| assert hyp1f2(a1,b1,b2,100000*j).ae(2.6398091437239324225e+193 + 4.1658080666870618332e+193j) | |
| assert hyp1f2(a1,b1,b2,1000000*j).ae('3.5999042951925965458e+613 + 1.5026014707128947992e+613j') | |
| assert hyp1f2(a1,b1,b2,10**7*j).ae('-8.3208715051623234801e+1939 - 3.6752883490851869429e+1941j') | |
| assert hyp1f2(a1,b1,b2,10**8*j).ae('2.0724195707891484454e+6140 - 1.3276619482724266387e+6141j') | |
| assert hyp1f2(a1,b1,b2,10**20*j).ae('-1.1734497974795488504e+6141851462 + 1.1498106965385471542e+6141851462j') | |
| def test_hyper_2f3(): | |
| mp.dps = 15 | |
| assert hyper([1,2],[3,4,5],6) == hyp2f3(1,2,3,4,5,6) | |
| a1,a2,b1,b2,b3 = (1,10),(2,3),(3,10), 2, 1./16 | |
| # Check asymptotic expansion | |
| assert hyp2f3(a1,a2,b1,b2,b3,10).ae(128.98207160698659976) | |
| assert hyp2f3(a1,a2,b1,b2,b3,1000).ae(6.6309632883131273141e25) | |
| assert hyp2f3(a1,a2,b1,b2,b3,10000).ae(4.6863639362713340539e84) | |
| assert hyp2f3(a1,a2,b1,b2,b3,100000).ae(8.6632451236103084119e271) | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**6).ae('2.0291718386574980641e865') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**7).ae('7.7639836665710030977e2742') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**8).ae('3.2537462584071268759e8681') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**20).ae('1.2966030542911614163e+8685889627') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10*j).ae(-18.551602185587547854 - 13.348031097874113552j) | |
| assert hyp2f3(a1,a2,b1,b2,b3,100*j).ae(78634.359124504488695 + 74459.535945281973996j) | |
| assert hyp2f3(a1,a2,b1,b2,b3,1000*j).ae(597682550276527901.59 - 65136194809352613.078j) | |
| assert hyp2f3(a1,a2,b1,b2,b3,10000*j).ae(-1.1779696326238582496e+59 + 1.2297607505213133872e+59j) | |
| assert hyp2f3(a1,a2,b1,b2,b3,100000*j).ae(2.9844228969804380301e+191 + 7.5587163231490273296e+190j) | |
| assert hyp2f3(a1,a2,b1,b2,b3,1000000*j).ae('7.4859161049322370311e+610 - 2.8467477015940090189e+610j') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**7*j).ae('-1.7477645579418800826e+1938 - 1.7606522995808116405e+1938j') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**8*j).ae('-1.6932731942958401784e+6137 - 2.4521909113114629368e+6137j') | |
| assert hyp2f3(a1,a2,b1,b2,b3,10**20*j).ae('-2.0988815677627225449e+6141851451 + 5.7708223542739208681e+6141851452j') | |
| def test_hyper_2f2(): | |
| mp.dps = 15 | |
| assert hyper([1,2],[3,4],5) == hyp2f2(1,2,3,4,5) | |
| a1,a2,b1,b2 = (3,10),4,(1,2),1./16 | |
| assert hyp2f2(a1,a2,b1,b2,10).ae(448225936.3377556696) | |
| assert hyp2f2(a1,a2,b1,b2,10000).ae('1.2012553712966636711e+4358') | |
| assert hyp2f2(a1,a2,b1,b2,-20000).ae(-0.04182343755661214626) | |
| assert hyp2f2(a1,a2,b1,b2,10**20).ae('1.1148680024303263661e+43429448190325182840') | |
| def test_orthpoly(): | |
| mp.dps = 15 | |
| assert jacobi(-4,2,3,0.7).ae(22800./4913) | |
| assert jacobi(3,2,4,5.5) == 4133.125 | |
| assert jacobi(1.5,5/6.,4,0).ae(-1.0851951434075508417) | |
| assert jacobi(-2, 1, 2, 4).ae(-0.16) | |
| assert jacobi(2, -1, 2.5, 4).ae(34.59375) | |
| #assert jacobi(2, -1, 2, 4) == 28.5 | |
| assert legendre(5, 7) == 129367 | |
| assert legendre(0.5,0).ae(0.53935260118837935667) | |
| assert legendre(-1,-1) == 1 | |
| assert legendre(0,-1) == 1 | |
| assert legendre(0, 1) == 1 | |
| assert legendre(1, -1) == -1 | |
| assert legendre(7, 1) == 1 | |
| assert legendre(7, -1) == -1 | |
| assert legendre(8,1.5).ae(15457523./32768) | |
| assert legendre(j,-j).ae(2.4448182735671431011 + 0.6928881737669934843j) | |
| assert chebyu(5,1) == 6 | |
| assert chebyt(3,2) == 26 | |
| assert legendre(3.5,-1) == inf | |
| assert legendre(4.5,-1) == -inf | |
| assert legendre(3.5+1j,-1) == mpc(inf,inf) | |
| assert legendre(4.5+1j,-1) == mpc(-inf,-inf) | |
| assert laguerre(4, -2, 3).ae(-1.125) | |
| assert laguerre(3, 1+j, 0.5).ae(0.2291666666666666667 + 2.5416666666666666667j) | |
| def test_hermite(): | |
| mp.dps = 15 | |
| assert hermite(-2, 0).ae(0.5) | |
| assert hermite(-1, 0).ae(0.88622692545275801365) | |
| assert hermite(0, 0).ae(1) | |
| assert hermite(1, 0) == 0 | |
| assert hermite(2, 0).ae(-2) | |
| assert hermite(0, 2).ae(1) | |
| assert hermite(1, 2).ae(4) | |
| assert hermite(1, -2).ae(-4) | |
| assert hermite(2, -2).ae(14) | |
| assert hermite(0.5, 0).ae(0.69136733903629335053) | |
| assert hermite(9, 0) == 0 | |
| assert hermite(4,4).ae(3340) | |
| assert hermite(3,4).ae(464) | |
| assert hermite(-4,4).ae(0.00018623860287512396181) | |
| assert hermite(-3,4).ae(0.0016540169879668766270) | |
| assert hermite(9, 2.5j).ae(13638725j) | |
| assert hermite(9, -2.5j).ae(-13638725j) | |
| assert hermite(9, 100).ae(511078883759363024000) | |
| assert hermite(9, -100).ae(-511078883759363024000) | |
| assert hermite(9, 100j).ae(512922083920643024000j) | |
| assert hermite(9, -100j).ae(-512922083920643024000j) | |
| assert hermite(-9.5, 2.5j).ae(-2.9004951258126778174e-6 + 1.7601372934039951100e-6j) | |
| assert hermite(-9.5, -2.5j).ae(-2.9004951258126778174e-6 - 1.7601372934039951100e-6j) | |
| assert hermite(-9.5, 100).ae(1.3776300722767084162e-22, abs_eps=0, rel_eps=eps) | |
| assert hermite(-9.5, -100).ae('1.3106082028470671626e4355') | |
| assert hermite(-9.5, 100j).ae(-9.7900218581864768430e-23 - 9.7900218581864768430e-23j, abs_eps=0, rel_eps=eps) | |
| assert hermite(-9.5, -100j).ae(-9.7900218581864768430e-23 + 9.7900218581864768430e-23j, abs_eps=0, rel_eps=eps) | |
| assert hermite(2+3j, -1-j).ae(851.3677063883687676 - 1496.4373467871007997j) | |
| def test_gegenbauer(): | |
| mp.dps = 15 | |
| assert gegenbauer(1,2,3).ae(12) | |
| assert gegenbauer(2,3,4).ae(381) | |
| assert gegenbauer(0,0,0) == 0 | |
| assert gegenbauer(2,-1,3) == 0 | |
| assert gegenbauer(-7, 0.5, 3).ae(8989) | |
| assert gegenbauer(1, -0.5, 3).ae(-3) | |
| assert gegenbauer(1, -1.5, 3).ae(-9) | |
| assert gegenbauer(1, -0.5, 3).ae(-3) | |
| assert gegenbauer(-0.5, -0.5, 3).ae(-2.6383553159023906245) | |
| assert gegenbauer(2+3j, 1-j, 3+4j).ae(14.880536623203696780 + 20.022029711598032898j) | |
| #assert gegenbauer(-2, -0.5, 3).ae(-12) | |
| def test_legenp(): | |
| mp.dps = 15 | |
| assert legenp(2,0,4) == legendre(2,4) | |
| assert legenp(-2, -1, 0.5).ae(0.43301270189221932338) | |
| assert legenp(-2, -1, 0.5, type=3).ae(0.43301270189221932338j) | |
| assert legenp(-2, 1, 0.5).ae(-0.86602540378443864676) | |
| assert legenp(2+j, 3+4j, -j).ae(134742.98773236786148 + 429782.72924463851745j) | |
| assert legenp(2+j, 3+4j, -j, type=3).ae(802.59463394152268507 - 251.62481308942906447j) | |
| assert legenp(2,4,3).ae(0) | |
| assert legenp(2,4,3,type=3).ae(0) | |
| assert legenp(2,1,0.5).ae(-1.2990381056766579701) | |
| assert legenp(2,1,0.5,type=3).ae(1.2990381056766579701j) | |
| assert legenp(3,2,3).ae(-360) | |
| assert legenp(3,3,3).ae(240j*2**0.5) | |
| assert legenp(3,4,3).ae(0) | |
| assert legenp(0,0.5,2).ae(0.52503756790433198939 - 0.52503756790433198939j) | |
| assert legenp(-1,-0.5,2).ae(0.60626116232846498110 + 0.60626116232846498110j) | |
| assert legenp(-2,0.5,2).ae(1.5751127037129959682 - 1.5751127037129959682j) | |
| assert legenp(-2,0.5,-0.5).ae(-0.85738275810499171286) | |
| def test_legenq(): | |
| mp.dps = 15 | |
| f = legenq | |
| # Evaluation at poles | |
| assert isnan(f(3,2,1)) | |
| assert isnan(f(3,2,-1)) | |
| assert isnan(f(3,2,1,type=3)) | |
| assert isnan(f(3,2,-1,type=3)) | |
| # Evaluation at 0 | |
| assert f(0,1,0,type=2).ae(-1) | |
| assert f(-2,2,0,type=2,zeroprec=200).ae(0) | |
| assert f(1.5,3,0,type=2).ae(-2.2239343475841951023) | |
| assert f(0,1,0,type=3).ae(j) | |
| assert f(-2,2,0,type=3,zeroprec=200).ae(0) | |
| assert f(1.5,3,0,type=3).ae(2.2239343475841951022*(1-1j)) | |
| # Standard case, degree 0 | |
| assert f(0,0,-1.5).ae(-0.8047189562170501873 + 1.5707963267948966192j) | |
| assert f(0,0,-0.5).ae(-0.54930614433405484570) | |
| assert f(0,0,0,zeroprec=200).ae(0) | |
| assert f(0,0,0.5).ae(0.54930614433405484570) | |
| assert f(0,0,1.5).ae(0.8047189562170501873 - 1.5707963267948966192j) | |
| assert f(0,0,-1.5,type=3).ae(-0.80471895621705018730) | |
| assert f(0,0,-0.5,type=3).ae(-0.5493061443340548457 - 1.5707963267948966192j) | |
| assert f(0,0,0,type=3).ae(-1.5707963267948966192j) | |
| assert f(0,0,0.5,type=3).ae(0.5493061443340548457 - 1.5707963267948966192j) | |
| assert f(0,0,1.5,type=3).ae(0.80471895621705018730) | |
| # Standard case, degree 1 | |
| assert f(1,0,-1.5).ae(0.2070784343255752810 - 2.3561944901923449288j) | |
| assert f(1,0,-0.5).ae(-0.72534692783297257715) | |
| assert f(1,0,0).ae(-1) | |
| assert f(1,0,0.5).ae(-0.72534692783297257715) | |
| assert f(1,0,1.5).ae(0.2070784343255752810 - 2.3561944901923449288j) | |
| # Standard case, degree 2 | |
| assert f(2,0,-1.5).ae(-0.0635669991240192885 + 4.5160394395353277803j) | |
| assert f(2,0,-0.5).ae(0.81866326804175685571) | |
| assert f(2,0,0,zeroprec=200).ae(0) | |
| assert f(2,0,0.5).ae(-0.81866326804175685571) | |
| assert f(2,0,1.5).ae(0.0635669991240192885 - 4.5160394395353277803j) | |
| # Misc orders and degrees | |
| assert f(2,3,1.5,type=2).ae(-5.7243340223994616228j) | |
| assert f(2,3,1.5,type=3).ae(-5.7243340223994616228) | |
| assert f(2,3,0.5,type=2).ae(-12.316805742712016310) | |
| assert f(2,3,0.5,type=3).ae(-12.316805742712016310j) | |
| assert f(2,3,-1.5,type=2).ae(-5.7243340223994616228j) | |
| assert f(2,3,-1.5,type=3).ae(5.7243340223994616228) | |
| assert f(2,3,-0.5,type=2).ae(-12.316805742712016310) | |
| assert f(2,3,-0.5,type=3).ae(-12.316805742712016310j) | |
| assert f(2+3j, 3+4j, 0.5, type=3).ae(0.0016119404873235186807 - 0.0005885900510718119836j) | |
| assert f(2+3j, 3+4j, -1.5, type=3).ae(0.008451400254138808670 + 0.020645193304593235298j) | |
| assert f(-2.5,1,-1.5).ae(3.9553395527435335749j) | |
| assert f(-2.5,1,-0.5).ae(1.9290561746445456908) | |
| assert f(-2.5,1,0).ae(1.2708196271909686299) | |
| assert f(-2.5,1,0.5).ae(-0.31584812990742202869) | |
| assert f(-2.5,1,1.5).ae(-3.9553395527435335742 + 0.2993235655044701706j) | |
| assert f(-2.5,1,-1.5,type=3).ae(0.29932356550447017254j) | |
| assert f(-2.5,1,-0.5,type=3).ae(-0.3158481299074220287 - 1.9290561746445456908j) | |
| assert f(-2.5,1,0,type=3).ae(1.2708196271909686292 - 1.2708196271909686299j) | |
| assert f(-2.5,1,0.5,type=3).ae(1.9290561746445456907 + 0.3158481299074220287j) | |
| assert f(-2.5,1,1.5,type=3).ae(-0.29932356550447017254) | |
| def test_agm(): | |
| mp.dps = 15 | |
| assert agm(0,0) == 0 | |
| assert agm(0,1) == 0 | |
| assert agm(1,1) == 1 | |
| assert agm(7,7) == 7 | |
| assert agm(j,j) == j | |
| assert (1/agm(1,sqrt(2))).ae(0.834626841674073186) | |
| assert agm(1,2).ae(1.4567910310469068692) | |
| assert agm(1,3).ae(1.8636167832448965424) | |
| assert agm(1,j).ae(0.599070117367796104+0.599070117367796104j) | |
| assert agm(2) == agm(1,2) | |
| assert agm(-3,4).ae(0.63468509766550907+1.3443087080896272j) | |
| def test_gammainc(): | |
| mp.dps = 15 | |
| assert gammainc(2,5).ae(6*exp(-5)) | |
| assert gammainc(2,0,5).ae(1-6*exp(-5)) | |
| assert gammainc(2,3,5).ae(-6*exp(-5)+4*exp(-3)) | |
| assert gammainc(-2.5,-0.5).ae(-0.9453087204829418812-5.3164237738936178621j) | |
| assert gammainc(0,2,4).ae(0.045121158298212213088) | |
| assert gammainc(0,3).ae(0.013048381094197037413) | |
| assert gammainc(0,2+j,1-j).ae(0.00910653685850304839-0.22378752918074432574j) | |
| assert gammainc(0,1-j).ae(0.00028162445198141833+0.17932453503935894015j) | |
| assert gammainc(3,4,5,True).ae(0.11345128607046320253) | |
| assert gammainc(3.5,0,inf).ae(gamma(3.5)) | |
| assert gammainc(-150.5,500).ae('6.9825435345798951153e-627') | |
| assert gammainc(-150.5,800).ae('4.6885137549474089431e-788') | |
| assert gammainc(-3.5, -20.5).ae(0.27008820585226911 - 1310.31447140574997636j) | |
| assert gammainc(-3.5, -200.5).ae(0.27008820585226911 - 5.3264597096208368435e76j) # XXX real part | |
| assert gammainc(0,0,2) == inf | |
| assert gammainc(1,b=1).ae(0.6321205588285576784) | |
| assert gammainc(3,2,2) == 0 | |
| assert gammainc(2,3+j,3-j).ae(-0.28135485191849314194j) | |
| assert gammainc(4+0j,1).ae(5.8860710587430771455) | |
| # GH issue #301 | |
| assert gammainc(-1,-1).ae(-0.8231640121031084799 + 3.1415926535897932385j) | |
| assert gammainc(-2,-1).ae(1.7707229202810768576 - 1.5707963267948966192j) | |
| assert gammainc(-3,-1).ae(-1.4963349162467073643 + 0.5235987755982988731j) | |
| assert gammainc(-4,-1).ae(1.05365418617643814992 - 0.13089969389957471827j) | |
| # Regularized upper gamma | |
| assert isnan(gammainc(0, 0, regularized=True)) | |
| assert gammainc(-1, 0, regularized=True) == inf | |
| assert gammainc(1, 0, regularized=True) == 1 | |
| assert gammainc(0, 5, regularized=True) == 0 | |
| assert gammainc(0, 2+3j, regularized=True) == 0 | |
| assert gammainc(0, 5000, regularized=True) == 0 | |
| assert gammainc(0, 10**30, regularized=True) == 0 | |
| assert gammainc(-1, 5, regularized=True) == 0 | |
| assert gammainc(-1, 5000, regularized=True) == 0 | |
| assert gammainc(-1, 10**30, regularized=True) == 0 | |
| assert gammainc(-1, -5, regularized=True) == 0 | |
| assert gammainc(-1, -5000, regularized=True) == 0 | |
| assert gammainc(-1, -10**30, regularized=True) == 0 | |
| assert gammainc(-1, 3+4j, regularized=True) == 0 | |
| assert gammainc(1, 5, regularized=True).ae(exp(-5)) | |
| assert gammainc(1, 5000, regularized=True).ae(exp(-5000)) | |
| assert gammainc(1, 10**30, regularized=True).ae(exp(-10**30)) | |
| assert gammainc(1, 3+4j, regularized=True).ae(exp(-3-4j)) | |
| assert gammainc(-1000000,2).ae('1.3669297209397347754e-301037', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(-1000000,2,regularized=True) == 0 | |
| assert gammainc(-1000000,3+4j).ae('-1.322575609404222361e-698979 - 4.9274570591854533273e-698978j', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(-1000000,3+4j,regularized=True) == 0 | |
| assert gammainc(2+3j, 4+5j, regularized=True).ae(0.085422013530993285774-0.052595379150390078503j) | |
| assert gammainc(1000j, 1000j, regularized=True).ae(0.49702647628921131761 + 0.00297355675013575341j) | |
| # Generalized | |
| assert gammainc(3,4,2) == -gammainc(3,2,4) | |
| assert gammainc(4, 2, 3).ae(1.2593494302978947396) | |
| assert gammainc(4, 2, 3, regularized=True).ae(0.20989157171631578993) | |
| assert gammainc(0, 2, 3).ae(0.035852129613864082155) | |
| assert gammainc(0, 2, 3, regularized=True) == 0 | |
| assert gammainc(-1, 2, 3).ae(0.015219822548487616132) | |
| assert gammainc(-1, 2, 3, regularized=True) == 0 | |
| assert gammainc(0, 2, 3).ae(0.035852129613864082155) | |
| assert gammainc(0, 2, 3, regularized=True) == 0 | |
| # Should use upper gammas | |
| assert gammainc(5, 10000, 12000).ae('1.1359381951461801687e-4327', abs_eps=0, rel_eps=8*eps) | |
| # Should use lower gammas | |
| assert gammainc(10000, 2, 3).ae('8.1244514125995785934e4765') | |
| # GH issue 306 | |
| assert gammainc(3,-1-1j) == 0 | |
| assert gammainc(3,-1+1j) == 0 | |
| assert gammainc(2,-1) == 0 | |
| assert gammainc(2,-1+0j) == 0 | |
| assert gammainc(2+0j,-1) == 0 | |
| def test_gammainc_expint_n(): | |
| # These tests are intended to check all cases of the low-level code | |
| # for upper gamma and expint with small integer index. | |
| # Need to cover positive/negative arguments; small/large/huge arguments | |
| # for both positive and negative indices, as well as indices 0 and 1 | |
| # which may be special-cased | |
| mp.dps = 15 | |
| assert expint(-3,3.5).ae(0.021456366563296693987) | |
| assert expint(-2,3.5).ae(0.014966633183073309405) | |
| assert expint(-1,3.5).ae(0.011092916359219041088) | |
| assert expint(0,3.5).ae(0.0086278238349481430685) | |
| assert expint(1,3.5).ae(0.0069701398575483929193) | |
| assert expint(2,3.5).ae(0.0058018939208991255223) | |
| assert expint(3,3.5).ae(0.0049453773495857807058) | |
| assert expint(-3,-3.5).ae(-4.6618170604073311319) | |
| assert expint(-2,-3.5).ae(-5.5996974157555515963) | |
| assert expint(-1,-3.5).ae(-6.7582555017739415818) | |
| assert expint(0,-3.5).ae(-9.4615577024835182145) | |
| assert expint(1,-3.5).ae(-13.925353995152335292 - 3.1415926535897932385j) | |
| assert expint(2,-3.5).ae(-15.62328702434085977 - 10.995574287564276335j) | |
| assert expint(3,-3.5).ae(-10.783026313250347722 - 19.242255003237483586j) | |
| assert expint(-3,350).ae(2.8614825451252838069e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(-2,350).ae(2.8532837224504675901e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(-1,350).ae(2.8451316155828634555e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(0,350).ae(2.8370258275042797989e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(1,350).ae(2.8289659656701459404e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(2,350).ae(2.8209516419468505006e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(3,350).ae(2.8129824725501272171e-155, abs_eps=0, rel_eps=8*eps) | |
| assert expint(-3,-350).ae(-2.8528796154044839443e+149) | |
| assert expint(-2,-350).ae(-2.8610072121701264351e+149) | |
| assert expint(-1,-350).ae(-2.8691813842677537647e+149) | |
| assert expint(0,-350).ae(-2.8774025343659421709e+149) | |
| u = expint(1,-350) | |
| assert u.ae(-2.8856710698020863568e+149) | |
| assert u.imag.ae(-3.1415926535897932385) | |
| u = expint(2,-350) | |
| assert u.ae(-2.8939874026504650534e+149) | |
| assert u.imag.ae(-1099.5574287564276335) | |
| u = expint(3,-350) | |
| assert u.ae(-2.9023519497915044349e+149) | |
| assert u.imag.ae(-192422.55003237483586) | |
| assert expint(-3,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(-2,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(-1,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(0,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(1,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(2,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(3,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert expint(-3,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') | |
| assert expint(-2,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') | |
| assert expint(-1,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') | |
| assert expint(0,-350000000000000000000000).ae('-3.7805306852415755699e+152003068666138139677871') | |
| u = expint(1,-350000000000000000000000) | |
| assert u.ae('-3.7805306852415755699e+152003068666138139677871') | |
| assert u.imag.ae(-3.1415926535897932385) | |
| u = expint(2,-350000000000000000000000) | |
| assert u.imag.ae(-1.0995574287564276335e+24) | |
| assert u.ae('-3.7805306852415755699e+152003068666138139677871') | |
| u = expint(3,-350000000000000000000000) | |
| assert u.imag.ae(-1.9242255003237483586e+47) | |
| assert u.ae('-3.7805306852415755699e+152003068666138139677871') | |
| # Small case; no branch cut | |
| assert gammainc(-3,3.5).ae(0.00010020262545203707109) | |
| assert gammainc(-2,3.5).ae(0.00040370427343557393517) | |
| assert gammainc(-1,3.5).ae(0.0016576839773997501492) | |
| assert gammainc(0,3.5).ae(0.0069701398575483929193) | |
| assert gammainc(1,3.5).ae(0.03019738342231850074) | |
| assert gammainc(2,3.5).ae(0.13588822540043325333) | |
| assert gammainc(3,3.5).ae(0.64169439772426814072) | |
| # Small case; with branch cut | |
| assert gammainc(-3,-3.5).ae(0.03595832954467563286 + 0.52359877559829887308j) | |
| assert gammainc(-2,-3.5).ae(-0.88024704597962022221 - 1.5707963267948966192j) | |
| assert gammainc(-1,-3.5).ae(4.4637962926688170771 + 3.1415926535897932385j) | |
| assert gammainc(0,-3.5).ae(-13.925353995152335292 - 3.1415926535897932385j) | |
| assert gammainc(1,-3.5).ae(33.115451958692313751) | |
| assert gammainc(2,-3.5).ae(-82.788629896730784377) | |
| assert gammainc(3,-3.5).ae(240.08702670051927469) | |
| # Asymptotic case; no branch cut | |
| assert gammainc(-3,350).ae(6.5424095113340358813e-163, abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(-2,350).ae(2.296312222489899769e-160, abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(-1,350).ae(8.059861834133858573e-158, abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(0,350).ae(2.8289659656701459404e-155, abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(1,350).ae(9.9295903962649792963e-153, abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(2,350).ae(3.485286229089007733e-150, abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(3,350).ae(1.2233453960006379793e-147, abs_eps=0, rel_eps=8*eps) | |
| # Asymptotic case; branch cut | |
| u = gammainc(-3,-350) | |
| assert u.ae(6.7889565783842895085e+141) | |
| assert u.imag.ae(0.52359877559829887308) | |
| u = gammainc(-2,-350) | |
| assert u.ae(-2.3692668977889832121e+144) | |
| assert u.imag.ae(-1.5707963267948966192) | |
| u = gammainc(-1,-350) | |
| assert u.ae(8.2685354361441858669e+146) | |
| assert u.imag.ae(3.1415926535897932385) | |
| u = gammainc(0,-350) | |
| assert u.ae(-2.8856710698020863568e+149) | |
| assert u.imag.ae(-3.1415926535897932385) | |
| u = gammainc(1,-350) | |
| assert u.ae(1.0070908870280797598e+152) | |
| assert u.imag == 0 | |
| u = gammainc(2,-350) | |
| assert u.ae(-3.5147471957279983618e+154) | |
| assert u.imag == 0 | |
| u = gammainc(3,-350) | |
| assert u.ae(1.2266568422179417091e+157) | |
| assert u.imag == 0 | |
| # Extreme asymptotic case | |
| assert gammainc(-3,350000000000000000000000).ae('5.0362468738874738859e-152003068666138139677990', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(-2,350000000000000000000000).ae('1.7626864058606158601e-152003068666138139677966', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(-1,350000000000000000000000).ae('6.1694024205121555102e-152003068666138139677943', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(0,350000000000000000000000).ae('2.1592908471792544286e-152003068666138139677919', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(1,350000000000000000000000).ae('7.5575179651273905e-152003068666138139677896', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(2,350000000000000000000000).ae('2.645131287794586675e-152003068666138139677872', abs_eps=0, rel_eps=8*eps) | |
| assert gammainc(3,350000000000000000000000).ae('9.2579595072810533625e-152003068666138139677849', abs_eps=0, rel_eps=8*eps) | |
| u = gammainc(-3,-350000000000000000000000) | |
| assert u.ae('8.8175642804468234866e+152003068666138139677800') | |
| assert u.imag.ae(0.52359877559829887308) | |
| u = gammainc(-2,-350000000000000000000000) | |
| assert u.ae('-3.0861474981563882203e+152003068666138139677824') | |
| assert u.imag.ae(-1.5707963267948966192) | |
| u = gammainc(-1,-350000000000000000000000) | |
| assert u.ae('1.0801516243547358771e+152003068666138139677848') | |
| assert u.imag.ae(3.1415926535897932385) | |
| u = gammainc(0,-350000000000000000000000) | |
| assert u.ae('-3.7805306852415755699e+152003068666138139677871') | |
| assert u.imag.ae(-3.1415926535897932385) | |
| assert gammainc(1,-350000000000000000000000).ae('1.3231857398345514495e+152003068666138139677895') | |
| assert gammainc(2,-350000000000000000000000).ae('-4.6311500894209300731e+152003068666138139677918') | |
| assert gammainc(3,-350000000000000000000000).ae('1.6209025312973255256e+152003068666138139677942') | |
| def test_incomplete_beta(): | |
| mp.dps = 15 | |
| assert betainc(-2,-3,0.5,0.75).ae(63.4305673311255413583969) | |
| assert betainc(4.5,0.5+2j,2.5,6).ae(0.2628801146130621387903065 + 0.5162565234467020592855378j) | |
| assert betainc(4,5,0,6).ae(90747.77142857142857142857) | |
| def test_erf(): | |
| mp.dps = 15 | |
| assert erf(0) == 0 | |
| assert erf(1).ae(0.84270079294971486934) | |
| assert erf(3+4j).ae(-120.186991395079444098 - 27.750337293623902498j) | |
| assert erf(-4-3j).ae(-0.99991066178539168236 + 0.00004972026054496604j) | |
| assert erf(pi).ae(0.99999112385363235839) | |
| assert erf(1j).ae(1.6504257587975428760j) | |
| assert erf(-1j).ae(-1.6504257587975428760j) | |
| assert isinstance(erf(1), mpf) | |
| assert isinstance(erf(-1), mpf) | |
| assert isinstance(erf(0), mpf) | |
| assert isinstance(erf(0j), mpc) | |
| assert erf(inf) == 1 | |
| assert erf(-inf) == -1 | |
| assert erfi(0) == 0 | |
| assert erfi(1/pi).ae(0.371682698493894314) | |
| assert erfi(inf) == inf | |
| assert erfi(-inf) == -inf | |
| assert erf(1+0j) == erf(1) | |
| assert erfc(1+0j) == erfc(1) | |
| assert erf(0.2+0.5j).ae(1 - erfc(0.2+0.5j)) | |
| assert erfc(0) == 1 | |
| assert erfc(1).ae(1-erf(1)) | |
| assert erfc(-1).ae(1-erf(-1)) | |
| assert erfc(1/pi).ae(1-erf(1/pi)) | |
| assert erfc(-10) == 2 | |
| assert erfc(-1000000) == 2 | |
| assert erfc(-inf) == 2 | |
| assert erfc(inf) == 0 | |
| assert isnan(erfc(nan)) | |
| assert (erfc(10**4)*mpf(10)**43429453).ae('3.63998738656420') | |
| assert erf(8+9j).ae(-1072004.2525062051158 + 364149.91954310255423j) | |
| assert erfc(8+9j).ae(1072005.2525062051158 - 364149.91954310255423j) | |
| assert erfc(-8-9j).ae(-1072003.2525062051158 + 364149.91954310255423j) | |
| mp.dps = 50 | |
| # This one does not use the asymptotic series | |
| assert (erfc(10)*10**45).ae('2.0884875837625447570007862949577886115608181193212') | |
| # This one does | |
| assert (erfc(50)*10**1088).ae('2.0709207788416560484484478751657887929322509209954') | |
| mp.dps = 15 | |
| assert str(erfc(10**50)) == '3.66744826532555e-4342944819032518276511289189166050822943970058036665661144537831658646492088707747292249493384317534' | |
| assert erfinv(0) == 0 | |
| assert erfinv(0.5).ae(0.47693627620446987338) | |
| assert erfinv(-0.5).ae(-0.47693627620446987338) | |
| assert erfinv(1) == inf | |
| assert erfinv(-1) == -inf | |
| assert erf(erfinv(0.95)).ae(0.95) | |
| assert erf(erfinv(0.999999999995)).ae(0.999999999995) | |
| assert erf(erfinv(-0.999999999995)).ae(-0.999999999995) | |
| mp.dps = 50 | |
| assert erf(erfinv('0.99999999999999999999999999999995')).ae('0.99999999999999999999999999999995') | |
| assert erf(erfinv('0.999999999999999999999999999999995')).ae('0.999999999999999999999999999999995') | |
| assert erf(erfinv('-0.999999999999999999999999999999995')).ae('-0.999999999999999999999999999999995') | |
| mp.dps = 15 | |
| # Complex asymptotic expansions | |
| v = erfc(50j) | |
| assert v.real == 1 | |
| assert v.imag.ae('-6.1481820666053078736e+1083') | |
| assert erfc(-100+5j).ae(2) | |
| assert (erfc(100+5j)*10**4335).ae(2.3973567853824133572 - 3.9339259530609420597j) | |
| assert erfc(100+100j).ae(0.00065234366376857698698 - 0.0039357263629214118437j) | |
| def test_pdf(): | |
| mp.dps = 15 | |
| assert npdf(-inf) == 0 | |
| assert npdf(inf) == 0 | |
| assert npdf(5,0,2).ae(npdf(5+4,4,2)) | |
| assert quadts(lambda x: npdf(x,-0.5,0.8), [-inf, inf]) == 1 | |
| assert ncdf(0) == 0.5 | |
| assert ncdf(3,3) == 0.5 | |
| assert ncdf(-inf) == 0 | |
| assert ncdf(inf) == 1 | |
| assert ncdf(10) == 1 | |
| # Verify that this is computed accurately | |
| assert (ncdf(-10)*10**24).ae(7.619853024160526) | |
| def test_lambertw(): | |
| mp.dps = 15 | |
| assert lambertw(0) == 0 | |
| assert lambertw(0+0j) == 0 | |
| assert lambertw(inf) == inf | |
| assert isnan(lambertw(nan)) | |
| assert lambertw(inf,1).real == inf | |
| assert lambertw(inf,1).imag.ae(2*pi) | |
| assert lambertw(-inf,1).real == inf | |
| assert lambertw(-inf,1).imag.ae(3*pi) | |
| assert lambertw(0,-1) == -inf | |
| assert lambertw(0,1) == -inf | |
| assert lambertw(0,3) == -inf | |
| assert lambertw(e).ae(1) | |
| assert lambertw(1).ae(0.567143290409783873) | |
| assert lambertw(-pi/2).ae(j*pi/2) | |
| assert lambertw(-log(2)/2).ae(-log(2)) | |
| assert lambertw(0.25).ae(0.203888354702240164) | |
| assert lambertw(-0.25).ae(-0.357402956181388903) | |
| assert lambertw(-1./10000,0).ae(-0.000100010001500266719) | |
| assert lambertw(-0.25,-1).ae(-2.15329236411034965) | |
| assert lambertw(0.25,-1).ae(-3.00899800997004620-4.07652978899159763j) | |
| assert lambertw(-0.25,-1).ae(-2.15329236411034965) | |
| assert lambertw(0.25,1).ae(-3.00899800997004620+4.07652978899159763j) | |
| assert lambertw(-0.25,1).ae(-3.48973228422959210+7.41405453009603664j) | |
| assert lambertw(-4).ae(0.67881197132094523+1.91195078174339937j) | |
| assert lambertw(-4,1).ae(-0.66743107129800988+7.76827456802783084j) | |
| assert lambertw(-4,-1).ae(0.67881197132094523-1.91195078174339937j) | |
| assert lambertw(1000).ae(5.24960285240159623) | |
| assert lambertw(1000,1).ae(4.91492239981054535+5.44652615979447070j) | |
| assert lambertw(1000,-1).ae(4.91492239981054535-5.44652615979447070j) | |
| assert lambertw(1000,5).ae(3.5010625305312892+29.9614548941181328j) | |
| assert lambertw(3+4j).ae(1.281561806123775878+0.533095222020971071j) | |
| assert lambertw(-0.4+0.4j).ae(-0.10396515323290657+0.61899273315171632j) | |
| assert lambertw(3+4j,1).ae(-0.11691092896595324+5.61888039871282334j) | |
| assert lambertw(3+4j,-1).ae(0.25856740686699742-3.85211668616143559j) | |
| assert lambertw(-0.5,-1).ae(-0.794023632344689368-0.770111750510379110j) | |
| assert lambertw(-1./10000,1).ae(-11.82350837248724344+6.80546081842002101j) | |
| assert lambertw(-1./10000,-1).ae(-11.6671145325663544) | |
| assert lambertw(-1./10000,-2).ae(-11.82350837248724344-6.80546081842002101j) | |
| assert lambertw(-1./100000,4).ae(-14.9186890769540539+26.1856750178782046j) | |
| assert lambertw(-1./100000,5).ae(-15.0931437726379218666+32.5525721210262290086j) | |
| assert lambertw((2+j)/10).ae(0.173704503762911669+0.071781336752835511j) | |
| assert lambertw((2+j)/10,1).ae(-3.21746028349820063+4.56175438896292539j) | |
| assert lambertw((2+j)/10,-1).ae(-3.03781405002993088-3.53946629633505737j) | |
| assert lambertw((2+j)/10,4).ae(-4.6878509692773249+23.8313630697683291j) | |
| assert lambertw(-(2+j)/10).ae(-0.226933772515757933-0.164986470020154580j) | |
| assert lambertw(-(2+j)/10,1).ae(-2.43569517046110001+0.76974067544756289j) | |
| assert lambertw(-(2+j)/10,-1).ae(-3.54858738151989450-6.91627921869943589j) | |
| assert lambertw(-(2+j)/10,4).ae(-4.5500846928118151+20.6672982215434637j) | |
| mp.dps = 50 | |
| assert lambertw(pi).ae('1.073658194796149172092178407024821347547745350410314531') | |
| mp.dps = 15 | |
| # Former bug in generated branch | |
| assert lambertw(-0.5+0.002j).ae(-0.78917138132659918344 + 0.76743539379990327749j) | |
| assert lambertw(-0.5-0.002j).ae(-0.78917138132659918344 - 0.76743539379990327749j) | |
| assert lambertw(-0.448+0.4j).ae(-0.11855133765652382241 + 0.66570534313583423116j) | |
| assert lambertw(-0.448-0.4j).ae(-0.11855133765652382241 - 0.66570534313583423116j) | |
| assert lambertw(-0.65475+0.0001j).ae(-0.61053421111385310898+1.0396534993944097723803j) | |
| # Huge branch index | |
| w = lambertw(1,10**20) | |
| assert w.real.ae(-47.889578926290259164) | |
| assert w.imag.ae(6.2831853071795864769e+20) | |
| def test_lambertw_hard(): | |
| def check(x,y): | |
| y = convert(y) | |
| type_ok = True | |
| if isinstance(y, mpf): | |
| type_ok = isinstance(x, mpf) | |
| real_ok = abs(x.real-y.real) <= abs(y.real)*8*eps | |
| imag_ok = abs(x.imag-y.imag) <= abs(y.imag)*8*eps | |
| #print x, y, abs(x.real-y.real), abs(x.imag-y.imag) | |
| return real_ok and imag_ok | |
| # Evaluation near 0 | |
| mp.dps = 15 | |
| assert check(lambertw(1e-10), 9.999999999000000000e-11) | |
| assert check(lambertw(-1e-10), -1.000000000100000000e-10) | |
| assert check(lambertw(1e-10j), 9.999999999999999999733e-21 + 9.99999999999999999985e-11j) | |
| assert check(lambertw(-1e-10j), 9.999999999999999999733e-21 - 9.99999999999999999985e-11j) | |
| assert check(lambertw(1e-10,1), -26.303186778379041559 + 3.265093911703828397j) | |
| assert check(lambertw(-1e-10,1), -26.326236166739163892 + 6.526183280686333315j) | |
| assert check(lambertw(1e-10j,1), -26.312931726911421551 + 4.896366881798013421j) | |
| assert check(lambertw(-1e-10j,1), -26.297238779529035066 + 1.632807161345576513j) | |
| assert check(lambertw(1e-10,-1), -26.303186778379041559 - 3.265093911703828397j) | |
| assert check(lambertw(-1e-10,-1), -26.295238819246925694) | |
| assert check(lambertw(1e-10j,-1), -26.297238779529035028 - 1.6328071613455765135j) | |
| assert check(lambertw(-1e-10j,-1), -26.312931726911421551 - 4.896366881798013421j) | |
| # Test evaluation very close to the branch point -1/e | |
| # on the -1, 0, and 1 branches | |
| add = lambda x, y: fadd(x,y,exact=True) | |
| sub = lambda x, y: fsub(x,y,exact=True) | |
| addj = lambda x, y: fadd(x,fmul(y,1j,exact=True),exact=True) | |
| subj = lambda x, y: fadd(x,fmul(y,-1j,exact=True),exact=True) | |
| mp.dps = 1500 | |
| a = -1/e + 10*eps | |
| d3 = mpf('1e-3') | |
| d10 = mpf('1e-10') | |
| d20 = mpf('1e-20') | |
| d40 = mpf('1e-40') | |
| d80 = mpf('1e-80') | |
| d300 = mpf('1e-300') | |
| d1000 = mpf('1e-1000') | |
| mp.dps = 15 | |
| # ---- Branch 0 ---- | |
| # -1/e + eps | |
| assert check(lambertw(add(a,d3)), -0.92802015005456704876) | |
| assert check(lambertw(add(a,d10)), -0.99997668374140088071) | |
| assert check(lambertw(add(a,d20)), -0.99999999976683560186) | |
| assert lambertw(add(a,d40)) == -1 | |
| assert lambertw(add(a,d80)) == -1 | |
| assert lambertw(add(a,d300)) == -1 | |
| assert lambertw(add(a,d1000)) == -1 | |
| # -1/e - eps | |
| assert check(lambertw(sub(a,d3)), -0.99819016149860989001+0.07367191188934638577j) | |
| assert check(lambertw(sub(a,d10)), -0.9999999998187812114595992+0.0000233164398140346109194j) | |
| assert check(lambertw(sub(a,d20)), -0.99999999999999999998187+2.331643981597124203344e-10j) | |
| assert check(lambertw(sub(a,d40)), -1.0+2.33164398159712420336e-20j) | |
| assert check(lambertw(sub(a,d80)), -1.0+2.33164398159712420336e-40j) | |
| assert check(lambertw(sub(a,d300)), -1.0+2.33164398159712420336e-150j) | |
| assert check(lambertw(sub(a,d1000)), mpc(-1,'2.33164398159712420336e-500')) | |
| # -1/e + eps*j | |
| assert check(lambertw(addj(a,d3)), -0.94790387486938526634+0.05036819639190132490j) | |
| assert check(lambertw(addj(a,d10)), -0.9999835127872943680999899+0.0000164870314895821225256j) | |
| assert check(lambertw(addj(a,d20)), -0.999999999835127872929987+1.64872127051890935830e-10j) | |
| assert check(lambertw(addj(a,d40)), -0.9999999999999999999835+1.6487212707001281468305e-20j) | |
| assert check(lambertw(addj(a,d80)), -1.0 + 1.64872127070012814684865e-40j) | |
| assert check(lambertw(addj(a,d300)), -1.0 + 1.64872127070012814684865e-150j) | |
| assert check(lambertw(addj(a,d1000)), mpc(-1.0,'1.64872127070012814684865e-500')) | |
| # -1/e - eps*j | |
| assert check(lambertw(subj(a,d3)), -0.94790387486938526634-0.05036819639190132490j) | |
| assert check(lambertw(subj(a,d10)), -0.9999835127872943680999899-0.0000164870314895821225256j) | |
| assert check(lambertw(subj(a,d20)), -0.999999999835127872929987-1.64872127051890935830e-10j) | |
| assert check(lambertw(subj(a,d40)), -0.9999999999999999999835-1.6487212707001281468305e-20j) | |
| assert check(lambertw(subj(a,d80)), -1.0 - 1.64872127070012814684865e-40j) | |
| assert check(lambertw(subj(a,d300)), -1.0 - 1.64872127070012814684865e-150j) | |
| assert check(lambertw(subj(a,d1000)), mpc(-1.0,'-1.64872127070012814684865e-500')) | |
| # ---- Branch 1 ---- | |
| assert check(lambertw(addj(a,d3),1), -3.088501303219933378005990 + 7.458676867597474813950098j) | |
| assert check(lambertw(addj(a,d80),1), -3.088843015613043855957087 + 7.461489285654254556906117j) | |
| assert check(lambertw(addj(a,d300),1), -3.088843015613043855957087 + 7.461489285654254556906117j) | |
| assert check(lambertw(addj(a,d1000),1), -3.088843015613043855957087 + 7.461489285654254556906117j) | |
| assert check(lambertw(subj(a,d3),1), -1.0520914180450129534365906 + 0.0539925638125450525673175j) | |
| assert check(lambertw(subj(a,d10),1), -1.0000164872127056318529390 + 0.000016487393927159250398333077j) | |
| assert check(lambertw(subj(a,d20),1), -1.0000000001648721270700128 + 1.64872127088134693542628e-10j) | |
| assert check(lambertw(subj(a,d40),1), -1.000000000000000000016487 + 1.64872127070012814686677e-20j) | |
| assert check(lambertw(subj(a,d80),1), -1.0 + 1.64872127070012814684865e-40j) | |
| assert check(lambertw(subj(a,d300),1), -1.0 + 1.64872127070012814684865e-150j) | |
| assert check(lambertw(subj(a,d1000),1), mpc(-1.0, '1.64872127070012814684865e-500')) | |
| # ---- Branch -1 ---- | |
| # -1/e + eps | |
| assert check(lambertw(add(a,d3),-1), -1.075608941186624989414945) | |
| assert check(lambertw(add(a,d10),-1), -1.000023316621036696460620) | |
| assert check(lambertw(add(a,d20),-1), -1.000000000233164398177834) | |
| assert lambertw(add(a,d40),-1) == -1 | |
| assert lambertw(add(a,d80),-1) == -1 | |
| assert lambertw(add(a,d300),-1) == -1 | |
| assert lambertw(add(a,d1000),-1) == -1 | |
| # -1/e - eps | |
| assert check(lambertw(sub(a,d3),-1), -0.99819016149860989001-0.07367191188934638577j) | |
| assert check(lambertw(sub(a,d10),-1), -0.9999999998187812114595992-0.0000233164398140346109194j) | |
| assert check(lambertw(sub(a,d20),-1), -0.99999999999999999998187-2.331643981597124203344e-10j) | |
| assert check(lambertw(sub(a,d40),-1), -1.0-2.33164398159712420336e-20j) | |
| assert check(lambertw(sub(a,d80),-1), -1.0-2.33164398159712420336e-40j) | |
| assert check(lambertw(sub(a,d300),-1), -1.0-2.33164398159712420336e-150j) | |
| assert check(lambertw(sub(a,d1000),-1), mpc(-1,'-2.33164398159712420336e-500')) | |
| # -1/e + eps*j | |
| assert check(lambertw(addj(a,d3),-1), -1.0520914180450129534365906 - 0.0539925638125450525673175j) | |
| assert check(lambertw(addj(a,d10),-1), -1.0000164872127056318529390 - 0.0000164873939271592503983j) | |
| assert check(lambertw(addj(a,d20),-1), -1.0000000001648721270700 - 1.64872127088134693542628e-10j) | |
| assert check(lambertw(addj(a,d40),-1), -1.00000000000000000001648 - 1.6487212707001281468667726e-20j) | |
| assert check(lambertw(addj(a,d80),-1), -1.0 - 1.64872127070012814684865e-40j) | |
| assert check(lambertw(addj(a,d300),-1), -1.0 - 1.64872127070012814684865e-150j) | |
| assert check(lambertw(addj(a,d1000),-1), mpc(-1.0,'-1.64872127070012814684865e-500')) | |
| # -1/e - eps*j | |
| assert check(lambertw(subj(a,d3),-1), -3.088501303219933378005990-7.458676867597474813950098j) | |
| assert check(lambertw(subj(a,d10),-1), -3.088843015579260686911033-7.461489285372968780020716j) | |
| assert check(lambertw(subj(a,d20),-1), -3.088843015613043855953708-7.461489285654254556877988j) | |
| assert check(lambertw(subj(a,d40),-1), -3.088843015613043855957087-7.461489285654254556906117j) | |
| assert check(lambertw(subj(a,d80),-1), -3.088843015613043855957087 - 7.461489285654254556906117j) | |
| assert check(lambertw(subj(a,d300),-1), -3.088843015613043855957087 - 7.461489285654254556906117j) | |
| assert check(lambertw(subj(a,d1000),-1), -3.088843015613043855957087 - 7.461489285654254556906117j) | |
| # One more case, testing higher precision | |
| mp.dps = 500 | |
| x = -1/e + mpf('1e-13') | |
| ans = "-0.99999926266961377166355784455394913638782494543377383"\ | |
| "744978844374498153493943725364881490261187530235150668593869563"\ | |
| "168276697689459394902153960200361935311512317183678882" | |
| mp.dps = 15 | |
| assert lambertw(x).ae(ans) | |
| mp.dps = 50 | |
| assert lambertw(x).ae(ans) | |
| mp.dps = 150 | |
| assert lambertw(x).ae(ans) | |
| def test_meijerg(): | |
| mp.dps = 15 | |
| assert meijerg([[2,3],[1]],[[0.5,2],[3,4]], 2.5).ae(4.2181028074787439386) | |
| assert meijerg([[],[1+j]],[[1],[1]], 3+4j).ae(271.46290321152464592 - 703.03330399954820169j) | |
| assert meijerg([[0.25],[1]],[[0.5],[2]],0) == 0 | |
| assert meijerg([[0],[]],[[0,0,'1/3','2/3'], []], '2/27').ae(2.2019391389653314120) | |
| # Verify 1/z series being used | |
| assert meijerg([[-3],[-0.5]], [[-1],[-2.5]], -0.5).ae(-1.338096165935754898687431) | |
| assert meijerg([[1-(-1)],[1-(-2.5)]], [[1-(-3)],[1-(-0.5)]], -2.0).ae(-1.338096165935754898687431) | |
| assert meijerg([[-3],[-0.5]], [[-1],[-2.5]], -1).ae(-(pi+4)/(4*pi)) | |
| a = 2.5 | |
| b = 1.25 | |
| for z in [mpf(0.25), mpf(2)]: | |
| x1 = hyp1f1(a,b,z) | |
| x2 = gamma(b)/gamma(a)*meijerg([[1-a],[]],[[0],[1-b]],-z) | |
| x3 = gamma(b)/gamma(a)*meijerg([[1-0],[1-(1-b)]],[[1-(1-a)],[]],-1/z) | |
| assert x1.ae(x2) | |
| assert x1.ae(x3) | |
| def test_appellf1(): | |
| mp.dps = 15 | |
| assert appellf1(2,-2,1,1,2,3).ae(-1.75) | |
| assert appellf1(2,1,-2,1,2,3).ae(-8) | |
| assert appellf1(2,1,-2,1,0.5,0.25).ae(1.5) | |
| assert appellf1(-2,1,3,2,3,3).ae(19) | |
| assert appellf1(1,2,3,4,0.5,0.125).ae( 1.53843285792549786518) | |
| def test_coulomb(): | |
| # Note: most tests are doctests | |
| # Test for a bug: | |
| mp.dps = 15 | |
| assert coulombg(mpc(-5,0),2,3).ae(20.087729487721430394) | |
| def test_hyper_param_accuracy(): | |
| mp.dps = 15 | |
| As = [n+1e-10 for n in range(-5,-1)] | |
| Bs = [n+1e-10 for n in range(-12,-5)] | |
| assert hyper(As,Bs,10).ae(-381757055858.652671927) | |
| assert legenp(0.5, 100, 0.25).ae(-2.4124576567211311755e+144) | |
| assert (hyp1f1(1000,1,-100)*10**24).ae(5.2589445437370169113) | |
| assert (hyp2f1(10, -900, 10.5, 0.99)*10**24).ae(1.9185370579660768203) | |
| assert (hyp2f1(1000,1.5,-3.5,-1.5)*10**385).ae(-2.7367529051334000764) | |
| assert hyp2f1(-5, 10, 3, 0.5, zeroprec=500) == 0 | |
| assert (hyp1f1(-10000, 1000, 100)*10**424).ae(-3.1046080515824859974) | |
| assert (hyp2f1(1000,1.5,-3.5,-0.75,maxterms=100000)*10**231).ae(-4.0534790813913998643) | |
| assert legenp(2, 3, 0.25) == 0 | |
| pytest.raises(ValueError, lambda: hypercomb(lambda a: [([],[],[],[],[a],[-a],0.5)], [3])) | |
| assert hypercomb(lambda a: [([],[],[],[],[a],[-a],0.5)], [3], infprec=200) == inf | |
| assert meijerg([[],[]],[[0,0,0,0],[]],0.1).ae(1.5680822343832351418) | |
| assert (besselk(400,400)*10**94).ae(1.4387057277018550583) | |
| mp.dps = 5 | |
| (hyp1f1(-5000.5, 1500, 100)*10**185).ae(8.5185229673381935522) | |
| (hyp1f1(-5000, 1500, 100)*10**185).ae(9.1501213424563944311) | |
| mp.dps = 15 | |
| (hyp1f1(-5000.5, 1500, 100)*10**185).ae(8.5185229673381935522) | |
| (hyp1f1(-5000, 1500, 100)*10**185).ae(9.1501213424563944311) | |
| assert hyp0f1(fadd(-20,'1e-100',exact=True), 0.25).ae(1.85014429040102783e+49) | |
| assert hyp0f1((-20*10**100+1, 10**100), 0.25).ae(1.85014429040102783e+49) | |
| def test_hypercomb_zero_pow(): | |
| # check that 0^0 = 1 | |
| assert hypercomb(lambda a: (([0],[a],[],[],[],[],0),), [0]) == 1 | |
| assert meijerg([[-1.5],[]],[[0],[-0.75]],0).ae(1.4464090846320771425) | |
| def test_spherharm(): | |
| mp.dps = 15 | |
| t = 0.5; r = 0.25 | |
| assert spherharm(0,0,t,r).ae(0.28209479177387814347) | |
| assert spherharm(1,-1,t,r).ae(0.16048941205971996369 - 0.04097967481096344271j) | |
| assert spherharm(1,0,t,r).ae(0.42878904414183579379) | |
| assert spherharm(1,1,t,r).ae(-0.16048941205971996369 - 0.04097967481096344271j) | |
| assert spherharm(2,-2,t,r).ae(0.077915886919031181734 - 0.042565643022253962264j) | |
| assert spherharm(2,-1,t,r).ae(0.31493387233497459884 - 0.08041582001959297689j) | |
| assert spherharm(2,0,t,r).ae(0.41330596756220761898) | |
| assert spherharm(2,1,t,r).ae(-0.31493387233497459884 - 0.08041582001959297689j) | |
| assert spherharm(2,2,t,r).ae(0.077915886919031181734 + 0.042565643022253962264j) | |
| assert spherharm(3,-3,t,r).ae(0.033640236589690881646 - 0.031339125318637082197j) | |
| assert spherharm(3,-2,t,r).ae(0.18091018743101461963 - 0.09883168583167010241j) | |
| assert spherharm(3,-1,t,r).ae(0.42796713930907320351 - 0.10927795157064962317j) | |
| assert spherharm(3,0,t,r).ae(0.27861659336351639787) | |
| assert spherharm(3,1,t,r).ae(-0.42796713930907320351 - 0.10927795157064962317j) | |
| assert spherharm(3,2,t,r).ae(0.18091018743101461963 + 0.09883168583167010241j) | |
| assert spherharm(3,3,t,r).ae(-0.033640236589690881646 - 0.031339125318637082197j) | |
| assert spherharm(0,-1,t,r) == 0 | |
| assert spherharm(0,-2,t,r) == 0 | |
| assert spherharm(0,1,t,r) == 0 | |
| assert spherharm(0,2,t,r) == 0 | |
| assert spherharm(1,2,t,r) == 0 | |
| assert spherharm(1,3,t,r) == 0 | |
| assert spherharm(1,-2,t,r) == 0 | |
| assert spherharm(1,-3,t,r) == 0 | |
| assert spherharm(2,3,t,r) == 0 | |
| assert spherharm(2,4,t,r) == 0 | |
| assert spherharm(2,-3,t,r) == 0 | |
| assert spherharm(2,-4,t,r) == 0 | |
| assert spherharm(3,4.5,0.5,0.25).ae(-22.831053442240790148 + 10.910526059510013757j) | |
| assert spherharm(2+3j, 1-j, 1+j, 3+4j).ae(-2.6582752037810116935 - 1.0909214905642160211j) | |
| assert spherharm(-6,2.5,t,r).ae(0.39383644983851448178 + 0.28414687085358299021j) | |
| assert spherharm(-3.5, 3, 0.5, 0.25).ae(0.014516852987544698924 - 0.015582769591477628495j) | |
| assert spherharm(-3, 3, 0.5, 0.25) == 0 | |
| assert spherharm(-6, 3, 0.5, 0.25).ae(-0.16544349818782275459 - 0.15412657723253924562j) | |
| assert spherharm(-6, 1.5, 0.5, 0.25).ae(0.032208193499767402477 + 0.012678000924063664921j) | |
| assert spherharm(3,0,0,1).ae(0.74635266518023078283) | |
| assert spherharm(3,-2,0,1) == 0 | |
| assert spherharm(3,-2,1,1).ae(-0.16270707338254028971 - 0.35552144137546777097j) | |
| def test_qfunctions(): | |
| mp.dps = 15 | |
| assert qp(2,3,100).ae('2.7291482267247332183e2391') | |
| def test_issue_239(): | |
| mp.prec = 150 | |
| x = ldexp(2476979795053773,-52) | |
| assert betainc(206, 385, 0, 0.55, 1).ae('0.99999999999999999999996570910644857895771110649954') | |
| mp.dps = 15 | |
| pytest.raises(ValueError, lambda: hyp2f1(-5,5,0.5,0.5)) | |
| # Extra stress testing for Bessel functions | |
| # Reference zeros generated with the aid of scipy.special | |
| # jn_zero, jnp_zero, yn_zero, ynp_zero | |
| V = 15 | |
| M = 15 | |
| jn_small_zeros = \ | |
| [[2.4048255576957728, | |
| 5.5200781102863106, | |
| 8.6537279129110122, | |
| 11.791534439014282, | |
| 14.930917708487786, | |
| 18.071063967910923, | |
| 21.211636629879259, | |
| 24.352471530749303, | |
| 27.493479132040255, | |
| 30.634606468431975, | |
| 33.775820213573569, | |
| 36.917098353664044, | |
| 40.058425764628239, | |
| 43.19979171317673, | |
| 46.341188371661814], | |
| [3.8317059702075123, | |
| 7.0155866698156188, | |
| 10.173468135062722, | |
| 13.323691936314223, | |
| 16.470630050877633, | |
| 19.615858510468242, | |
| 22.760084380592772, | |
| 25.903672087618383, | |
| 29.046828534916855, | |
| 32.189679910974404, | |
| 35.332307550083865, | |
| 38.474766234771615, | |
| 41.617094212814451, | |
| 44.759318997652822, | |
| 47.901460887185447], | |
| [5.1356223018406826, | |
| 8.4172441403998649, | |
| 11.619841172149059, | |
| 14.795951782351261, | |
| 17.959819494987826, | |
| 21.116997053021846, | |
| 24.270112313573103, | |
| 27.420573549984557, | |
| 30.569204495516397, | |
| 33.7165195092227, | |
| 36.86285651128381, | |
| 40.008446733478192, | |
| 43.153453778371463, | |
| 46.297996677236919, | |
| 49.442164110416873], | |
| [6.3801618959239835, | |
| 9.7610231299816697, | |
| 13.015200721698434, | |
| 16.223466160318768, | |
| 19.409415226435012, | |
| 22.582729593104442, | |
| 25.748166699294978, | |
| 28.908350780921758, | |
| 32.064852407097709, | |
| 35.218670738610115, | |
| 38.370472434756944, | |
| 41.520719670406776, | |
| 44.669743116617253, | |
| 47.817785691533302, | |
| 50.965029906205183], | |
| [7.5883424345038044, | |
| 11.064709488501185, | |
| 14.37253667161759, | |
| 17.615966049804833, | |
| 20.826932956962388, | |
| 24.01901952477111, | |
| 27.199087765981251, | |
| 30.371007667117247, | |
| 33.537137711819223, | |
| 36.699001128744649, | |
| 39.857627302180889, | |
| 43.01373772335443, | |
| 46.167853512924375, | |
| 49.320360686390272, | |
| 52.471551398458023], | |
| [8.771483815959954, | |
| 12.338604197466944, | |
| 15.700174079711671, | |
| 18.980133875179921, | |
| 22.217799896561268, | |
| 25.430341154222704, | |
| 28.626618307291138, | |
| 31.811716724047763, | |
| 34.988781294559295, | |
| 38.159868561967132, | |
| 41.326383254047406, | |
| 44.489319123219673, | |
| 47.649399806697054, | |
| 50.80716520300633, | |
| 53.963026558378149], | |
| [9.9361095242176849, | |
| 13.589290170541217, | |
| 17.003819667816014, | |
| 20.320789213566506, | |
| 23.58608443558139, | |
| 26.820151983411405, | |
| 30.033722386570469, | |
| 33.233041762847123, | |
| 36.422019668258457, | |
| 39.603239416075404, | |
| 42.778481613199507, | |
| 45.949015998042603, | |
| 49.11577372476426, | |
| 52.279453903601052, | |
| 55.440592068853149], | |
| [11.086370019245084, | |
| 14.821268727013171, | |
| 18.287582832481726, | |
| 21.641541019848401, | |
| 24.934927887673022, | |
| 28.191188459483199, | |
| 31.42279419226558, | |
| 34.637089352069324, | |
| 37.838717382853611, | |
| 41.030773691585537, | |
| 44.21540850526126, | |
| 47.394165755570512, | |
| 50.568184679795566, | |
| 53.738325371963291, | |
| 56.905249991978781], | |
| [12.225092264004655, | |
| 16.037774190887709, | |
| 19.554536430997055, | |
| 22.94517313187462, | |
| 26.266814641176644, | |
| 29.54565967099855, | |
| 32.795800037341462, | |
| 36.025615063869571, | |
| 39.240447995178135, | |
| 42.443887743273558, | |
| 45.638444182199141, | |
| 48.825930381553857, | |
| 52.007691456686903, | |
| 55.184747939289049, | |
| 58.357889025269694], | |
| [13.354300477435331, | |
| 17.241220382489128, | |
| 20.807047789264107, | |
| 24.233885257750552, | |
| 27.583748963573006, | |
| 30.885378967696675, | |
| 34.154377923855096, | |
| 37.400099977156589, | |
| 40.628553718964528, | |
| 43.843801420337347, | |
| 47.048700737654032, | |
| 50.245326955305383, | |
| 53.435227157042058, | |
| 56.619580266508436, | |
| 59.799301630960228], | |
| [14.475500686554541, | |
| 18.433463666966583, | |
| 22.046985364697802, | |
| 25.509450554182826, | |
| 28.887375063530457, | |
| 32.211856199712731, | |
| 35.499909205373851, | |
| 38.761807017881651, | |
| 42.004190236671805, | |
| 45.231574103535045, | |
| 48.447151387269394, | |
| 51.653251668165858, | |
| 54.851619075963349, | |
| 58.043587928232478, | |
| 61.230197977292681], | |
| [15.589847884455485, | |
| 19.61596690396692, | |
| 23.275853726263409, | |
| 26.773322545509539, | |
| 30.17906117878486, | |
| 33.526364075588624, | |
| 36.833571341894905, | |
| 40.111823270954241, | |
| 43.368360947521711, | |
| 46.608132676274944, | |
| 49.834653510396724, | |
| 53.050498959135054, | |
| 56.257604715114484, | |
| 59.457456908388002, | |
| 62.651217388202912], | |
| [16.698249933848246, | |
| 20.789906360078443, | |
| 24.494885043881354, | |
| 28.026709949973129, | |
| 31.45996003531804, | |
| 34.829986990290238, | |
| 38.156377504681354, | |
| 41.451092307939681, | |
| 44.721943543191147, | |
| 47.974293531269048, | |
| 51.211967004101068, | |
| 54.437776928325074, | |
| 57.653844811906946, | |
| 60.8618046824805, | |
| 64.062937824850136], | |
| [17.801435153282442, | |
| 21.95624406783631, | |
| 25.705103053924724, | |
| 29.270630441874802, | |
| 32.731053310978403, | |
| 36.123657666448762, | |
| 39.469206825243883, | |
| 42.780439265447158, | |
| 46.06571091157561, | |
| 49.330780096443524, | |
| 52.579769064383396, | |
| 55.815719876305778, | |
| 59.040934037249271, | |
| 62.257189393731728, | |
| 65.465883797232125], | |
| [18.899997953174024, | |
| 23.115778347252756, | |
| 26.907368976182104, | |
| 30.505950163896036, | |
| 33.993184984781542, | |
| 37.408185128639695, | |
| 40.772827853501868, | |
| 44.100590565798301, | |
| 47.400347780543231, | |
| 50.678236946479898, | |
| 53.93866620912693, | |
| 57.184898598119301, | |
| 60.419409852130297, | |
| 63.644117508962281, | |
| 66.860533012260103]] | |
| jnp_small_zeros = \ | |
| [[0.0, | |
| 3.8317059702075123, | |
| 7.0155866698156188, | |
| 10.173468135062722, | |
| 13.323691936314223, | |
| 16.470630050877633, | |
| 19.615858510468242, | |
| 22.760084380592772, | |
| 25.903672087618383, | |
| 29.046828534916855, | |
| 32.189679910974404, | |
| 35.332307550083865, | |
| 38.474766234771615, | |
| 41.617094212814451, | |
| 44.759318997652822], | |
| [1.8411837813406593, | |
| 5.3314427735250326, | |
| 8.5363163663462858, | |
| 11.706004902592064, | |
| 14.863588633909033, | |
| 18.015527862681804, | |
| 21.16436985918879, | |
| 24.311326857210776, | |
| 27.457050571059246, | |
| 30.601922972669094, | |
| 33.746182898667383, | |
| 36.889987409236811, | |
| 40.033444053350675, | |
| 43.176628965448822, | |
| 46.319597561173912], | |
| [3.0542369282271403, | |
| 6.7061331941584591, | |
| 9.9694678230875958, | |
| 13.170370856016123, | |
| 16.347522318321783, | |
| 19.512912782488205, | |
| 22.671581772477426, | |
| 25.826037141785263, | |
| 28.977672772993679, | |
| 32.127327020443474, | |
| 35.275535050674691, | |
| 38.422654817555906, | |
| 41.568934936074314, | |
| 44.714553532819734, | |
| 47.859641607992093], | |
| [4.2011889412105285, | |
| 8.0152365983759522, | |
| 11.345924310743006, | |
| 14.585848286167028, | |
| 17.78874786606647, | |
| 20.9724769365377, | |
| 24.144897432909265, | |
| 27.310057930204349, | |
| 30.470268806290424, | |
| 33.626949182796679, | |
| 36.781020675464386, | |
| 39.933108623659488, | |
| 43.083652662375079, | |
| 46.232971081836478, | |
| 49.381300092370349], | |
| [5.3175531260839944, | |
| 9.2823962852416123, | |
| 12.681908442638891, | |
| 15.964107037731551, | |
| 19.196028800048905, | |
| 22.401032267689004, | |
| 25.589759681386733, | |
| 28.767836217666503, | |
| 31.938539340972783, | |
| 35.103916677346764, | |
| 38.265316987088158, | |
| 41.423666498500732, | |
| 44.579623137359257, | |
| 47.733667523865744, | |
| 50.886159153182682], | |
| [6.4156163757002403, | |
| 10.519860873772308, | |
| 13.9871886301403, | |
| 17.312842487884625, | |
| 20.575514521386888, | |
| 23.803581476593863, | |
| 27.01030789777772, | |
| 30.20284907898166, | |
| 33.385443901010121, | |
| 36.560777686880356, | |
| 39.730640230067416, | |
| 42.896273163494417, | |
| 46.058566273567043, | |
| 49.218174614666636, | |
| 52.375591529563596], | |
| [7.501266144684147, | |
| 11.734935953042708, | |
| 15.268181461097873, | |
| 18.637443009666202, | |
| 21.931715017802236, | |
| 25.183925599499626, | |
| 28.409776362510085, | |
| 31.617875716105035, | |
| 34.81339298429743, | |
| 37.999640897715301, | |
| 41.178849474321413, | |
| 44.352579199070217, | |
| 47.521956905768113, | |
| 50.687817781723741, | |
| 53.85079463676896], | |
| [8.5778364897140741, | |
| 12.932386237089576, | |
| 16.529365884366944, | |
| 19.941853366527342, | |
| 23.268052926457571, | |
| 26.545032061823576, | |
| 29.790748583196614, | |
| 33.015178641375142, | |
| 36.224380548787162, | |
| 39.422274578939259, | |
| 42.611522172286684, | |
| 45.793999658055002, | |
| 48.971070951900596, | |
| 52.143752969301988, | |
| 55.312820330403446], | |
| [9.6474216519972168, | |
| 14.115518907894618, | |
| 17.774012366915256, | |
| 21.229062622853124, | |
| 24.587197486317681, | |
| 27.889269427955092, | |
| 31.155326556188325, | |
| 34.39662855427218, | |
| 37.620078044197086, | |
| 40.830178681822041, | |
| 44.030010337966153, | |
| 47.221758471887113, | |
| 50.407020967034367, | |
| 53.586995435398319, | |
| 56.762598475105272], | |
| [10.711433970699945, | |
| 15.28673766733295, | |
| 19.004593537946053, | |
| 22.501398726777283, | |
| 25.891277276839136, | |
| 29.218563499936081, | |
| 32.505247352375523, | |
| 35.763792928808799, | |
| 39.001902811514218, | |
| 42.224638430753279, | |
| 45.435483097475542, | |
| 48.636922645305525, | |
| 51.830783925834728, | |
| 55.01844255063594, | |
| 58.200955824859509], | |
| [11.770876674955582, | |
| 16.447852748486498, | |
| 20.223031412681701, | |
| 23.760715860327448, | |
| 27.182021527190532, | |
| 30.534504754007074, | |
| 33.841965775135715, | |
| 37.118000423665604, | |
| 40.371068905333891, | |
| 43.606764901379516, | |
| 46.828959446564562, | |
| 50.040428970943456, | |
| 53.243223214220535, | |
| 56.438892058982552, | |
| 59.628631306921512], | |
| [12.826491228033465, | |
| 17.600266557468326, | |
| 21.430854238060294, | |
| 25.008518704644261, | |
| 28.460857279654847, | |
| 31.838424458616998, | |
| 35.166714427392629, | |
| 38.460388720328256, | |
| 41.728625562624312, | |
| 44.977526250903469, | |
| 48.211333836373288, | |
| 51.433105171422278, | |
| 54.645106240447105, | |
| 57.849056857839799, | |
| 61.046288512821078], | |
| [13.878843069697276, | |
| 18.745090916814406, | |
| 22.629300302835503, | |
| 26.246047773946584, | |
| 29.72897816891134, | |
| 33.131449953571661, | |
| 36.480548302231658, | |
| 39.791940718940855, | |
| 43.075486800191012, | |
| 46.337772104541405, | |
| 49.583396417633095, | |
| 52.815686826850452, | |
| 56.037118687012179, | |
| 59.249577075517968, | |
| 62.454525995970462], | |
| [14.928374492964716, | |
| 19.88322436109951, | |
| 23.81938909003628, | |
| 27.474339750968247, | |
| 30.987394331665278, | |
| 34.414545662167183, | |
| 37.784378506209499, | |
| 41.113512376883377, | |
| 44.412454519229281, | |
| 47.688252845993366, | |
| 50.945849245830813, | |
| 54.188831071035124, | |
| 57.419876154678179, | |
| 60.641030026538746, | |
| 63.853885828967512], | |
| [15.975438807484321, | |
| 21.015404934568315, | |
| 25.001971500138194, | |
| 28.694271223110755, | |
| 32.236969407878118, | |
| 35.688544091185301, | |
| 39.078998185245057, | |
| 42.425854432866141, | |
| 45.740236776624833, | |
| 49.029635055514276, | |
| 52.299319390331728, | |
| 55.553127779547459, | |
| 58.793933759028134, | |
| 62.02393848337554, | |
| 65.244860767043859]] | |
| yn_small_zeros = \ | |
| [[0.89357696627916752, | |
| 3.9576784193148579, | |
| 7.0860510603017727, | |
| 10.222345043496417, | |
| 13.361097473872763, | |
| 16.500922441528091, | |
| 19.64130970088794, | |
| 22.782028047291559, | |
| 25.922957653180923, | |
| 29.064030252728398, | |
| 32.205204116493281, | |
| 35.346452305214321, | |
| 38.487756653081537, | |
| 41.629104466213808, | |
| 44.770486607221993], | |
| [2.197141326031017, | |
| 5.4296810407941351, | |
| 8.5960058683311689, | |
| 11.749154830839881, | |
| 14.897442128336725, | |
| 18.043402276727856, | |
| 21.188068934142213, | |
| 24.331942571356912, | |
| 27.475294980449224, | |
| 30.618286491641115, | |
| 33.761017796109326, | |
| 36.90355531614295, | |
| 40.045944640266876, | |
| 43.188218097393211, | |
| 46.330399250701687], | |
| [3.3842417671495935, | |
| 6.7938075132682675, | |
| 10.023477979360038, | |
| 13.209986710206416, | |
| 16.378966558947457, | |
| 19.539039990286384, | |
| 22.69395593890929, | |
| 25.845613720902269, | |
| 28.995080395650151, | |
| 32.143002257627551, | |
| 35.289793869635804, | |
| 38.435733485446343, | |
| 41.581014867297885, | |
| 44.725777117640461, | |
| 47.870122696676504], | |
| [4.5270246611496439, | |
| 8.0975537628604907, | |
| 11.396466739595867, | |
| 14.623077742393873, | |
| 17.81845523294552, | |
| 20.997284754187761, | |
| 24.166235758581828, | |
| 27.328799850405162, | |
| 30.486989604098659, | |
| 33.642049384702463, | |
| 36.794791029185579, | |
| 39.945767226378749, | |
| 43.095367507846703, | |
| 46.2438744334407, | |
| 49.391498015725107], | |
| [5.6451478942208959, | |
| 9.3616206152445429, | |
| 12.730144474090465, | |
| 15.999627085382479, | |
| 19.22442895931681, | |
| 22.424810599698521, | |
| 25.610267054939328, | |
| 28.785893657666548, | |
| 31.954686680031668, | |
| 35.118529525584828, | |
| 38.278668089521758, | |
| 41.435960629910073, | |
| 44.591018225353424, | |
| 47.744288086361052, | |
| 50.896105199722123], | |
| [6.7471838248710219, | |
| 10.597176726782031, | |
| 14.033804104911233, | |
| 17.347086393228382, | |
| 20.602899017175335, | |
| 23.826536030287532, | |
| 27.030134937138834, | |
| 30.220335654231385, | |
| 33.401105611047908, | |
| 36.574972486670962, | |
| 39.743627733020277, | |
| 42.908248189569535, | |
| 46.069679073215439, | |
| 49.228543693445843, | |
| 52.385312123112282], | |
| [7.8377378223268716, | |
| 11.811037107609447, | |
| 15.313615118517857, | |
| 18.670704965906724, | |
| 21.958290897126571, | |
| 25.206207715021249, | |
| 28.429037095235496, | |
| 31.634879502950644, | |
| 34.828638524084437, | |
| 38.013473399691765, | |
| 41.19151880917741, | |
| 44.364272633271975, | |
| 47.53281875312084, | |
| 50.697961822183806, | |
| 53.860312300118388], | |
| [8.919605734873789, | |
| 13.007711435388313, | |
| 16.573915129085334, | |
| 19.974342312352426, | |
| 23.293972585596648, | |
| 26.5667563757203, | |
| 29.809531451608321, | |
| 33.031769327150685, | |
| 36.239265816598239, | |
| 39.435790312675323, | |
| 42.623910919472727, | |
| 45.805442883111651, | |
| 48.981708325514764, | |
| 52.153694518185572, | |
| 55.322154420959698], | |
| [9.9946283820824834, | |
| 14.190361295800141, | |
| 17.817887841179873, | |
| 21.26093227125945, | |
| 24.612576377421522, | |
| 27.910524883974868, | |
| 31.173701563441602, | |
| 34.412862242025045, | |
| 37.634648706110989, | |
| 40.843415321050884, | |
| 44.04214994542435, | |
| 47.232978012841169, | |
| 50.417456447370186, | |
| 53.596753874948731, | |
| 56.771765754432457], | |
| [11.064090256031013, | |
| 15.361301343575925, | |
| 19.047949646361388, | |
| 22.532765416313869, | |
| 25.91620496332662, | |
| 29.2394205079349, | |
| 32.523270869465881, | |
| 35.779715464475261, | |
| 39.016196664616095, | |
| 42.237627509803703, | |
| 45.4474001519274, | |
| 48.647941127433196, | |
| 51.841036928216499, | |
| 55.028034667184916, | |
| 58.209970905250097], | |
| [12.128927704415439, | |
| 16.522284394784426, | |
| 20.265984501212254, | |
| 23.791669719454272, | |
| 27.206568881574774, | |
| 30.555020011020762, | |
| 33.859683872746356, | |
| 37.133649760307504, | |
| 40.385117593813002, | |
| 43.619533085646856, | |
| 46.840676630553575, | |
| 50.051265851897857, | |
| 53.253310556711732, | |
| 56.448332488918971, | |
| 59.637507005589829], | |
| [13.189846995683845, | |
| 17.674674253171487, | |
| 21.473493977824902, | |
| 25.03913093040942, | |
| 28.485081336558058, | |
| 31.858644293774859, | |
| 35.184165245422787, | |
| 38.475796636190897, | |
| 41.742455848758449, | |
| 44.990096293791186, | |
| 48.222870660068338, | |
| 51.443777308699826, | |
| 54.655042589416311, | |
| 57.858358441436511, | |
| 61.055036135780528], | |
| [14.247395665073945, | |
| 18.819555894710682, | |
| 22.671697117872794, | |
| 26.276375544903892, | |
| 29.752925495549038, | |
| 33.151412708998983, | |
| 36.497763772987645, | |
| 39.807134090704376, | |
| 43.089121522203808, | |
| 46.350163579538652, | |
| 49.594769786270069, | |
| 52.82620892320143, | |
| 56.046916910756961, | |
| 59.258751140598783, | |
| 62.463155567737854], | |
| [15.30200785858925, | |
| 19.957808654258601, | |
| 23.861599172945054, | |
| 27.504429642227545, | |
| 31.011103429019229, | |
| 34.434283425782942, | |
| 37.801385632318459, | |
| 41.128514139788358, | |
| 44.425913324440663, | |
| 47.700482714581842, | |
| 50.957073905278458, | |
| 54.199216028087261, | |
| 57.429547607017405, | |
| 60.65008661807661, | |
| 63.862406280068586], | |
| [16.354034360047551, | |
| 21.090156519983806, | |
| 25.044040298785627, | |
| 28.724161640881914, | |
| 32.260472459522644, | |
| 35.708083982611664, | |
| 39.095820003878235, | |
| 42.440684315990936, | |
| 45.75353669045622, | |
| 49.041718113283529, | |
| 52.310408280968073, | |
| 55.56338698149062, | |
| 58.803488508906895, | |
| 62.032886550960831, | |
| 65.253280088312461]] | |
| ynp_small_zeros = \ | |
| [[2.197141326031017, | |
| 5.4296810407941351, | |
| 8.5960058683311689, | |
| 11.749154830839881, | |
| 14.897442128336725, | |
| 18.043402276727856, | |
| 21.188068934142213, | |
| 24.331942571356912, | |
| 27.475294980449224, | |
| 30.618286491641115, | |
| 33.761017796109326, | |
| 36.90355531614295, | |
| 40.045944640266876, | |
| 43.188218097393211, | |
| 46.330399250701687], | |
| [3.6830228565851777, | |
| 6.9414999536541757, | |
| 10.123404655436613, | |
| 13.285758156782854, | |
| 16.440058007293282, | |
| 19.590241756629495, | |
| 22.738034717396327, | |
| 25.884314618788867, | |
| 29.029575819372535, | |
| 32.174118233366201, | |
| 35.318134458192094, | |
| 38.461753870997549, | |
| 41.605066618873108, | |
| 44.74813744908079, | |
| 47.891014070791065], | |
| [5.0025829314460639, | |
| 8.3507247014130795, | |
| 11.574195465217647, | |
| 14.760909306207676, | |
| 17.931285939466855, | |
| 21.092894504412739, | |
| 24.249231678519058, | |
| 27.402145837145258, | |
| 30.552708880564553, | |
| 33.70158627151572, | |
| 36.849213419846257, | |
| 39.995887376143356, | |
| 43.141817835750686, | |
| 46.287157097544201, | |
| 49.432018469138281], | |
| [6.2536332084598136, | |
| 9.6987879841487711, | |
| 12.972409052292216, | |
| 16.19044719506921, | |
| 19.38238844973613, | |
| 22.559791857764261, | |
| 25.728213194724094, | |
| 28.890678419054777, | |
| 32.048984005266337, | |
| 35.204266606440635, | |
| 38.357281675961019, | |
| 41.508551443818436, | |
| 44.658448731963676, | |
| 47.807246956681162, | |
| 50.95515126455207], | |
| [7.4649217367571329, | |
| 11.005169149809189, | |
| 14.3317235192331, | |
| 17.58443601710272, | |
| 20.801062338411128, | |
| 23.997004122902644, | |
| 27.179886689853435, | |
| 30.353960608554323, | |
| 33.521797098666792, | |
| 36.685048382072301, | |
| 39.844826969405863, | |
| 43.001910515625288, | |
| 46.15685955107263, | |
| 49.310088614282257, | |
| 52.461911043685864], | |
| [8.6495562436971983, | |
| 12.280868725807848, | |
| 15.660799304540377, | |
| 18.949739756016503, | |
| 22.192841809428241, | |
| 25.409072788867674, | |
| 28.608039283077593, | |
| 31.795195353138159, | |
| 34.973890634255288, | |
| 38.14630522169358, | |
| 41.313923188794905, | |
| 44.477791768537617, | |
| 47.638672065035628, | |
| 50.797131066967842, | |
| 53.953600129601663], | |
| [9.8147970120105779, | |
| 13.532811875789828, | |
| 16.965526446046053, | |
| 20.291285512443867, | |
| 23.56186260680065, | |
| 26.799499736027237, | |
| 30.015665481543419, | |
| 33.216968050039509, | |
| 36.407516858984748, | |
| 39.590015243560459, | |
| 42.766320595957378, | |
| 45.937754257017323, | |
| 49.105283450953203, | |
| 52.269633324547373, | |
| 55.431358715604255], | |
| [10.965152105242974, | |
| 14.765687379508912, | |
| 18.250123150217555, | |
| 21.612750053384621, | |
| 24.911310600813573, | |
| 28.171051927637585, | |
| 31.40518108895689, | |
| 34.621401012564177, | |
| 37.824552065973114, | |
| 41.017847386464902, | |
| 44.203512240871601, | |
| 47.3831408366063, | |
| 50.557907466622796, | |
| 53.728697478957026, | |
| 56.896191727313342], | |
| [12.103641941939539, | |
| 15.982840905145284, | |
| 19.517731005559611, | |
| 22.916962141504605, | |
| 26.243700855690533, | |
| 29.525960140695407, | |
| 32.778568197561124, | |
| 36.010261572392516, | |
| 39.226578757802172, | |
| 42.43122493258747, | |
| 45.626783824134354, | |
| 48.815117837929515, | |
| 51.997606404328863, | |
| 55.175294723956816, | |
| 58.348990221754937], | |
| [13.232403808592215, | |
| 17.186756572616758, | |
| 20.770762917490496, | |
| 24.206152448722253, | |
| 27.561059462697153, | |
| 30.866053571250639, | |
| 34.137476603379774, | |
| 37.385039772270268, | |
| 40.614946085165892, | |
| 43.831373184731238, | |
| 47.037251786726299, | |
| 50.234705848765229, | |
| 53.425316228549359, | |
| 56.610286079882087, | |
| 59.790548623216652], | |
| [14.35301374369987, | |
| 18.379337301642568, | |
| 22.011118775283494, | |
| 25.482116178696707, | |
| 28.865046588695164, | |
| 32.192853922166294, | |
| 35.483296655830277, | |
| 38.747005493021857, | |
| 41.990815194320955, | |
| 45.219355876831731, | |
| 48.435892856078888, | |
| 51.642803925173029, | |
| 54.84186659475857, | |
| 58.034439083840155, | |
| 61.221578745109862], | |
| [15.466672066554263, | |
| 19.562077985759503, | |
| 23.240325531101082, | |
| 26.746322986645901, | |
| 30.157042415639891, | |
| 33.507642948240263, | |
| 36.817212798512775, | |
| 40.097251300178642, | |
| 43.355193847719752, | |
| 46.596103410173672, | |
| 49.823567279972794, | |
| 53.040208868780832, | |
| 56.247996968470062, | |
| 59.448441365714251, | |
| 62.642721301357187], | |
| [16.574317035530872, | |
| 20.73617763753932, | |
| 24.459631728238804, | |
| 27.999993668839644, | |
| 31.438208790267783, | |
| 34.811512070805535, | |
| 38.140243708611251, | |
| 41.436725143893739, | |
| 44.708963264433333, | |
| 47.962435051891027, | |
| 51.201037321915983, | |
| 54.427630745992975, | |
| 57.644369734615238, | |
| 60.852911791989989, | |
| 64.054555435720397], | |
| [17.676697936439624, | |
| 21.9026148697762, | |
| 25.670073356263225, | |
| 29.244155124266438, | |
| 32.709534477396028, | |
| 36.105399554497548, | |
| 39.453272918267025, | |
| 42.766255701958017, | |
| 46.052899215578358, | |
| 49.319076602061401, | |
| 52.568982147952547, | |
| 55.805705507386287, | |
| 59.031580956740466, | |
| 62.248409689597653, | |
| 65.457606670836759], | |
| [18.774423978290318, | |
| 23.06220035979272, | |
| 26.872520985976736, | |
| 30.479680663499762, | |
| 33.971869047372436, | |
| 37.390118854896324, | |
| 40.757072537673599, | |
| 44.086572292170345, | |
| 47.387688809191869, | |
| 50.66667461073936, | |
| 53.928009929563275, | |
| 57.175005343085052, | |
| 60.410169281219877, | |
| 63.635442539153021, | |
| 66.85235358587768]] | |
| def test_bessel_zeros_extra(): | |
| mp.dps = 15 | |
| for v in range(V): | |
| for m in range(1,M+1): | |
| print(v, m, "of", V, M) | |
| # Twice to test cache (if used) | |
| assert besseljzero(v,m).ae(jn_small_zeros[v][m-1]) | |
| assert besseljzero(v,m).ae(jn_small_zeros[v][m-1]) | |
| assert besseljzero(v,m,1).ae(jnp_small_zeros[v][m-1]) | |
| assert besseljzero(v,m,1).ae(jnp_small_zeros[v][m-1]) | |
| assert besselyzero(v,m).ae(yn_small_zeros[v][m-1]) | |
| assert besselyzero(v,m).ae(yn_small_zeros[v][m-1]) | |
| assert besselyzero(v,m,1).ae(ynp_small_zeros[v][m-1]) | |
| assert besselyzero(v,m,1).ae(ynp_small_zeros[v][m-1]) | |