#this program corresponds to special.py ### Means test is not done yet #E Means test is giving error (E) #F Means test is failing (F) #EF Means test is giving error and Failing #! Means test is segfaulting #8 Means test runs forever ### test_besselpoly ### test_jnjnp_zeros ### test_mathieu_a ### test_mathieu_even_coef ### test_mathieu_odd_coef ### test_modfresnelp ### test_modfresnelm # test_pbdv_seq ### test_pbvv_seq ### test_sph_harm # test_sph_in # test_sph_jn # test_sph_kn import numpy as np from numpy import array from numpy.testing import * from scipy.special import * import scipy.special._cephes as cephes import numpy as np def assert_tol_equal(a, b, rtol=1e-7, atol=0, err_msg='', verbose=True): """Assert that `a` and `b` are equal to tolerance ``atol + rtol*abs(b)``""" def compare(x, y): return allclose(x, y, rtol=rtol, atol=atol) a, b = asanyarray(a), asanyarray(b) header = 'Not equal to tolerance rtol=%g, atol=%g' % (rtol, atol) np.testing.utils.assert_array_compare(compare, a, b, err_msg=str(err_msg), verbose=verbose, header=header) class TestCephes(TestCase): def test_airy(self): cephes.airy(0) def test_airye(self): cephes.airye(0) def test_bdtr(self): assert_equal(cephes.bdtr(1,1,0.5),1.0) def test_bdtri(self): assert_equal(cephes.bdtri(1,3,0.5),0.5) def test_bdtrc(self): assert_equal(cephes.bdtrc(1,3,0.5),0.5) def test_bdtrin(self): assert_equal(cephes.bdtrin(1,0,1),5.0) def test_bdtrik(self): cephes.bdtrik(1,3,0.5) def test_bei(self): assert_equal(cephes.bei(0),0.0) def test_beip(self): assert_equal(cephes.beip(0),0.0) def test_ber(self): assert_equal(cephes.ber(0),1.0) def test_berp(self): assert_equal(cephes.berp(0),0.0) def test_besselpoly(self): assert_equal(cephes.besselpoly(0,0,0),1.0) def test_beta(self): assert_equal(cephes.beta(1,1),1.0) def test_betainc(self): assert_equal(cephes.betainc(1,1,1),1.0) def test_betaln(self): assert_equal(cephes.betaln(1,1),0.0) def test_betaincinv(self): assert_equal(cephes.betaincinv(1,1,1),1.0) def test_btdtr(self): assert_equal(cephes.btdtr(1,1,1),1.0) def test_btdtri(self): assert_equal(cephes.btdtri(1,1,1),1.0) def test_btdtria(self): assert_equal(cephes.btdtria(1,1,1),5.0) def test_btdtrib(self): assert_equal(cephes.btdtrib(1,1,1),5.0) def test_cbrt(self): assert_approx_equal(cephes.cbrt(1),1.0) def test_chdtr(self): assert_equal(cephes.chdtr(1,0),0.0) def test_chdtrc(self): assert_equal(cephes.chdtrc(1,0),1.0) def test_chdtri(self): assert_equal(cephes.chdtri(1,1),0.0) def test_chdtriv(self): assert_equal(cephes.chdtriv(0,0),5.0) def test_chndtr(self): assert_equal(cephes.chndtr(0,1,0),0.0) def test_chndtridf(self): assert_equal(cephes.chndtridf(0,0,1),5.0) def test_chndtrinc(self): assert_equal(cephes.chndtrinc(0,1,0),5.0) def test_chndtrix(self): assert_equal(cephes.chndtrix(0,1,0),0.0) def test_cosdg(self): assert_equal(cephes.cosdg(0),1.0) def test_cosm1(self): assert_equal(cephes.cosm1(0),0.0) def test_cotdg(self): assert_almost_equal(cephes.cotdg(45),1.0) def test_dawsn(self): assert_equal(cephes.dawsn(0),0.0) def test_ellipe(self): assert_equal(cephes.ellipe(1),1.0) def test_ellipeinc(self): assert_equal(cephes.ellipeinc(0,1),0.0) def test_ellipj(self): cephes.ellipj(0,1) def test_ellipk(self): cephes.ellipk(0)#==pi/2 def test_ellipkinc(self): assert_equal(cephes.ellipkinc(0,0),0.0) def test_erf(self): assert_equal(cephes.erf(0),0.0) def test_erfc(self): assert_equal(cephes.erfc(0),1.0) def test_exp1(self): cephes.exp1(1) def test_expi(self): cephes.expi(1) def test_expn(self): cephes.expn(1,1) def test_exp1_reg(self): # Regression for #834 a = cephes.exp1(-complex(19.9999990)) b = cephes.exp1(-complex(19.9999991)) assert_array_almost_equal(a.imag, b.imag) def test_exp10(self): assert_approx_equal(cephes.exp10(2),100.0) def test_exp2(self): assert_equal(cephes.exp2(2),4.0) def test_expm1(self): assert_equal(cephes.expm1(0),0.0) def test_fdtr(self): assert_equal(cephes.fdtr(1,1,0),0.0) def test_fdtrc(self): assert_equal(cephes.fdtrc(1,1,0),1.0) def test_fdtri(self): cephes.fdtri(1,1,0.5) def test_fdtridfd(self): assert_equal(cephes.fdtridfd(1,0,0),5.0) def test_fresnel(self): assert_equal(cephes.fresnel(0),(0.0,0.0)) def test_gamma(self): assert_equal(cephes.gamma(5),24.0) def test_gammainc(self): assert_equal(cephes.gammainc(5,0),0.0) def test_gammaincc(self): assert_equal(cephes.gammaincc(5,0),1.0) def test_gammainccinv(self): assert_equal(cephes.gammainccinv(5,1),0.0) def test_gammaln(self): cephes.gammaln(10) def test_gdtr(self): assert_equal(cephes.gdtr(1,1,0),0.0) def test_gdtrc(self): assert_equal(cephes.gdtrc(1,1,0),1.0) def test_gdtria(self): assert_equal(cephes.gdtria(0,1,1),0.0) def test_gdtrib(self): cephes.gdtrib(1,0,1) #assert_equal(cephes.gdtrib(1,0,1),5.0) def test_gdtrix(self): cephes.gdtrix(1,1,.1) def test_hankel1(self): cephes.hankel1(1,1) def test_hankel1e(self): cephes.hankel1e(1,1) def test_hankel2(self): cephes.hankel2(1,1) def test_hankel2e(self): cephes.hankel2e(1,1) def test_hyp1f1(self): assert_approx_equal(cephes.hyp1f1(1,1,1), exp(1.0)) assert_approx_equal(cephes.hyp1f1(3,4,-6), 0.026056422099537251095) cephes.hyp1f1(1,1,1) def test_hyp1f2(self): cephes.hyp1f2(1,1,1,1) def test_hyp2f0(self): cephes.hyp2f0(1,1,1,1) def test_hyp2f1(self): assert_equal(cephes.hyp2f1(1,1,1,0),1.0) def test_hyp3f0(self): assert_equal(cephes.hyp3f0(1,1,1,0),(1.0,0.0)) def test_hyperu(self): assert_equal(cephes.hyperu(0,1,1),1.0) def test_i0(self): assert_equal(cephes.i0(0),1.0) def test_i0e(self): assert_equal(cephes.i0e(0),1.0) def test_i1(self): assert_equal(cephes.i1(0),0.0) def test_i1e(self): assert_equal(cephes.i1e(0),0.0) def test_it2i0k0(self): cephes.it2i0k0(1) def test_it2j0y0(self): cephes.it2j0y0(1) def test_it2struve0(self): cephes.it2struve0(1) def test_itairy(self): cephes.itairy(1) def test_iti0k0(self): assert_equal(cephes.iti0k0(0),(0.0,0.0)) def test_itj0y0(self): assert_equal(cephes.itj0y0(0),(0.0,0.0)) def test_itmodstruve0(self): assert_equal(cephes.itmodstruve0(0),0.0) def test_itstruve0(self): assert_equal(cephes.itstruve0(0),0.0) def test_iv(self): assert_equal(cephes.iv(1,0),0.0) def _check_ive(self): assert_equal(cephes.ive(1,0),0.0) def test_j0(self): assert_equal(cephes.j0(0),1.0) def test_j1(self): assert_equal(cephes.j1(0),0.0) def test_jn(self): assert_equal(cephes.jn(0,0),1.0) def test_jv(self): assert_equal(cephes.jv(0,0),1.0) def _check_jve(self): assert_equal(cephes.jve(0,0),1.0) def test_k0(self): cephes.k0(2) def test_k0e(self): cephes.k0e(2) def test_k1(self): cephes.k1(2) def test_k1e(self): cephes.k1e(2) def test_kei(self): cephes.kei(2) def test_keip(self): assert_equal(cephes.keip(0),0.0) def test_ker(self): cephes.ker(2) def test_kerp(self): cephes.kerp(2) def _check_kelvin(self): cephes.kelvin(2) def test_kn(self): cephes.kn(1,1) def test_kolmogi(self): assert_equal(cephes.kolmogi(1),0.0) def test_kolmogorov(self): assert_equal(cephes.kolmogorov(0),1.0) def _check_kv(self): cephes.kv(1,1) def _check_kve(self): cephes.kve(1,1) def test_log1p(self): assert_equal(cephes.log1p(0),0.0) def test_lpmv(self): assert_equal(cephes.lpmv(0,0,1),1.0) def test_mathieu_a(self): assert_equal(cephes.mathieu_a(1,0),1.0) def test_mathieu_b(self): assert_equal(cephes.mathieu_b(1,0),1.0) def test_mathieu_cem(self): assert_equal(cephes.mathieu_cem(1,0,0),(1.0,0.0)) def test_mathieu_modcem1(self): assert_equal(cephes.mathieu_modcem1(1,0,0),(0.0,0.0)) def test_mathieu_modcem2(self): cephes.mathieu_modcem2(1,1,1) def test_mathieu_sem(self): assert_equal(cephes.mathieu_sem(1,0,0),(0.0,1.0)) def test_mathieu_modsem1(self): assert_equal(cephes.mathieu_modsem1(1,0,0),(0.0,0.0)) def test_mathieu_modsem2(self): cephes.mathieu_modsem2(1,1,1) def test_modfresnelm(self): cephes.modfresnelm(0) def test_modfresnelp(self): cephes.modfresnelp(0) def _check_modstruve(self): assert_equal(cephes.modstruve(1,0),0.0) def test_nbdtr(self): assert_equal(cephes.nbdtr(1,1,1),1.0) def test_nbdtrc(self): assert_equal(cephes.nbdtrc(1,1,1),0.0) def test_nbdtri(self): assert_equal(cephes.nbdtri(1,1,1),1.0) def __check_nbdtrik(self): cephes.nbdtrik(1,.4,.5) def test_nbdtrin(self): assert_equal(cephes.nbdtrin(1,0,0),5.0) def test_ncfdtr(self): assert_equal(cephes.ncfdtr(1,1,1,0),0.0) def test_ncfdtri(self): assert_equal(cephes.ncfdtri(1,1,1,0),0.0) def test_ncfdtridfd(self): cephes.ncfdtridfd(1,0.5,0,1) def __check_ncfdtridfn(self): cephes.ncfdtridfn(1,0.5,0,1) def __check_ncfdtrinc(self): cephes.ncfdtrinc(1,0.5,0,1) def test_nctdtr(self): assert_equal(cephes.nctdtr(1,0,0),0.5) def __check_nctdtridf(self): cephes.nctdtridf(1,0.5,0) def test_nctdtrinc(self): cephes.nctdtrinc(1,0,0) def test_nctdtrit(self): cephes.nctdtrit(.1,0.2,.5) def test_ndtr(self): assert_equal(cephes.ndtr(0),0.5) def test_ndtri(self): assert_equal(cephes.ndtri(0.5),0.0) def test_nrdtrimn(self): assert_approx_equal(cephes.nrdtrimn(0.5,1,1),1.0) def test_nrdtrisd(self): # abs() because nrdtrisd(0.5,0.5,0.5) returns -0.0, should be +0.0 assert_equal(np.abs(cephes.nrdtrisd(0.5,0.5,0.5)), 0.0) def test_obl_ang1(self): cephes.obl_ang1(1,1,1,0) def test_obl_ang1_cv(self): result = cephes.obl_ang1_cv(1,1,1,1,0) assert_almost_equal(result[0],1.0) assert_almost_equal(result[1],0.0) def _check_obl_cv(self): assert_equal(cephes.obl_cv(1,1,0),2.0) def test_obl_rad1(self): cephes.obl_rad1(1,1,1,0) def test_obl_rad1_cv(self): cephes.obl_rad1_cv(1,1,1,1,0) def test_obl_rad2(self): cephes.obl_rad2(1,1,1,0) def test_obl_rad2_cv(self): cephes.obl_rad2_cv(1,1,1,1,0) def test_pbdv(self): assert_equal(cephes.pbdv(1,0),(0.0,1.0)) def test_pbvv(self): cephes.pbvv(1,0) def test_pbwa(self): cephes.pbwa(1,0) def test_pdtr(self): cephes.pdtr(0,1) def test_pdtrc(self): cephes.pdtrc(0,1) def test_pdtri(self): cephes.pdtri(0.5,0.5) def test_pdtrik(self): cephes.pdtrik(0.5,1) def test_pro_ang1(self): cephes.pro_ang1(1,1,1,0) def test_pro_ang1_cv(self): assert_array_almost_equal(cephes.pro_ang1_cv(1,1,1,1,0), array((1.0,0.0))) def _check_pro_cv(self): assert_equal(cephes.pro_cv(1,1,0),2.0) def test_pro_rad1(self): cephes.pro_rad1(1,1,1,0.1) def test_pro_rad1_cv(self): cephes.pro_rad1_cv(1,1,1,1,0) def test_pro_rad2(self): cephes.pro_rad2(1,1,1,0) def test_pro_rad2_cv(self): cephes.pro_rad2_cv(1,1,1,1,0) def test_psi(self): cephes.psi(1) def test_radian(self): assert_equal(cephes.radian(0,0,0),0) def test_rgamma(self): assert_equal(cephes.rgamma(1),1.0) def test_round(self): assert_equal(cephes.round(3.4),3.0) assert_equal(cephes.round(-3.4),-3.0) assert_equal(cephes.round(3.6),4.0) assert_equal(cephes.round(-3.6),-4.0) assert_equal(cephes.round(3.5),4.0) assert_equal(cephes.round(-3.5),-4.0) def test_shichi(self): cephes.shichi(1) def test_sici(self): cephes.sici(1) def test_sindg(self): assert_equal(cephes.sindg(90),1.0) def test_smirnov(self): assert_equal(cephes.smirnov(1,.1),0.9) def test_smirnovi(self): assert_almost_equal(cephes.smirnov(1,cephes.smirnovi(1,0.4)),0.4) assert_almost_equal(cephes.smirnov(1,cephes.smirnovi(1,0.6)),0.6) def test_spence(self): assert_equal(cephes.spence(1),0.0) def test_stdtr(self): assert_equal(cephes.stdtr(1,0),0.5) def test_stdtridf(self): cephes.stdtridf(0.7,1) def test_stdtrit(self): cephes.stdtrit(1,0.7) def test_struve(self): assert_equal(cephes.struve(0,0),0.0) def test_tandg(self): assert_equal(cephes.tandg(45),1.0) def test_tklmbda(self): assert_almost_equal(cephes.tklmbda(1,1),1.0) def test_y0(self): cephes.y0(1) def test_y1(self): cephes.y1(1) def test_yn(self): cephes.yn(1,1) def test_yv(self): cephes.yv(1,1) def _check_yve(self): cephes.yve(1,1) def test_zeta(self): cephes.zeta(2,2) def test_zetac(self): assert_equal(cephes.zetac(0),-1.5) def test_wofz(self): cephes.wofz(0) class TestAiry(TestCase): def test_airy(self): #This tests the airy function to ensure 8 place accuracy in computation x = airy(.99) assert_array_almost_equal(x,array([0.13689066,-0.16050153,1.19815925,0.92046818]),8) x = airy(.41) assert_array_almost_equal(x,array([0.25238916,-.23480512,0.80686202,0.51053919]),8) x = airy(-.36) assert_array_almost_equal(x,array([0.44508477,-0.23186773,0.44939534,0.48105354]),8) def test_airye(self): a = airye(0.01) b = airy(0.01) b1 = [None]*4 for n in range(2): b1[n] = b[n]*exp(2.0/3.0*0.01*sqrt(0.01)) for n in range(2,4): b1[n] = b[n]*exp(-abs(real(2.0/3.0*0.01*sqrt(0.01)))) assert_array_almost_equal(a,b1,6) def test_bi_zeros(self): bi = bi_zeros(2) bia = (array([-1.17371322, -3.2710930]), array([-2.29443968, -4.07315509]), array([-0.45494438, 0.39652284]), array([ 0.60195789 , -0.76031014])) assert_array_almost_equal(bi,bia,4) def test_ai_zeros(self): ai = ai_zeros(1) assert_array_almost_equal(ai,(array([-2.33810741]), array([-1.01879297]), array([ 0.5357]), array([ 0.7012])),4) class TestAssocLaguerre(TestCase): def test_assoc_laguerre(self): a1 = genlaguerre(11,1) a2 = assoc_laguerre(.2,11,1) assert_array_almost_equal(a2,a1(.2),8) a2 = assoc_laguerre(1,11,1) assert_array_almost_equal(a2,a1(1),8) class TestBesselpoly(TestCase): def test_besselpoly(self): pass class TestKelvin(TestCase): def test_bei(self): mbei = bei(2) assert_almost_equal(mbei, 0.9722916273066613,5)#this may not be exact def test_beip(self): mbeip = beip(2) assert_almost_equal(mbeip,0.91701361338403631,5)#this may not be exact def test_ber(self): mber = ber(2) assert_almost_equal(mber,0.75173418271380821,5)#this may not be exact def test_berp(self): mberp = berp(2) assert_almost_equal(mberp,-0.49306712470943909,5)#this may not be exact def test_bei_zeros(self): bi = bi_zeros(5) assert_array_almost_equal(bi[0],array([-1.173713222709127, -3.271093302836352, -4.830737841662016, -6.169852128310251, -7.376762079367764]),11) assert_array_almost_equal(bi[1],array([-2.294439682614122, -4.073155089071828, -5.512395729663599, -6.781294445990305, -7.940178689168587]),10) assert_array_almost_equal(bi[2],array([-0.454944383639657, 0.396522836094465, -0.367969161486959, 0.349499116831805, -0.336026240133662]),11) assert_array_almost_equal(bi[3],array([0.601957887976239, -0.760310141492801, 0.836991012619261, -0.88947990142654, 0.929983638568022]),11) def test_beip_zeros(self): bip = beip_zeros(5) assert_array_almost_equal(bip,array([ 3.772673304934953, 8.280987849760042, 12.742147523633703, 17.193431752512542, 21.641143941167325]),4) def test_ber_zeros(self): ber = ber_zeros(5) assert_array_almost_equal(ber,array([2.84892, 7.23883, 11.67396, 16.11356, 20.55463]),4) def test_berp_zeros(self): brp = berp_zeros(5) assert_array_almost_equal(brp,array([6.03871, 10.51364, 14.96844, 19.41758, 23.86430]),4) def test_kelvin(self): mkelv = kelvin(2) assert_array_almost_equal(mkelv,(ber(2)+bei(2)*1j, ker(2)+kei(2)*1j, berp(2)+beip(2)*1j, kerp(2)+keip(2)*1j),8) def test_kei(self): mkei = kei(2) assert_almost_equal(mkei,-0.20240006776470432,5) def test_keip(self): mkeip = keip(2) assert_almost_equal(mkeip,0.21980790991960536,5) def test_ker(self): mker = ker(2) assert_almost_equal(mker,-0.041664513991509472,5) def test_kerp(self): mkerp = kerp(2) assert_almost_equal(mkerp,-0.10660096588105264,5) def test_kei_zeros(self): kei = kei_zeros(5) assert_array_almost_equal(kei,array([ 3.91467, 8.34422, 12.78256, 17.22314, 21.66464]),4) def test_keip_zeros(self): keip = keip_zeros(5) assert_array_almost_equal(keip,array([ 4.93181, 9.40405, 13.85827, 18.30717, 22.75379]),4) # numbers come from 9.9 of A&S pg. 381 def test_kelvin_zeros(self): tmp = kelvin_zeros(5) berz,beiz,kerz,keiz,berpz,beipz,kerpz,keipz = tmp assert_array_almost_equal(berz,array([ 2.84892, 7.23883, 11.67396, 16.11356, 20.55463]),4) assert_array_almost_equal(beiz,array([ 5.02622, 9.45541, 13.89349, 18.33398, 22.77544]),4) assert_array_almost_equal(kerz,array([ 1.71854, 6.12728, 10.56294, 15.00269, 19.44382]),4) assert_array_almost_equal(keiz,array([ 3.91467, 8.34422, 12.78256, 17.22314, 21.66464]),4) assert_array_almost_equal(berpz,array([ 6.03871, 10.51364, 14.96844, 19.41758, 23.86430]),4) assert_array_almost_equal(beipz,array([ 3.77267, # table from 1927 had 3.77320 # but this is more accurate 8.28099, 12.74215, 17.19343, 21.64114]),4) assert_array_almost_equal(kerpz,array([ 2.66584, 7.17212, 11.63218, 16.08312, 20.53068]),4) assert_array_almost_equal(keipz,array([ 4.93181, 9.40405, 13.85827, 18.30717, 22.75379]),4) def test_ker_zeros(self): ker = ker_zeros(5) assert_array_almost_equal(ker,array([ 1.71854, 6.12728, 10.56294, 15.00269, 19.44381]),4) def test_kerp_zeros(self): kerp = kerp_zeros(5) assert_array_almost_equal(kerp,array([ 2.66584, 7.17212, 11.63218, 16.08312, 20.53068]),4) class TestBernoulli(TestCase): def test_bernoulli(self): brn = bernoulli(5) assert_array_almost_equal(brn,array([1.0000, -0.5000, 0.1667, 0.0000, -0.0333, 0.0000]),4) class TestBeta(TestCase): def test_beta(self): bet = beta(2,4) betg = (gamma(2)*gamma(4))/gamma(6) assert_almost_equal(bet,betg,8) def test_betaln(self): betln = betaln(2,4) bet = log(abs(beta(2,4))) assert_almost_equal(betln,bet,8) def test_betainc(self): btinc = betainc(1,1,.2) assert_almost_equal(btinc,0.2,8) def test_betaincinv(self): y = betaincinv(2,4,.5) comp = betainc(2,4,y) assert_almost_equal(comp,.5,5) class TestCheby(TestCase): def test_chebyc(self): C0 = chebyc(0) C1 = chebyc(1) C2 = chebyc(2) C3 = chebyc(3) C4 = chebyc(4) C5 = chebyc(5) assert_array_almost_equal(C0.c,[2],13) assert_array_almost_equal(C1.c,[1,0],13) assert_array_almost_equal(C2.c,[1,0,-2],13) assert_array_almost_equal(C3.c,[1,0,-3,0],13) assert_array_almost_equal(C4.c,[1,0,-4,0,2],13) assert_array_almost_equal(C5.c,[1,0,-5,0,5,0],13) def test_chebys(self): S0 = chebys(0) S1 = chebys(1) S2 = chebys(2) S3 = chebys(3) S4 = chebys(4) S5 = chebys(5) assert_array_almost_equal(S0.c,[1],13) assert_array_almost_equal(S1.c,[1,0],13) assert_array_almost_equal(S2.c,[1,0,-1],13) assert_array_almost_equal(S3.c,[1,0,-2,0],13) assert_array_almost_equal(S4.c,[1,0,-3,0,1],13) assert_array_almost_equal(S5.c,[1,0,-4,0,3,0],13) def test_chebyt(self): T0 = chebyt(0) T1 = chebyt(1) T2 = chebyt(2) T3 = chebyt(3) T4 = chebyt(4) T5 = chebyt(5) assert_array_almost_equal(T0.c,[1],13) assert_array_almost_equal(T1.c,[1,0],13) assert_array_almost_equal(T2.c,[2,0,-1],13) assert_array_almost_equal(T3.c,[4,0,-3,0],13) assert_array_almost_equal(T4.c,[8,0,-8,0,1],13) assert_array_almost_equal(T5.c,[16,0,-20,0,5,0],13) def test_chebyu(self): U0 = chebyu(0) U1 = chebyu(1) U2 = chebyu(2) U3 = chebyu(3) U4 = chebyu(4) U5 = chebyu(5) assert_array_almost_equal(U0.c,[1],13) assert_array_almost_equal(U1.c,[2,0],13) assert_array_almost_equal(U2.c,[4,0,-1],13) assert_array_almost_equal(U3.c,[8,0,-4,0],13) assert_array_almost_equal(U4.c,[16,0,-12,0,1],13) assert_array_almost_equal(U5.c,[32,0,-32,0,6,0],13) class TestTrigonometric(TestCase): def test_cbrt(self): cb = cbrt(27) cbrl = 27**(1.0/3.0) assert_approx_equal(cb,cbrl) def test_cbrtmore(self): cb1 = cbrt(27.9) cbrl1 = 27.9**(1.0/3.0) assert_almost_equal(cb1,cbrl1,8) def test_cosdg(self): cdg = cosdg(90) cdgrl = cos(pi/2.0) assert_almost_equal(cdg,cdgrl,8) def test_cosdgmore(self): cdgm = cosdg(30) cdgmrl = cos(pi/6.0) assert_almost_equal(cdgm,cdgmrl,8) def test_cosm1(self): cs = (cosm1(0),cosm1(.3),cosm1(pi/10)) csrl = (cos(0)-1,cos(.3)-1,cos(pi/10)-1) assert_array_almost_equal(cs,csrl,8) def test_cotdg(self): ct = cotdg(30) ctrl = tan(pi/6.0)**(-1) assert_almost_equal(ct,ctrl,8) def test_cotdgmore(self): ct1 = cotdg(45) ctrl1 = tan(pi/4.0)**(-1) assert_almost_equal(ct1,ctrl1,8) def test_specialpoints(self): assert_almost_equal(cotdg(45), 1.0, 14) assert_almost_equal(cotdg(-45), -1.0, 14) assert_almost_equal(cotdg(90), 0.0, 14) assert_almost_equal(cotdg(-90), 0.0, 14) assert_almost_equal(cotdg(135), -1.0, 14) assert_almost_equal(cotdg(-135), 1.0, 14) assert_almost_equal(cotdg(225), 1.0, 14) assert_almost_equal(cotdg(-225), -1.0, 14) assert_almost_equal(cotdg(270), 0.0, 14) assert_almost_equal(cotdg(-270), 0.0, 14) assert_almost_equal(cotdg(315), -1.0, 14) assert_almost_equal(cotdg(-315), 1.0, 14) assert_almost_equal(cotdg(765), 1.0, 14) def test_sinc(self): c = arange(-2,2,.1) y = sinc(c) yre = sin(pi*c)/(pi*c) yre[20] = 1.0 assert_array_almost_equal(y, yre, 4) def test_0(self): x = 0.0 assert_equal(sinc(x),1.0) def test_sindg(self): sn = sindg(90) assert_equal(sn,1.0) def test_sindgmore(self): snm = sindg(30) snmrl = sin(pi/6.0) assert_almost_equal(snm,snmrl,8) snm1 = sindg(45) snmrl1 = sin(pi/4.0) assert_almost_equal(snm1,snmrl1,8) class TestTandg(TestCase): def test_tandg(self): tn = tandg(30) tnrl = tan(pi/6.0) assert_almost_equal(tn,tnrl,8) def test_tandgmore(self): tnm = tandg(45) tnmrl = tan(pi/4.0) assert_almost_equal(tnm,tnmrl,8) tnm1 = tandg(60) tnmrl1 = tan(pi/3.0) assert_almost_equal(tnm1,tnmrl1,8) def test_specialpoints(self): assert_almost_equal(tandg(0), 0.0, 14) assert_almost_equal(tandg(45), 1.0, 14) assert_almost_equal(tandg(-45), -1.0, 14) assert_almost_equal(tandg(135), -1.0, 14) assert_almost_equal(tandg(-135), 1.0, 14) assert_almost_equal(tandg(180), 0.0, 14) assert_almost_equal(tandg(-180), 0.0, 14) assert_almost_equal(tandg(225), 1.0, 14) assert_almost_equal(tandg(-225), -1.0, 14) assert_almost_equal(tandg(315), -1.0, 14) assert_almost_equal(tandg(-315), 1.0, 14) class TestEllip(TestCase): def test_ellipj_nan(self): """Regression test for #946.""" ellipj(0.5, np.nan) def test_ellipj(self): el = ellipj(0.2,0) rel = [sin(0.2),cos(0.2),1.0,0.20] assert_array_almost_equal(el,rel,13) def test_ellipk(self): elk = ellipk(.2) assert_almost_equal(elk,1.659623598610528,11) def test_ellipkinc(self): elkinc = ellipkinc(pi/2,.2) elk = ellipk(0.2) assert_almost_equal(elkinc,elk,15) alpha = 20*pi/180 phi = 45*pi/180 m = sin(alpha)**2 elkinc = ellipkinc(phi,m) assert_almost_equal(elkinc,0.79398143,8) # From pg. 614 of A & S def test_ellipe(self): ele = ellipe(.2) assert_almost_equal(ele,1.4890350580958529,8) def test_ellipeinc(self): eleinc = ellipeinc(pi/2,.2) ele = ellipe(0.2) assert_almost_equal(eleinc,ele,14) # pg 617 of A & S alpha, phi = 52*pi/180,35*pi/180 m = sin(alpha)**2 eleinc = ellipeinc(phi,m) assert_almost_equal(eleinc, 0.58823065, 8) class TestErf(TestCase): def test_erf(self): er = erf(.25) assert_almost_equal(er,0.2763263902,8) def test_erf_zeros(self): erz = erf_zeros(5) erzr= array([1.45061616+1.88094300j, 2.24465928+2.61657514j, 2.83974105+3.17562810j, 3.33546074+3.64617438j, 3.76900557+4.06069723j]) assert_array_almost_equal(erz,erzr,4) def test_erfcinv(self): i = erfcinv(1) assert_equal(i,0) def test_erfinv(self): i = erfinv(0) assert_equal(i,0) def test_errprint(self): a = errprint() b = 1-a #a is the state 1-a inverts state c = errprint(b) #returns last state 'a' assert_equal(a,c) d = errprint(a) #returns to original state assert_equal(d,b) #makes sure state was returned #assert_equal(d,1-a) class TestEuler(TestCase): def test_euler(self): eu0 = euler(0) eu1 = euler(1) eu2 = euler(2) # just checking segfaults assert_almost_equal(eu0[0],1,8) assert_almost_equal(eu2[2],-1,8) eu24 = euler(24) mathworld = [1,1,5,61,1385,50521,2702765,199360981, 19391512145l,2404879675441l, 370371188237525l,69348874393137901l, 15514534163557086905l] correct = zeros((25,),'d') for k in range(0,13): if (k % 2): correct[2*k] = -float(mathworld[k]) else: correct[2*k] = float(mathworld[k]) err = nan_to_num((eu24-correct)/correct) errmax = max(err) assert_almost_equal(errmax, 0.0, 14) class TestExp(TestCase): def test_exp2(self): ex = exp2(2) exrl = 2**2 assert_equal(ex,exrl) def test_exp2more(self): exm = exp2(2.5) exmrl = 2**(2.5) assert_almost_equal(exm,exmrl,8) def test_exp10(self): ex = exp10(2) exrl = 10**2 assert_approx_equal(ex,exrl) def test_exp10more(self): exm = exp10(2.5) exmrl = 10**(2.5) assert_almost_equal(exm,exmrl,8) def test_expm1(self): ex = (expm1(2),expm1(3),expm1(4)) exrl = (exp(2)-1,exp(3)-1,exp(4)-1) assert_array_almost_equal(ex,exrl,8) def test_expm1more(self): ex1 = (expm1(2),expm1(2.1),expm1(2.2)) exrl1 = (exp(2)-1,exp(2.1)-1,exp(2.2)-1) assert_array_almost_equal(ex1,exrl1,8) class TestFresnel(TestCase): def test_fresnel(self): frs = array(fresnel(.5)) assert_array_almost_equal(frs,array([0.064732432859999287, 0.49234422587144644]),8) # values from pg 329 Table 7.11 of A & S # slightly corrected in 4th decimal place def test_fresnel_zeros(self): szo, czo = fresnel_zeros(5) assert_array_almost_equal(szo, array([ 2.0093+0.2885j, 2.8335+0.2443j, 3.4675+0.2185j, 4.0026+0.2009j, 4.4742+0.1877j]),3) assert_array_almost_equal(czo, array([ 1.7437+0.3057j, 2.6515+0.2529j, 3.3204+0.2240j, 3.8757+0.2047j, 4.3611+0.1907j]),3) vals1 = fresnel(szo)[0] vals2 = fresnel(czo)[1] assert_array_almost_equal(vals1,0,14) assert_array_almost_equal(vals2,0,14) def test_fresnelc_zeros(self): szo, czo = fresnel_zeros(6) frc = fresnelc_zeros(6) assert_array_almost_equal(frc,czo,12) def test_fresnels_zeros(self): szo, czo = fresnel_zeros(5) frs = fresnels_zeros(5) assert_array_almost_equal(frs,szo,12) class TestGamma(TestCase): def test_gamma(self): gam = gamma(5) assert_equal(gam,24.0) def test_gammaln(self): gamln = gammaln(3) lngam = log(gamma(3)) assert_almost_equal(gamln,lngam,8) def test_gammainc(self): gama = gammainc(.5,.5) assert_almost_equal(gama,.7,1) def test_gammaincc(self): gicc = gammaincc(.5,.5) greal = 1 - gammainc(.5,.5) assert_almost_equal(gicc,greal,8) def test_gammainccinv(self): gccinv = gammainccinv(.5,.5) gcinv = gammaincinv(.5,.5) assert_almost_equal(gccinv,gcinv,8) def test_gammaincinv(self): y = gammaincinv(.4,.4) x = gammainc(.4,y) assert_almost_equal(x,0.4,1) y = gammainc(10, 0.05) x = gammaincinv(10, 2.5715803516000736e-20) assert_almost_equal(0.05, x, decimal=10) assert_almost_equal(y, 2.5715803516000736e-20, decimal=10) x = gammaincinv(50, 8.20754777388471303050299243573393e-18) assert_almost_equal(11.0, x, decimal=10) def test_rgamma(self): rgam = rgamma(8) rlgam = 1/gamma(8) assert_almost_equal(rgam,rlgam,8) class TestHankel(TestCase): def test_negv(self): assert_almost_equal(hankel1(-3,2), -hankel1(3,2), 14) def test_hankel1(self): hank1 = hankel1(1,.1) hankrl = (jv(1,.1)+yv(1,.1)*1j) assert_almost_equal(hank1,hankrl,8) def test_negv(self): assert_almost_equal(hankel1e(-3,2), -hankel1e(3,2), 14) def test_hankel1e(self): hank1e = hankel1e(1,.1) hankrle = hankel1(1,.1)*exp(-.1j) assert_almost_equal(hank1e,hankrle,8) def test_negv(self): assert_almost_equal(hankel2(-3,2), -hankel2(3,2), 14) def test_hankel2(self): hank2 = hankel2(1,.1) hankrl2 = (jv(1,.1)-yv(1,.1)*1j) assert_almost_equal(hank2,hankrl2,8) def test_negv(self): assert_almost_equal(hankel2e(-3,2), -hankel2e(3,2), 14) def test_hankl2e(self): hank2e = hankel2e(1,.1) hankrl2e = hankel2e(1,.1) assert_almost_equal(hank2e,hankrl2e,8) class TestHermite(TestCase): def test_hermite(self): H0 = hermite(0) H1 = hermite(1) H2 = hermite(2) H3 = hermite(3) H4 = hermite(4) H5 = hermite(5) assert_array_almost_equal(H0.c,[1],13) assert_array_almost_equal(H1.c,[2,0],13) assert_array_almost_equal(H2.c,[4,0,-2],13) assert_array_almost_equal(H3.c,[8,0,-12,0],13) assert_array_almost_equal(H4.c,[16,0,-48,0,12],12) assert_array_almost_equal(H5.c,[32,0,-160,0,120,0],12) def test_hermitenorm(self): # He_n(x) = 2**(-n/2) H_n(x/sqrt(2)) psub = poly1d([1.0/sqrt(2),0]) H0 = hermitenorm(0) H1 = hermitenorm(1) H2 = hermitenorm(2) H3 = hermitenorm(3) H4 = hermitenorm(4) H5 = hermitenorm(5) he0 = hermite(0)(psub) he1 = hermite(1)(psub) / sqrt(2) he2 = hermite(2)(psub) / 2.0 he3 = hermite(3)(psub) / (2*sqrt(2)) he4 = hermite(4)(psub) / 4.0 he5 = hermite(5)(psub) / (4.0*sqrt(2)) assert_array_almost_equal(H0.c,he0.c,13) assert_array_almost_equal(H1.c,he1.c,13) assert_array_almost_equal(H2.c,he2.c,13) assert_array_almost_equal(H3.c,he3.c,13) assert_array_almost_equal(H4.c,he4.c,13) assert_array_almost_equal(H5.c,he5.c,13) _gam = cephes.gamma class TestGegenbauer(TestCase): def test_gegenbauer(self): a = 5*rand()-0.5 if any(a==0): a = -0.2 Ca0 = gegenbauer(0,a) Ca1 = gegenbauer(1,a) Ca2 = gegenbauer(2,a) Ca3 = gegenbauer(3,a) Ca4 = gegenbauer(4,a) Ca5 = gegenbauer(5,a) assert_array_almost_equal(Ca0.c,array([1]),13) assert_array_almost_equal(Ca1.c,array([2*a,0]),13) assert_array_almost_equal(Ca2.c,array([2*a*(a+1),0,-a]),13) assert_array_almost_equal(Ca3.c,array([4*poch(a,3),0,-6*a*(a+1), 0])/3.0,11) assert_array_almost_equal(Ca4.c,array([4*poch(a,4),0,-12*poch(a,3), 0,3*a*(a+1)])/6.0,11) assert_array_almost_equal(Ca5.c,array([4*poch(a,5),0,-20*poch(a,4), 0,15*poch(a,3),0])/15.0,11) class TestHyper(TestCase): def test_h1vp(self): h1 = h1vp(1,.1) h1real = (jvp(1,.1)+yvp(1,.1)*1j) assert_almost_equal(h1,h1real,8) def test_h2vp(self): h2 = h2vp(1,.1) h2real = (jvp(1,.1)-yvp(1,.1)*1j) assert_almost_equal(h2,h2real,8) def test_hyp0f1(self): pass def test_hyp1f1(self): hyp1 = hyp1f1(.1,.1,.3) assert_almost_equal(hyp1, 1.3498588075760032,7) # test contributed by Moritz Deger (2008-05-29) # http://projects.scipy.org/scipy/scipy/ticket/659 # reference data obtained from mathematica [ a, b, x, m(a,b,x)]: # produced with test_hyp1f1.nb ref_data = array([[ -8.38132975e+00, -1.28436461e+01, -2.91081397e+01, 1.04178330e+04], [ 2.91076882e+00, -6.35234333e+00, -1.27083993e+01, 6.68132725e+00], [ -1.42938258e+01, 1.80869131e-01, 1.90038728e+01, 1.01385897e+05], [ 5.84069088e+00, 1.33187908e+01, 2.91290106e+01, 1.59469411e+08], [ -2.70433202e+01, -1.16274873e+01, -2.89582384e+01, 1.39900152e+24], [ 4.26344966e+00, -2.32701773e+01, 1.91635759e+01, 6.13816915e+21], [ 1.20514340e+01, -3.40260240e+00, 7.26832235e+00, 1.17696112e+13], [ 2.77372955e+01, -1.99424687e+00, 3.61332246e+00, 3.07419615e+13], [ 1.50310939e+01, -2.91198675e+01, -1.53581080e+01, -3.79166033e+02], [ 1.43995827e+01, 9.84311196e+00, 1.93204553e+01, 2.55836264e+10], [ -4.08759686e+00, 1.34437025e+01, -1.42072843e+01, 1.70778449e+01], [ 8.05595738e+00, -1.31019838e+01, 1.52180721e+01, 3.06233294e+21], [ 1.81815804e+01, -1.42908793e+01, 9.57868793e+00, -2.84771348e+20], [ -2.49671396e+01, 1.25082843e+01, -1.71562286e+01, 2.36290426e+07], [ 2.67277673e+01, 1.70315414e+01, 6.12701450e+00, 7.77917232e+03], [ 2.49565476e+01, 2.91694684e+01, 6.29622660e+00, 2.35300027e+02], [ 6.11924542e+00, -1.59943768e+00, 9.57009289e+00, 1.32906326e+11], [ -1.47863653e+01, 2.41691301e+01, -1.89981821e+01, 2.73064953e+03], [ 2.24070483e+01, -2.93647433e+00, 8.19281432e+00, -6.42000372e+17], [ 8.04042600e-01, 1.82710085e+01, -1.97814534e+01, 5.48372441e-01], [ 1.39590390e+01, 1.97318686e+01, 2.37606635e+00, 5.51923681e+00], [ -4.66640483e+00, -2.00237930e+01, 7.40365095e+00, 4.50310752e+00], [ 2.76821999e+01, -6.36563968e+00, 1.11533984e+01, -9.28725179e+23], [ -2.56764457e+01, 1.24544906e+00, 1.06407572e+01, 1.25922076e+01], [ 3.20447808e+00, 1.30874383e+01, 2.26098014e+01, 2.03202059e+04], [ -1.24809647e+01, 4.15137113e+00, -2.92265700e+01, 2.39621411e+08], [ 2.14778108e+01, -2.35162960e+00, -1.13758664e+01, 4.46882152e-01], [ -9.85469168e+00, -3.28157680e+00, 1.67447548e+01, -1.07342390e+07], [ 1.08122310e+01, -2.47353236e+01, -1.15622349e+01, -2.91733796e+03], [ -2.67933347e+01, -3.39100709e+00, 2.56006986e+01, -5.29275382e+09], [ -8.60066776e+00, -8.02200924e+00, 1.07231926e+01, 1.33548320e+06], [ -1.01724238e-01, -1.18479709e+01, -2.55407104e+01, 1.55436570e+00], [ -3.93356771e+00, 2.11106818e+01, -2.57598485e+01, 2.13467840e+01], [ 3.74750503e+00, 1.55687633e+01, -2.92841720e+01, 1.43873509e-02], [ 6.99726781e+00, 2.69855571e+01, -1.63707771e+01, 3.08098673e-02], [ -2.31996011e+01, 3.47631054e+00, 9.75119815e-01, 1.79971073e-02], [ 2.38951044e+01, -2.91460190e+01, -2.50774708e+00, 9.56934814e+00], [ 1.52730825e+01, 5.77062507e+00, 1.21922003e+01, 1.32345307e+09], [ 1.74673917e+01, 1.89723426e+01, 4.94903250e+00, 9.90859484e+01], [ 1.88971241e+01, 2.86255413e+01, 5.52360109e-01, 1.44165360e+00], [ 1.02002319e+01, -1.66855152e+01, -2.55426235e+01, 6.56481554e+02], [ -1.79474153e+01, 1.22210200e+01, -1.84058212e+01, 8.24041812e+05], [ -1.36147103e+01, 1.32365492e+00, -7.22375200e+00, 9.92446491e+05], [ 7.57407832e+00, 2.59738234e+01, -1.34139168e+01, 3.64037761e-02], [ 2.21110169e+00, 1.28012666e+01, 1.62529102e+01, 1.33433085e+02], [ -2.64297569e+01, -1.63176658e+01, -1.11642006e+01, -2.44797251e+13], [ -2.46622944e+01, -3.02147372e+00, 8.29159315e+00, -3.21799070e+05], [ -1.37215095e+01, -1.96680183e+01, 2.91940118e+01, 3.21457520e+12], [ -5.45566105e+00, 2.81292086e+01, 1.72548215e-01, 9.66973000e-01], [ -1.55751298e+00, -8.65703373e+00, 2.68622026e+01, -3.17190834e+16], [ 2.45393609e+01, -2.70571903e+01, 1.96815505e+01, 1.80708004e+37], [ 5.77482829e+00, 1.53203143e+01, 2.50534322e+01, 1.14304242e+06], [ -1.02626819e+01, 2.36887658e+01, -2.32152102e+01, 7.28965646e+02], [ -1.30833446e+00, -1.28310210e+01, 1.87275544e+01, -9.33487904e+12], [ 5.83024676e+00, -1.49279672e+01, 2.44957538e+01, -7.61083070e+27], [ -2.03130747e+01, 2.59641715e+01, -2.06174328e+01, 4.54744859e+04], [ 1.97684551e+01, -2.21410519e+01, -2.26728740e+01, 3.53113026e+06], [ 2.73673444e+01, 2.64491725e+01, 1.57599882e+01, 1.07385118e+07], [ 5.73287971e+00, 1.21111904e+01, 1.33080171e+01, 2.63220467e+03], [ -2.82751072e+01, 2.08605881e+01, 9.09838900e+00, -6.60957033e-07], [ 1.87270691e+01, -1.74437016e+01, 1.52413599e+01, 6.59572851e+27], [ 6.60681457e+00, -2.69449855e+00, 9.78972047e+00, -2.38587870e+12], [ 1.20895561e+01, -2.51355765e+01, 2.30096101e+01, 7.58739886e+32], [ -2.44682278e+01, 2.10673441e+01, -1.36705538e+01, 4.54213550e+04], [ -4.50665152e+00, 3.72292059e+00, -4.83403707e+00, 2.68938214e+01], [ -7.46540049e+00, -1.08422222e+01, -1.72203805e+01, -2.09402162e+02], [ -2.00307551e+01, -7.50604431e+00, -2.78640020e+01, 4.15985444e+19], [ 1.99890876e+01, 2.20677419e+01, -2.51301778e+01, 1.23840297e-09], [ 2.03183823e+01, -7.66942559e+00, 2.10340070e+01, 1.46285095e+31], [ -2.90315825e+00, -2.55785967e+01, -9.58779316e+00, 2.65714264e-01], [ 2.73960829e+01, -1.80097203e+01, -2.03070131e+00, 2.52908999e+02], [ -2.11708058e+01, -2.70304032e+01, 2.48257944e+01, 3.09027527e+08], [ 2.21959758e+01, 4.00258675e+00, -1.62853977e+01, -9.16280090e-09], [ 1.61661840e+01, -2.26845150e+01, 2.17226940e+01, -8.24774394e+33], [ -3.35030306e+00, 1.32670581e+00, 9.39711214e+00, -1.47303163e+01], [ 7.23720726e+00, -2.29763909e+01, 2.34709682e+01, -9.20711735e+29], [ 2.71013568e+01, 1.61951087e+01, -7.11388906e-01, 2.98750911e-01], [ 8.40057933e+00, -7.49665220e+00, 2.95587388e+01, 6.59465635e+29], [ -1.51603423e+01, 1.94032322e+01, -7.60044357e+00, 1.05186941e+02], [ -8.83788031e+00, -2.72018313e+01, 1.88269907e+00, 1.81687019e+00], [ -1.87283712e+01, 5.87479570e+00, -1.91210203e+01, 2.52235612e+08], [ -5.61338513e-01, 2.69490237e+01, 1.16660111e-01, 9.97567783e-01], [ -5.44354025e+00, -1.26721408e+01, -4.66831036e+00, 1.06660735e-01], [ -2.18846497e+00, 2.33299566e+01, 9.62564397e+00, 3.03842061e-01], [ 6.65661299e+00, -2.39048713e+01, 1.04191807e+01, 4.73700451e+13], [ -2.57298921e+01, -2.60811296e+01, 2.74398110e+01, -5.32566307e+11], [ -1.11431826e+01, -1.59420160e+01, -1.84880553e+01, -1.01514747e+02], [ 6.50301931e+00, 2.59859051e+01, -2.33270137e+01, 1.22760500e-02], [ -1.94987891e+01, -2.62123262e+01, 3.90323225e+00, 1.71658894e+01], [ 7.26164601e+00, -1.41469402e+01, 2.81499763e+01, -2.50068329e+31], [ -1.52424040e+01, 2.99719005e+01, -2.85753678e+01, 1.31906693e+04], [ 5.24149291e+00, -1.72807223e+01, 2.22129493e+01, 2.50748475e+25], [ 3.63207230e-01, -9.54120862e-02, -2.83874044e+01, 9.43854939e-01], [ -2.11326457e+00, -1.25707023e+01, 1.17172130e+00, 1.20812698e+00], [ 2.48513582e+00, 1.03652647e+01, -1.84625148e+01, 6.47910997e-02], [ 2.65395942e+01, 2.74794672e+01, 1.29413428e+01, 2.89306132e+05], [ -9.49445460e+00, 1.59930921e+01, -1.49596331e+01, 3.27574841e+02], [ -5.89173945e+00, 9.96742426e+00, 2.60318889e+01, -3.15842908e-01], [ -1.15387239e+01, -2.21433107e+01, -2.17686413e+01, 1.56724718e-01], [ -5.30592244e+00, -2.42752190e+01, 1.29734035e+00, 1.31985534e+00]]) for a,b,c,expected in ref_data: result = hyp1f1(a,b,c) assert(abs(expected - result)/expected < 1e-4) def test_hyp1f2(self): pass def test_hyp2f0(self): pass def test_hyp2f1(self): # a collection of special cases taken from AMS 55 values = [[0.5, 1, 1.5, 0.2**2, 0.5/0.2*log((1+0.2)/(1-0.2))], [0.5, 1, 1.5, -0.2**2, 1./0.2*arctan(0.2)], [1, 1, 2, 0.2, -1/0.2*log(1-0.2)], [3, 3.5, 1.5, 0.2**2, 0.5/0.2/(-5)*((1+0.2)**(-5)-(1-0.2)**(-5))], [-3, 3, 0.5, sin(0.2)**2, cos(2*3*0.2)], [3, 4, 8, 1, gamma(8)*gamma(8-4-3)/gamma(8-3)/gamma(8-4)], [3, 2, 3-2+1, -1, 1./2**3*sqrt(pi)* gamma(1+3-2)/gamma(1+0.5*3-2)/gamma(0.5+0.5*3)], [5, 2, 5-2+1, -1, 1./2**5*sqrt(pi)* gamma(1+5-2)/gamma(1+0.5*5-2)/gamma(0.5+0.5*5)], [4, 0.5+4, 1.5-2*4, -1./3, (8./9)**(-2*4)*gamma(4./3)* gamma(1.5-2*4)/gamma(3./2)/gamma(4./3-2*4)], # and some others # ticket #424 [1.5, -0.5, 1.0, -10.0, 4.1300097765277476484], ] for i, (a, b, c, x, v) in enumerate(values): cv = hyp2f1(a, b, c, x) assert_almost_equal(cv, v, 8, err_msg='test #%d' % i) def test_hyp3f0(self): pass def test_hyperu(self): val1 = hyperu(1,0.1,100) assert_almost_equal(val1,0.0098153,7) a,b = [0.3,0.6,1.2,-2.7],[1.5,3.2,-0.4,-3.2] a,b = asarray(a), asarray(b) z = 0.5 hypu = hyperu(a,b,z) hprl = (pi/sin(pi*b))*(hyp1f1(a,b,z)/ \ (gamma(1+a-b)*gamma(b))- \ z**(1-b)*hyp1f1(1+a-b,2-b,z) \ /(gamma(a)*gamma(2-b))) assert_array_almost_equal(hypu,hprl,12) class TestBessel(TestCase): def test_itj0y0(self): it0 = array(itj0y0(.2)) assert_array_almost_equal(it0,array([0.19933433254006822, -0.34570883800412566]),8) def test_it2j0y0(self): it2 = array(it2j0y0(.2)) assert_array_almost_equal(it2,array([0.0049937546274601858, -0.43423067011231614]),8) def test_negv(self): assert_equal(iv(3,2), iv(-3,2)) def test_j0(self): oz = j0(.1) ozr = jn(0,.1) assert_almost_equal(oz,ozr,8) def test_j1(self): o1 = j1(.1) o1r = jn(1,.1) assert_almost_equal(o1,o1r,8) def test_jn(self): jnnr = jn(1,.2) assert_almost_equal(jnnr,0.099500832639235995,8) def test_negv(self): assert_almost_equal(jv(-3,2), -jv(3,2), 14) def test_jv(self): values = [[0, 0.1, 0.99750156206604002], [2./3, 1e-8, 0.3239028506761532e-5], [2./3, 1e-10, 0.1503423854873779e-6], [3.1, 1e-10, 0.1711956265409013e-32], [2./3, 4.0, -0.2325440850267039], ] for i, (v, x, y) in enumerate(values): yc = jv(v, x) assert_almost_equal(yc, y, 8, err_msg='test #%d' % i) def test_negv(self): assert_almost_equal(jve(-3,2), -jve(3,2), 14) def test_jve(self): jvexp = jve(1,.2) assert_almost_equal(jvexp,0.099500832639235995,8) jvexp1 = jve(1,.2+1j) z = .2+1j jvexpr = jv(1,z)*exp(-abs(z.imag)) assert_almost_equal(jvexp1,jvexpr,8) def test_jn_zeros(self): jn0 = jn_zeros(0,5) jn1 = jn_zeros(1,5) assert_array_almost_equal(jn0,array([ 2.4048255577, 5.5200781103, 8.6537279129, 11.7915344391, 14.9309177086]),4) assert_array_almost_equal(jn1,array([ 3.83171, 7.01559, 10.17347, 13.32369, 16.47063]),4) jn102 = jn_zeros(102,5) assert_tol_equal(jn102, array([110.89174935992040343, 117.83464175788308398, 123.70194191713507279, 129.02417238949092824, 134.00114761868422559]), rtol=1e-13) jn301 = jn_zeros(301,5) assert_tol_equal(jn301, array([313.59097866698830153, 323.21549776096288280, 331.22338738656748796, 338.39676338872084500, 345.03284233056064157]), rtol=1e-13) def test_jn_zeros_slow(self): jn0 = jn_zeros(0, 300) assert_tol_equal(jn0[260-1], 816.02884495068867280, rtol=1e-13) assert_tol_equal(jn0[280-1], 878.86068707124422606, rtol=1e-13) assert_tol_equal(jn0[300-1], 941.69253065317954064, rtol=1e-13) jn10 = jn_zeros(10, 300) assert_tol_equal(jn10[260-1], 831.67668514305631151, rtol=1e-13) assert_tol_equal(jn10[280-1], 894.51275095371316931, rtol=1e-13) assert_tol_equal(jn10[300-1], 957.34826370866539775, rtol=1e-13) jn3010 = jn_zeros(3010,5) assert_tol_equal(jn3010, array([3036.86590780927, 3057.06598526482, 3073.66360690272, 3088.37736494778, 3101.86438139042]), rtol=1e-8) def test_jnjnp_zeros(self): pass #jnjp = jnjnp(3) #assert_array_almost_equal(jnjp,(array([ #I don't think specfun jdzo is working properly the outputs do not seem to correlate #to the inputs def test_jnp_zeros(self): jnp = jnp_zeros(1,5) assert_array_almost_equal(jnp, array([ 1.84118, 5.33144, 8.53632, 11.70600, 14.86359]),4) jnp = jnp_zeros(443,5) assert_tol_equal(jvp(443, jnp), 0, atol=1e-15) def test_jnyn_zeros(self): jnz = jnyn_zeros(1,5) assert_array_almost_equal(jnz,(array([ 3.83171, 7.01559, 10.17347, 13.32369, 16.47063]), array([ 1.84118, 5.33144, 8.53632, 11.70600, 14.86359]), array([ 2.19714, 5.42968, 8.59601, 11.74915, 14.89744]), array([ 3.68302, 6.94150, 10.12340, 13.28576, 16.44006])),5) def test_jvp(self): jvprim = jvp(2,2) jv0 = (jv(1,2)-jv(3,2))/2 assert_almost_equal(jvprim,jv0,10) def test_k0(self): ozk = k0(.1) ozkr = kv(0,.1) assert_almost_equal(ozk,ozkr,8) def test_k0e(self): ozke = k0e(.1) ozker = kve(0,.1) assert_almost_equal(ozke,ozker,8) def test_k1(self): o1k = k1(.1) o1kr = kv(1,.1) assert_almost_equal(o1k,o1kr,8) def test_k1e(self): o1ke = k1e(.1) o1ker = kve(1,.1) assert_almost_equal(o1ke,o1ker,8) def test_jacobi(self): a = 5*rand() - 1 b = 5*rand() - 1 P0 = jacobi(0,a,b) P1 = jacobi(1,a,b) P2 = jacobi(2,a,b) P3 = jacobi(3,a,b) assert_array_almost_equal(P0.c,[1],13) assert_array_almost_equal(P1.c,array([a+b+2,a-b])/2.0,13) cp = [(a+b+3)*(a+b+4), 4*(a+b+3)*(a+2), 4*(a+1)*(a+2)] p2c = [cp[0],cp[1]-2*cp[0],cp[2]-cp[1]+cp[0]] assert_array_almost_equal(P2.c,array(p2c)/8.0,13) cp = [(a+b+4)*(a+b+5)*(a+b+6),6*(a+b+4)*(a+b+5)*(a+3), 12*(a+b+4)*(a+2)*(a+3),8*(a+1)*(a+2)*(a+3)] p3c = [cp[0],cp[1]-3*cp[0],cp[2]-2*cp[1]+3*cp[0],cp[3]-cp[2]+cp[1]-cp[0]] assert_array_almost_equal(P3.c,array(p3c)/48.0,13) def test_kn(self): kn1 = kn(0,.2) assert_almost_equal(kn1,1.7527038555281462,8) def test_negv(self): assert_equal(kv(3.0, 2.2), kv(-3.0, 2.2)) def test_kv0(self): kv0 = kv(0,.2) assert_almost_equal(kv0, 1.7527038555281462, 10) def test_kv1(self): kv1 = kv(1,0.2) assert_almost_equal(kv1, 4.775972543220472, 10) def test_kv2(self): kv2 = kv(2,0.2) assert_almost_equal(kv2, 49.51242928773287, 10) def test_negv(self): assert_equal(kve(3.0, 2.2), kve(-3.0, 2.2)) def test_kve(self): kve1 = kve(0,.2) kv1 = kv(0,.2)*exp(.2) assert_almost_equal(kve1,kv1,8) z = .2+1j kve2 = kve(0,z) kv2 = kv(0,z)*exp(z) assert_almost_equal(kve2,kv2,8) def test_kvp_v0n1(self): z = 2.2 assert_almost_equal(-kv(1,z), kvp(0,z, n=1), 10) def test_kvp_n1(self): v = 3. z = 2.2 xc = -kv(v+1,z) + v/z*kv(v,z) x = kvp(v,z, n=1) assert_almost_equal(xc, x, 10) #this function (kvp) is broken def test_kvp_n2(self): v = 3. z = 2.2 xc = (z**2+v**2-v)/z**2 * kv(v,z) + kv(v+1,z)/z x = kvp(v, z, n=2) assert_almost_equal(xc, x, 10) def test_y0(self): oz = y0(.1) ozr = yn(0,.1) assert_almost_equal(oz,ozr,8) def test_y1(self): o1 = y1(.1) o1r = yn(1,.1) assert_almost_equal(o1,o1r,8) def test_y0_zeros(self): yo,ypo = y0_zeros(2) zo,zpo = y0_zeros(2,complex=1) all = r_[yo,zo] allval = r_[ypo,zpo] assert_array_almost_equal(abs(yv(0.0,all)),0.0,11) assert_array_almost_equal(abs(yv(1,all)-allval),0.0,11) def test_y1_zeros(self): y1 = y1_zeros(1) assert_array_almost_equal(y1,(array([2.19714]),array([0.52079])),5) def test_y1p_zeros(self): y1p = y1p_zeros(1,complex=1) assert_array_almost_equal(y1p,(array([ 0.5768+0.904j]), array([-0.7635+0.5892j])),3) def test_yn_zeros(self): an = yn_zeros(4,2) assert_array_almost_equal(an,array([ 5.64515, 9.36162]),5) an = yn_zeros(443,5) assert_tol_equal(an, [450.13573091578090314, 463.05692376675001542, 472.80651546418663566, 481.27353184725625838, 488.98055964441374646], rtol=1e-15) def test_ynp_zeros(self): ao = ynp_zeros(0,2) assert_array_almost_equal(ao,array([ 2.19714133, 5.42968104]),6) ao = ynp_zeros(43,5) assert_tol_equal(yvp(43, ao), 0, atol=1e-15) ao = ynp_zeros(443,5) assert_tol_equal(yvp(443, ao), 0, atol=1e-9) @dec.knownfailureif(True, "cephes/yv is not eps accurate for large orders on " "all platforms, and has nan/inf issues") def test_ynp_zeros_large_order(self): ao = ynp_zeros(443,5) assert_tol_equal(yvp(443, ao), 0, atol=1e-15) def test_yn(self): yn2n = yn(1,.2) assert_almost_equal(yn2n,-3.3238249881118471,8) def test_negv(self): assert_almost_equal(yv(-3,2), -yv(3,2), 14) def test_yv(self): yv2 = yv(1,.2) assert_almost_equal(yv2,-3.3238249881118471,8) def test_negv(self): assert_almost_equal(yve(-3,2), -yve(3,2), 14) def test_yve(self): yve2 = yve(1,.2) assert_almost_equal(yve2,-3.3238249881118471,8) yve2r = yv(1,.2+1j)*exp(-1) yve22 = yve(1,.2+1j) assert_almost_equal(yve22,yve2r,8) def test_yvp(self): yvpr = (yv(1,.2) - yv(3,.2))/2.0 yvp1 = yvp(2,.2) assert_array_almost_equal(yvp1,yvpr,10) def _cephes_vs_amos_points(self): """Yield points at which to compare Cephes implementation to AMOS""" # check several points, including large-amplitude ones for v in [-120, -100.3, -20., -10., -1., -.5, 0., 1., 12.49, 120., 301]: for z in [-1300, -11, -10, -1, 1., 10., 200.5, 401., 600.5, 700.6, 1300, 10003]: yield v, z # check half-integers; these are problematic points at least # for cephes/iv for v in 0.5 + arange(-60, 60): yield v, 3.5 def check_cephes_vs_amos(self, f1, f2, rtol=1e-11, atol=0, skip=None): for v, z in self._cephes_vs_amos_points(): if skip is not None and skip(v, z): continue c1, c2, c3 = f1(v, z), f1(v,z+0j), f2(int(v), z) if np.isinf(c1): assert np.abs(c2) >= 1e300, (v, z) elif np.isnan(c1): assert c2.imag != 0, (v, z) else: assert_tol_equal(c1, c2, err_msg=(v, z), rtol=rtol, atol=atol) if v == int(v): assert_tol_equal(c3, c2, err_msg=(v, z), rtol=rtol, atol=atol) def test_jv_cephes_vs_amos(self): self.check_cephes_vs_amos(jv, jn, rtol=1e-10, atol=1e-305) @dec.knownfailureif(True, "cephes/yv is not eps accurate for large orders on " "all platforms, and has nan/inf issues") def test_yv_cephes_vs_amos(self): self.check_cephes_vs_amos(yv, yn, rtol=1e-11, atol=1e-305) def test_yv_cephes_vs_amos_only_small_orders(self): skipper = lambda v, z: (abs(v) > 50) self.check_cephes_vs_amos(yv, yn, rtol=1e-11, atol=1e-305, skip=skipper) def test_iv_cephes_vs_amos(self): self.check_cephes_vs_amos(iv, iv, rtol=1e-12, atol=1e-305) @dec.slow def test_iv_cephes_vs_amos_mass_test(self): N = 1000000 np.random.seed(1) v = np.random.pareto(0.5, N) * (-1)**np.random.randint(2, size=N) x = np.random.pareto(0.2, N) * (-1)**np.random.randint(2, size=N) imsk = (np.random.randint(8, size=N) == 0) v[imsk] = v.astype(int) c1 = iv(v, x) c2 = iv(v, x+0j) dc = abs(c1/c2 - 1) dc[np.isnan(dc)] = 0 k = np.argmax(dc) # Most error apparently comes from AMOS and not our implementation; # there are some problems near integer orders there assert dc[k] < 1e-9, (iv(v[k], x[k]), iv(v[k], x[k]+0j)) def test_kv_cephes_vs_amos(self): #self.check_cephes_vs_amos(kv, kn, rtol=1e-9, atol=1e-305) self.check_cephes_vs_amos(kv, kv, rtol=1e-9, atol=1e-305) def test_ticket_623(self): assert_tol_equal(jv(3, 4), 0.43017147387562193) assert_tol_equal(jv(301, 1300), 0.0183487151115275) assert_tol_equal(jv(301, 1296.0682), -0.0224174325312048) def test_ticket_853(self): """Negative-order Bessels""" # cephes assert_tol_equal(jv(-1, 1 ), -0.4400505857449335) assert_tol_equal(jv(-2, 1 ), 0.1149034849319005) assert_tol_equal(yv(-1, 1 ), 0.7812128213002887) assert_tol_equal(yv(-2, 1 ), -1.650682606816255) assert_tol_equal(iv(-1, 1 ), 0.5651591039924851) assert_tol_equal(iv(-2, 1 ), 0.1357476697670383) assert_tol_equal(kv(-1, 1 ), 0.6019072301972347) assert_tol_equal(kv(-2, 1 ), 1.624838898635178) assert_tol_equal(jv(-0.5, 1 ), 0.43109886801837607952) assert_tol_equal(yv(-0.5, 1 ), 0.6713967071418031) assert_tol_equal(iv(-0.5, 1 ), 1.231200214592967) assert_tol_equal(kv(-0.5, 1 ), 0.4610685044478945) # amos assert_tol_equal(jv(-1, 1+0j), -0.4400505857449335) assert_tol_equal(jv(-2, 1+0j), 0.1149034849319005) assert_tol_equal(yv(-1, 1+0j), 0.7812128213002887) assert_tol_equal(yv(-2, 1+0j), -1.650682606816255) assert_tol_equal(iv(-1, 1+0j), 0.5651591039924851) assert_tol_equal(iv(-2, 1+0j), 0.1357476697670383) assert_tol_equal(kv(-1, 1+0j), 0.6019072301972347) assert_tol_equal(kv(-2, 1+0j), 1.624838898635178) assert_tol_equal(jv(-0.5, 1+0j), 0.43109886801837607952) assert_tol_equal(jv(-0.5, 1+1j), 0.2628946385649065-0.827050182040562j) assert_tol_equal(yv(-0.5, 1+0j), 0.6713967071418031) assert_tol_equal(yv(-0.5, 1+1j), 0.967901282890131+0.0602046062142816j) assert_tol_equal(iv(-0.5, 1+0j), 1.231200214592967) assert_tol_equal(iv(-0.5, 1+1j), 0.77070737376928+0.39891821043561j) assert_tol_equal(kv(-0.5, 1+0j), 0.4610685044478945) assert_tol_equal(kv(-0.5, 1+1j), 0.06868578341999-0.38157825981268j) assert_tol_equal(jve(-0.5,1+0.3j), jv(-0.5, 1+0.3j)*exp(-0.3)) assert_tol_equal(yve(-0.5,1+0.3j), yv(-0.5, 1+0.3j)*exp(-0.3)) assert_tol_equal(ive(-0.5,0.3+1j), iv(-0.5, 0.3+1j)*exp(-0.3)) assert_tol_equal(kve(-0.5,0.3+1j), kv(-0.5, 0.3+1j)*exp(0.3+1j)) assert_tol_equal(hankel1(-0.5, 1+1j), jv(-0.5, 1+1j) + 1j*yv(-0.5,1+1j)) assert_tol_equal(hankel2(-0.5, 1+1j), jv(-0.5, 1+1j) - 1j*yv(-0.5,1+1j)) def test_ticket_854(self): """Real-valued Bessel domains""" assert isnan(jv(0.5, -1)) assert isnan(iv(0.5, -1)) assert isnan(yv(0.5, -1)) assert isnan(yv(1, -1)) assert isnan(kv(0.5, -1)) assert isnan(kv(1, -1)) assert isnan(jve(0.5, -1)) assert isnan(ive(0.5, -1)) assert isnan(yve(0.5, -1)) assert isnan(yve(1, -1)) assert isnan(kve(0.5, -1)) assert isnan(kve(1, -1)) assert isnan(airye(-1)[0:2]).all(), airye(-1) assert not isnan(airye(-1)[2:4]).any(), airye(-1) def test_ticket_503(self): """Real-valued Bessel I overflow""" assert_tol_equal(iv(1, 700), 1.528500390233901e302) assert_tol_equal(iv(1000, 1120), 1.301564549405821e301) def test_iv_hyperg_poles(self): assert_tol_equal(iv(-0.5, 1), 1.231200214592967) def iv_series(self, v, z, n=200): k = arange(0, n).astype(float_) r = (v+2*k)*log(.5*z) - gammaln(k+1) - gammaln(v+k+1) r[isnan(r)] = inf r = exp(r) err = abs(r).max() * finfo(float_).eps * n + abs(r[-1])*10 return r.sum(), err def test_i0_series(self): for z in [1., 10., 200.5]: value, err = self.iv_series(0, z) assert_tol_equal(i0(z), value, atol=err, err_msg=z) def test_i1_series(self): for z in [1., 10., 200.5]: value, err = self.iv_series(1, z) assert_tol_equal(i1(z), value, atol=err, err_msg=z) def test_iv_series(self): for v in [-20., -10., -1., 0., 1., 12.49, 120.]: for z in [1., 10., 200.5, -1+2j]: value, err = self.iv_series(v, z) assert_tol_equal(iv(v, z), value, atol=err, err_msg=(v, z)) def test_i0(self): values = [[0.0, 1.0], [1e-10, 1.0], [0.1, 0.9071009258], [0.5, 0.6450352706], [1.0, 0.4657596077], [2.5, 0.2700464416], [5.0, 0.1835408126], [20.0, 0.0897803119], ] for i, (x, v) in enumerate(values): cv = i0(x) * exp(-x) assert_almost_equal(cv, v, 8, err_msg='test #%d' % i) def test_i0e(self): oize = i0e(.1) oizer = ive(0,.1) assert_almost_equal(oize,oizer,8) def test_i1(self): values = [[0.0, 0.0], [1e-10, 0.4999999999500000e-10], [0.1, 0.0452984468], [0.5, 0.1564208032], [1.0, 0.2079104154], [5.0, 0.1639722669], [20.0, 0.0875062222], ] for i, (x, v) in enumerate(values): cv = i1(x) * exp(-x) assert_almost_equal(cv, v, 8, err_msg='test #%d' % i) def test_i1e(self): oi1e = i1e(.1) oi1er = ive(1,.1) assert_almost_equal(oi1e,oi1er,8) def test_iti0k0(self): iti0 = array(iti0k0(5)) assert_array_almost_equal(iti0,array([31.848667776169801, 1.5673873907283657]),5) def test_it2i0k0(self): it2k = it2i0k0(.1) assert_array_almost_equal(it2k,array([0.0012503906973464409, 3.3309450354686687]),6) def test_iv(self): iv1 = iv(0,.1)*exp(-.1) assert_almost_equal(iv1,0.90710092578230106,10) def test_negv(self): assert_equal(ive(3,2), ive(-3,2)) def test_ive(self): ive1 = ive(0,.1) iv1 = iv(0,.1)*exp(-.1) assert_almost_equal(ive1,iv1,10) def test_ivp0(self): assert_almost_equal(iv(1,2), ivp(0,2), 10) def test_ivp(self): y=(iv(0,2)+iv(2,2))/2 x = ivp(1,2) assert_almost_equal(x,y,10) class TestLaguerre(TestCase): def test_laguerre(self): lag0 = laguerre(0) lag1 = laguerre(1) lag2 = laguerre(2) lag3 = laguerre(3) lag4 = laguerre(4) lag5 = laguerre(5) assert_array_almost_equal(lag0.c,[1],13) assert_array_almost_equal(lag1.c,[-1,1],13) assert_array_almost_equal(lag2.c,array([1,-4,2])/2.0,13) assert_array_almost_equal(lag3.c,array([-1,9,-18,6])/6.0,13) assert_array_almost_equal(lag4.c,array([1,-16,72,-96,24])/24.0,13) assert_array_almost_equal(lag5.c,array([-1,25,-200,600,-600,120])/120.0,13) def test_genlaguerre(self): k = 5*rand()-0.9 lag0 = genlaguerre(0,k) lag1 = genlaguerre(1,k) lag2 = genlaguerre(2,k) lag3 = genlaguerre(3,k) assert_equal(lag0.c,[1]) assert_equal(lag1.c,[-1,k+1]) assert_almost_equal(lag2.c,array([1,-2*(k+2),(k+1.)*(k+2.)])/2.0) assert_almost_equal(lag3.c,array([-1,3*(k+3),-3*(k+2)*(k+3),(k+1)*(k+2)*(k+3)])/6.0) # Base polynomials come from Abrahmowitz and Stegan class TestLegendre(TestCase): def test_legendre(self): leg0 = legendre(0) leg1 = legendre(1) leg2 = legendre(2) leg3 = legendre(3) leg4 = legendre(4) leg5 = legendre(5) assert_equal(leg0.c,[1]) assert_equal(leg1.c,[1,0]) assert_equal(leg2.c,array([3,0,-1])/2.0) assert_almost_equal(leg3.c,array([5,0,-3,0])/2.0) assert_almost_equal(leg4.c,array([35,0,-30,0,3])/8.0) assert_almost_equal(leg5.c,array([63,0,-70,0,15,0])/8.0) class TestLambda(TestCase): def test_lmbda(self): lam = lmbda(1,.1) lamr = (array([jn(0,.1), 2*jn(1,.1)/.1]), array([jvp(0,.1), -2*jv(1,.1)/.01 + 2*jvp(1,.1)/.1])) assert_array_almost_equal(lam,lamr,8) class TestLog1p(TestCase): def test_log1p(self): l1p = (log1p(10),log1p(11),log1p(12)) l1prl = (log(11),log(12),log(13)) assert_array_almost_equal(l1p,l1prl,8) def test_log1pmore(self): l1pm = (log1p(1),log1p(1.1),log1p(1.2)) l1pmrl = (log(2),log(2.1),log(2.2)) assert_array_almost_equal(l1pm,l1pmrl,8) class TestLegendreFunctions(TestCase): def test_lpmn(self): lp = lpmn(0,2,.5) assert_array_almost_equal(lp,(array([ [ 1.00000 , 0.50000, -0.12500]]), array([ [ 0.00000 , 1.00000 , 1.50000]])),4) def test_lpn(self): lpnf = lpn(2,.5) assert_array_almost_equal(lpnf,(array( [ 1.00000 , 0.50000, -0.12500]), array( [ 0.00000 , 1.00000 , 1.50000])),4) def test_lpmv(self): lp = lpmv(0,2,.5) assert_almost_equal(lp,-0.125,3) def test_lqmn(self): lqmnf = lqmn(0,2,.5) lqmnf = lqmn(0,2,.5) lqf = lqn(2,.5) assert_array_almost_equal(lqmnf[0][0],lqf[0],4) assert_array_almost_equal(lqmnf[1][0],lqf[1],4) def test_lqmn_shape(self): a, b = lqmn(4, 4, 1.1) assert_equal(a.shape, (5, 5)) assert_equal(b.shape, (5, 5)) a, b = lqmn(4, 0, 1.1) assert_equal(a.shape, (5, 1)) assert_equal(b.shape, (5, 1)) def test_lqn(self): lqf = lqn(2,.5) assert_array_almost_equal(lqf,(array([ 0.5493, -0.7253, -0.8187]), array([ 1.3333, 1.216 , -0.8427])),4) class TestMathieu(TestCase): def test_mathieu_a(self): pass def test_mathieu_even_coef(self): mc = mathieu_even_coef(2,5) #Q not defined broken and cannot figure out proper reporting order def test_mathieu_odd_coef(self): pass #same problem as above class TestFresnelIntegral(TestCase): def test_modfresnelp(self): pass def test_modfresnelm(self): pass class TestOblCvSeq(TestCase): def test_obl_cv_seq(self): obl = obl_cv_seq(0,3,1) assert_array_almost_equal(obl,array([ -0.348602, 1.393206, 5.486800, 11.492120]),5) class TestParabolicCylinder(TestCase): def test_pbdn_seq(self): pb = pbdn_seq(1,.1) assert_array_almost_equal(pb,(array([ 0.9975, 0.0998]), array([-0.0499, 0.9925])),4) def test_pbdv(self): pbv = pbdv(1,.2) derrl = 1/2*(.2)*pbdv(1,.2)[0] - pbdv(0,.2)[0] def test_pbdv_seq(self): pbn = pbdn_seq(1,.1) pbv = pbdv_seq(1,.1) assert_array_almost_equal(pbv,(real(pbn[0]),real(pbn[1])),4) def test_pbdv_points(self): # simple case eta = np.linspace(-10, 10, 5) z = 2**(eta/2)*np.sqrt(np.pi)/gamma(.5-.5*eta) assert_tol_equal(pbdv(eta, 0.)[0], z, rtol=1e-14, atol=1e-14) # some points assert_tol_equal(pbdv(10.34, 20.44)[0], 1.3731383034455e-32, rtol=1e-12) assert_tol_equal(pbdv(-9.53, 3.44)[0], 3.166735001119246e-8, rtol=1e-12) def test_pbdv_gradient(self): x = np.linspace(-4, 4, 8)[:,None] eta = np.linspace(-10, 10, 5)[None,:] p = pbdv(eta, x) eps = 1e-7 + 1e-7*abs(x) dp = (pbdv(eta, x + eps)[0] - pbdv(eta, x - eps)[0]) / eps / 2. assert_tol_equal(p[1], dp, rtol=1e-6, atol=1e-6) def test_pbvv_gradient(self): x = np.linspace(-4, 4, 8)[:,None] eta = np.linspace(-10, 10, 5)[None,:] p = pbvv(eta, x) eps = 1e-7 + 1e-7*abs(x) dp = (pbvv(eta, x + eps)[0] - pbvv(eta, x - eps)[0]) / eps / 2. assert_tol_equal(p[1], dp, rtol=1e-6, atol=1e-6) class TestPolygamma(TestCase): # from Table 6.2 (pg. 271) of A&S def test_polygamma(self): poly2 = polygamma(2,1) poly3 = polygamma(3,1) assert_almost_equal(poly2,-2.4041138063,10) assert_almost_equal(poly3,6.4939394023,10) class TestProCvSeq(TestCase): def test_pro_cv_seq(self): prol = pro_cv_seq(0,3,1) assert_array_almost_equal(prol,array([ 0.319000, 2.593084, 6.533471, 12.514462]),5) class TestPsi(TestCase): def test_psi(self): ps = psi(1) assert_almost_equal(ps,-0.57721566490153287,8) class TestRadian(TestCase): def test_radian(self): rad = radian(90,0,0) assert_almost_equal(rad,pi/2.0,5) def test_radianmore(self): rad1 = radian(90,1,60) assert_almost_equal(rad1,pi/2+0.0005816135199345904,5) class TestRiccati(TestCase): def test_riccati_jn(self): jnrl = (sph_jn(1,.2)[0]*.2,sph_jn(1,.2)[0]+sph_jn(1,.2)[1]*.2) ricjn = riccati_jn(1,.2) assert_array_almost_equal(ricjn,jnrl,8) def test_riccati_yn(self): ynrl = (sph_yn(1,.2)[0]*.2,sph_yn(1,.2)[0]+sph_yn(1,.2)[1]*.2) ricyn = riccati_yn(1,.2) assert_array_almost_equal(ricyn,ynrl,8) class TestRound(TestCase): def test_round(self): rnd = map(int,(round(10.1),round(10.4),round(10.5),round(10.6))) # Note: According to the documentation, scipy.special.round is # supposed to round to the nearest even number if the fractional # part is exactly 0.5. On some platforms, this does not appear # to work and thus this test may fail. However, this unit test is # correctly written. rndrl = (10,10,10,11) assert_array_equal(rnd,rndrl) class _test_sh_legendre(TestCase): def test_sh_legendre(self): # P*_n(x) = P_n(2x-1) psub = poly1d([2,-1]) Ps0 = sh_legendre(0) Ps1 = sh_legendre(1) Ps2 = sh_legendre(2) Ps3 = sh_legendre(3) Ps4 = sh_legendre(4) Ps5 = sh_legendre(5) pse0 = legendre(0)(psub) pse1 = legendre(1)(psub) pse2 = legendre(2)(psub) pse3 = legendre(3)(psub) pse4 = legendre(4)(psub) pse5 = legendre(5)(psub) assert_array_almost_equal(Ps0.c,pse0.c,13) assert_array_almost_equal(Ps1.c,pse1.c,13) assert_array_almost_equal(Ps2.c,pse2.c,13) assert_array_almost_equal(Ps3.c,pse3.c,13) assert_array_almost_equal(Ps4.c,pse4.c,12) assert_array_almost_equal(Ps5.c,pse5.c,12) class _test_sh_chebyt(TestCase): def test_sh_chebyt(self): # T*_n(x) = T_n(2x-1) psub = poly1d([2,-1]) Ts0 = sh_chebyt(0) Ts1 = sh_chebyt(1) Ts2 = sh_chebyt(2) Ts3 = sh_chebyt(3) Ts4 = sh_chebyt(4) Ts5 = sh_chebyt(5) tse0 = chebyt(0)(psub) tse1 = chebyt(1)(psub) tse2 = chebyt(2)(psub) tse3 = chebyt(3)(psub) tse4 = chebyt(4)(psub) tse5 = chebyt(5)(psub) assert_array_almost_equal(Ts0.c,tse0.c,13) assert_array_almost_equal(Ts1.c,tse1.c,13) assert_array_almost_equal(Ts2.c,tse2.c,13) assert_array_almost_equal(Ts3.c,tse3.c,13) assert_array_almost_equal(Ts4.c,tse4.c,12) assert_array_almost_equal(Ts5.c,tse5.c,12) class _test_sh_chebyu(TestCase): def test_sh_chebyu(self): # U*_n(x) = U_n(2x-1) psub = poly1d([2,-1]) Us0 = sh_chebyu(0) Us1 = sh_chebyu(1) Us2 = sh_chebyu(2) Us3 = sh_chebyu(3) Us4 = sh_chebyu(4) Us5 = sh_chebyu(5) use0 = chebyu(0)(psub) use1 = chebyu(1)(psub) use2 = chebyu(2)(psub) use3 = chebyu(3)(psub) use4 = chebyu(4)(psub) use5 = chebyu(5)(psub) assert_array_almost_equal(Us0.c,use0.c,13) assert_array_almost_equal(Us1.c,use1.c,13) assert_array_almost_equal(Us2.c,use2.c,13) assert_array_almost_equal(Us3.c,use3.c,13) assert_array_almost_equal(Us4.c,use4.c,12) assert_array_almost_equal(Us5.c,use5.c,11) class _test_sh_jacobi(TestCase): def test_sh_jacobi(self): # G^(p,q)_n(x) = n! gamma(n+p)/gamma(2*n+p) * P^(p-q,q-1)_n(2*x-1) conv = lambda n,p: _gam(n+1)*_gam(n+p)/_gam(2*n+p) psub = poly1d([2,-1]) q = 4*rand() p = q-1 + 2*rand() #print "shifted jacobi p,q = ", p, q G0 = sh_jacobi(0,p,q) G1 = sh_jacobi(1,p,q) G2 = sh_jacobi(2,p,q) G3 = sh_jacobi(3,p,q) G4 = sh_jacobi(4,p,q) G5 = sh_jacobi(5,p,q) ge0 = jacobi(0,p-q,q-1)(psub) * conv(0,p) ge1 = jacobi(1,p-q,q-1)(psub) * conv(1,p) ge2 = jacobi(2,p-q,q-1)(psub) * conv(2,p) ge3 = jacobi(3,p-q,q-1)(psub) * conv(3,p) ge4 = jacobi(4,p-q,q-1)(psub) * conv(4,p) ge5 = jacobi(5,p-q,q-1)(psub) * conv(5,p) assert_array_almost_equal(G0.c,ge0.c,13) assert_array_almost_equal(G1.c,ge1.c,13) assert_array_almost_equal(G2.c,ge2.c,13) assert_array_almost_equal(G3.c,ge3.c,13) assert_array_almost_equal(G4.c,ge4.c,13) assert_array_almost_equal(G5.c,ge5.c,13) class TestSpherical(TestCase): def test_sph_harm(self): pass def test_sph_in(self): i1n = sph_in(1,.2) inp0 = (i1n[0][1]) inp1 = (i1n[0][0] - 2.0/0.2 * i1n[0][1]) assert_array_almost_equal(i1n[0],array([1.0066800127054699381, 0.066933714568029540839]),12) assert_array_almost_equal(i1n[1],[inp0,inp1],12) def test_sph_inkn(self): spikn = r_[sph_in(1,.2)+sph_kn(1,.2)] inkn = r_[sph_inkn(1,.2)] assert_array_almost_equal(inkn,spikn,10) def test_sph_jn(self): s1 = sph_jn(2,.2) s10 = -s1[0][1] s11 = s1[0][0]-2.0/0.2*s1[0][1] s12 = s1[0][1]-3.0/0.2*s1[0][2] assert_array_almost_equal(s1[0],[0.99334665397530607731, 0.066400380670322230863, 0.0026590560795273856680],12) assert_array_almost_equal(s1[1],[s10,s11,s12],12) def test_sph_jnyn(self): jnyn = r_[sph_jn(1,.2) + sph_yn(1,.2)] # tuple addition jnyn1 = r_[sph_jnyn(1,.2)] assert_array_almost_equal(jnyn1,jnyn,9) def test_sph_kn(self): kn = sph_kn(2,.2) kn0 = -kn[0][1] kn1 = -kn[0][0]-2.0/0.2*kn[0][1] kn2 = -kn[0][1]-3.0/0.2*kn[0][2] assert_array_almost_equal(kn[0],[6.4302962978445670140, 38.581777787067402086, 585.15696310385559829],12) assert_array_almost_equal(kn[1],[kn0,kn1,kn2],9) def test_sph_yn(self): sy1 = sph_yn(2,.2)[0][2] sy2 = sph_yn(0,.2)[0][0] sphpy = (sph_yn(1,.2)[0][0]-2*sph_yn(2,.2)[0][2])/3 #correct derivative value assert_almost_equal(sy1,-377.52483,5)#previous values in the system assert_almost_equal(sy2,-4.9003329,5) sy3 = sph_yn(1,.2)[1][1] assert_almost_equal(sy3,sphpy,4) #compare correct derivative val. (correct =-system val). class TestStruve(object): def _series(self, v, z, n=100): """Compute Struve function & error estimate from its power series.""" k = arange(0, n) r = (-1)**k * (.5*z)**(2*k+v+1)/gamma(k+1.5)/gamma(k+v+1.5) err = abs(r).max() * finfo(float_).eps * n return r.sum(), err def test_vs_series(self): """Check Struve function versus its power series""" for v in [-20, -10, -7.99, -3.4, -1, 0, 1, 3.4, 12.49, 16]: for z in [1, 10, 19, 21, 30]: value, err = self._series(v, z) assert_tol_equal(struve(v, z), value, rtol=0, atol=err), (v, z) def test_some_values(self): assert_tol_equal(struve(-7.99, 21), 0.0467547614113, rtol=1e-7) assert_tol_equal(struve(-8.01, 21), 0.0398716951023, rtol=1e-8) assert_tol_equal(struve(-3.0, 200), 0.0142134427432, rtol=1e-12) assert_tol_equal(struve(-8.0, -41), 0.0192469727846, rtol=1e-11) assert_equal(struve(-12, -41), -struve(-12, 41)) assert_equal(struve(+12, -41), -struve(+12, 41)) assert_equal(struve(-11, -41), +struve(-11, 41)) assert_equal(struve(+11, -41), +struve(+11, 41)) assert isnan(struve(-7.1, -1)) assert isnan(struve(-10.1, -1)) def test_regression_679(self): """Regression test for #679""" assert_tol_equal(struve(-1.0, 20 - 1e-8), struve(-1.0, 20 + 1e-8)) assert_tol_equal(struve(-2.0, 20 - 1e-8), struve(-2.0, 20 + 1e-8)) assert_tol_equal(struve(-4.3, 20 - 1e-8), struve(-4.3, 20 + 1e-8)) if __name__ == "__main__": run_module_suite()