import pytest from mpmath import * def test_approximation(): mp.dps = 15 f = lambda x: cos(2-2*x)/x p, err = chebyfit(f, [2, 4], 8, error=True) assert err < 1e-5 for i in range(10): x = 2 + i/5. assert abs(polyval(p, x) - f(x)) < err def test_limits(): mp.dps = 15 assert limit(lambda x: (x-sin(x))/x**3, 0).ae(mpf(1)/6) assert limit(lambda n: (1+1/n)**n, inf).ae(e) def test_polyval(): assert polyval([], 3) == 0 assert polyval([0], 3) == 0 assert polyval([5], 3) == 5 # 4x^3 - 2x + 5 p = [4, 0, -2, 5] assert polyval(p,4) == 253 assert polyval(p,4,derivative=True) == (253, 190) def test_polyroots(): p = polyroots([1,-4]) assert p[0].ae(4) p, q = polyroots([1,2,3]) assert p.ae(-1 - sqrt(2)*j) assert q.ae(-1 + sqrt(2)*j) #this is not a real test, it only tests a specific case assert polyroots([1]) == [] pytest.raises(ValueError, lambda: polyroots([0])) def test_polyroots_legendre(): n = 64 coeffs = [11975573020964041433067793888190275875, 0, -190100434726484311252477736051902332000, 0, 1437919688271127330313741595496589239248, 0, -6897338342113537600691931230430793911840, 0, 23556405536185284408974715545252277554280, 0, -60969520211303089058522793175947071316960, 0, 124284021969194758465450309166353645376880, 0, -204721258548015217049921875719981284186016, 0, 277415422258095841688223780704620656114900, 0, -313237834141273382807123548182995095192800, 0, 297432255354328395601259515935229287637200, 0, -239057700565161140389797367947941296605600, 0, 163356095386193445933028201431093219347160, 0, -95158890516229191805647495979277603503200, 0, 47310254620162038075933656063247634556400, 0, -20071017111583894941305187420771723751200, 0, 7255051932731034189479516844750603752850, 0, -2228176940331017311443863996901733412640, 0, 579006552594977616773047095969088431600, 0, -126584428502545713788439446082310831200, 0, 23112325428835593809686977515028663000, 0, -3491517141958743235617737161547844000, 0, 431305058712550634988073414073557200, 0, -42927166660756742088912492757452000, 0, 3378527005707706553294038781836500, 0, -205277590220215081719131470288800, 0, 9330799555464321896324157740400, 0, -304114948474392713657972548576, 0, 6695289961520387531608984680, 0, -91048139350447232095702560, 0, 659769125727878493447120, 0, -1905929106580294155360, 0, 916312070471295267] with mp.workdps(3): with pytest.raises(mp.NoConvergence): polyroots(coeffs, maxsteps=5, cleanup=True, error=False, extraprec=n*10) roots = polyroots(coeffs, maxsteps=50, cleanup=True, error=False, extraprec=n*10) roots = [str(r) for r in roots] assert roots == \ ['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961', '-0.946', '-0.93', '-0.911', '-0.889', '-0.866', '-0.841', '-0.813', '-0.784', '-0.753', '-0.72', '-0.685', '-0.649', '-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402', '-0.357', '-0.311', '-0.265', '-0.217', '-0.17', '-0.121', '-0.073', '-0.0243', '0.0243', '0.073', '0.121', '0.17', '0.217', '0.265', '0.311', '0.357', '0.402', '0.446', '0.489', '0.531', '0.572', '0.611', '0.649', '0.685', '0.72', '0.753', '0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.93', '0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999'] def test_polyroots_legendre_init(): extra_prec = 100 coeffs = [11975573020964041433067793888190275875, 0, -190100434726484311252477736051902332000, 0, 1437919688271127330313741595496589239248, 0, -6897338342113537600691931230430793911840, 0, 23556405536185284408974715545252277554280, 0, -60969520211303089058522793175947071316960, 0, 124284021969194758465450309166353645376880, 0, -204721258548015217049921875719981284186016, 0, 277415422258095841688223780704620656114900, 0, -313237834141273382807123548182995095192800, 0, 297432255354328395601259515935229287637200, 0, -239057700565161140389797367947941296605600, 0, 163356095386193445933028201431093219347160, 0, -95158890516229191805647495979277603503200, 0, 47310254620162038075933656063247634556400, 0, -20071017111583894941305187420771723751200, 0, 7255051932731034189479516844750603752850, 0, -2228176940331017311443863996901733412640, 0, 579006552594977616773047095969088431600, 0, -126584428502545713788439446082310831200, 0, 23112325428835593809686977515028663000, 0, -3491517141958743235617737161547844000, 0, 431305058712550634988073414073557200, 0, -42927166660756742088912492757452000, 0, 3378527005707706553294038781836500, 0, -205277590220215081719131470288800, 0, 9330799555464321896324157740400, 0, -304114948474392713657972548576, 0, 6695289961520387531608984680, 0, -91048139350447232095702560, 0, 659769125727878493447120, 0, -1905929106580294155360, 0, 916312070471295267] roots_init = matrix(['-0.999', '-0.996', '-0.991', '-0.983', '-0.973', '-0.961', '-0.946', '-0.93', '-0.911', '-0.889', '-0.866', '-0.841', '-0.813', '-0.784', '-0.753', '-0.72', '-0.685', '-0.649', '-0.611', '-0.572', '-0.531', '-0.489', '-0.446', '-0.402', '-0.357', '-0.311', '-0.265', '-0.217', '-0.17', '-0.121', '-0.073', '-0.0243', '0.0243', '0.073', '0.121', '0.17', '0.217', '0.265', ' 0.311', '0.357', '0.402', '0.446', '0.489', '0.531', '0.572', '0.611', '0.649', '0.685', '0.72', '0.753', '0.784', '0.813', '0.841', '0.866', '0.889', '0.911', '0.93', '0.946', '0.961', '0.973', '0.983', '0.991', '0.996', '0.999', '1.0']) with mp.workdps(2*mp.dps): roots_exact = polyroots(coeffs, maxsteps=50, cleanup=True, error=False, extraprec=2*extra_prec) with pytest.raises(mp.NoConvergence): polyroots(coeffs, maxsteps=5, cleanup=True, error=False, extraprec=extra_prec) roots,err = polyroots(coeffs, maxsteps=5, cleanup=True, error=True, extraprec=extra_prec,roots_init=roots_init) assert max(matrix(roots_exact)-matrix(roots).apply(abs)) < err roots1,err1 = polyroots(coeffs, maxsteps=25, cleanup=True, error=True, extraprec=extra_prec,roots_init=roots_init[:60]) assert max(matrix(roots_exact)-matrix(roots1).apply(abs)) < err1 def test_pade(): one = mpf(1) mp.dps = 20 N = 10 a = [one] k = 1 for i in range(1, N+1): k *= i a.append(one/k) p, q = pade(a, N//2, N//2) for x in arange(0, 1, 0.1): r = polyval(p[::-1], x)/polyval(q[::-1], x) assert(r.ae(exp(x), 1.0e-10)) mp.dps = 15 def test_fourier(): mp.dps = 15 c, s = fourier(lambda x: x+1, [-1, 2], 2) #plot([lambda x: x+1, lambda x: fourierval((c, s), [-1, 2], x)], [-1, 2]) assert c[0].ae(1.5) assert c[1].ae(-3*sqrt(3)/(2*pi)) assert c[2].ae(3*sqrt(3)/(4*pi)) assert s[0] == 0 assert s[1].ae(3/(2*pi)) assert s[2].ae(3/(4*pi)) assert fourierval((c, s), [-1, 2], 1).ae(1.9134966715663442) def test_differint(): mp.dps = 15 assert differint(lambda t: t, 2, -0.5).ae(8*sqrt(2/pi)/3) def test_invlap(): mp.dps = 15 t = 0.01 fp = lambda p: 1/(p+1)**2 ft = lambda t: t*exp(-t) ftt = ft(t) assert invertlaplace(fp,t,method='talbot').ae(ftt) assert invertlaplace(fp,t,method='stehfest').ae(ftt) assert invertlaplace(fp,t,method='dehoog').ae(ftt) assert invertlaplace(fp,t,method='cohen').ae(ftt) t = 1.0 ftt = ft(t) assert invertlaplace(fp,t,method='talbot').ae(ftt) assert invertlaplace(fp,t,method='stehfest').ae(ftt) assert invertlaplace(fp,t,method='dehoog').ae(ftt) assert invertlaplace(fp,t,method='cohen').ae(ftt) t = 0.01 fp = lambda p: log(p)/p ft = lambda t: -euler-log(t) ftt = ft(t) assert invertlaplace(fp,t,method='talbot').ae(ftt) assert invertlaplace(fp,t,method='stehfest').ae(ftt) assert invertlaplace(fp,t,method='dehoog').ae(ftt) assert invertlaplace(fp,t,method='cohen').ae(ftt) t = 1.0 ftt = ft(t) assert invertlaplace(fp,t,method='talbot').ae(ftt) assert invertlaplace(fp,t,method='stehfest').ae(ftt) assert invertlaplace(fp,t,method='dehoog').ae(ftt) assert invertlaplace(fp,t,method='cohen').ae(ftt)