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