def test_alpha():
# Test data for the alpha_k. See DLMF 8.12.14.
with mp.workdps(30):
alpha = [mp.mpf(0), mp.mpf(1), mp.mpf(1)/3, mp.mpf(1)/36,
-mp.mpf(1)/270, mp.mpf(1)/4320, mp.mpf(1)/17010,
-mp.mpf(139)/5443200, mp.mpf(1)/204120]
mp_assert_allclose(compute_alpha(9), alpha)
def test_g():
# Test data for the g_k. See DLMF 5.11.4.
with mp.workdps(30):
g = [mp.mpf(1), mp.mpf(1)/12, mp.mpf(1)/288,
-mp.mpf(139)/51840, -mp.mpf(571)/2488320,
mp.mpf(163879)/209018880, mp.mpf(5246819)/75246796800]
mp_assert_allclose(compute_g(7), g)
def test_d():
# Compare the d_{k, n} to the results in appendix F of [1].
#
# Sources
# -------
# [1] DiDonato and Morris, Computation of the Incomplete Gamma
# Function Ratios and their Inverse, ACM Transactions on
# Mathematical Software, 1986.
with mp.workdps(50):
dataset = [(0, 0, -mp.mpf('0.333333333333333333333333333333')),
(0, 12, mp.mpf('0.102618097842403080425739573227e-7')),
(1, 0, -mp.mpf('0.185185185185185185185185185185e-2')),
(1, 12, mp.mpf('0.119516285997781473243076536700e-7')),
(2, 0, mp.mpf('0.413359788359788359788359788360e-2')),
(2, 12, -mp.mpf('0.140925299108675210532930244154e-7')),
(3, 0, mp.mpf('0.649434156378600823045267489712e-3')),
(3, 12, -mp.mpf('0.191111684859736540606728140873e-7')),
(4, 0, -mp.mpf('0.861888290916711698604702719929e-3')),
(4, 12, mp.mpf('0.288658297427087836297341274604e-7')),
(5, 0, -mp.mpf('0.336798553366358150308767592718e-3')),
(5, 12, mp.mpf('0.482409670378941807563762631739e-7')),
(6, 0, mp.mpf('0.531307936463992223165748542978e-3')),
(6, 12, -mp.mpf('0.882860074633048352505085243179e-7')),
(7, 0, mp.mpf('0.344367606892377671254279625109e-3')),
(7, 12, -mp.mpf('0.175629733590604619378669693914e-6')),
(8, 0, -mp.mpf('0.652623918595309418922034919727e-3')),
(8, 12, mp.mpf('0.377358774161109793380344937299e-6')),
(9, 0, -mp.mpf('0.596761290192746250124390067179e-3')),
(9, 12, mp.mpf('0.870823417786464116761231237189e-6'))]
d = compute_d(10, 13)
res = [d[k][n] for k, n, std in dataset]
std = map(lambda x: x[2], dataset)
mp_assert_allclose(res, std)
def test_alpha():
# Test data for the alpha_k. See DLMF 8.12.14.
with mp.workdps(30):
alpha = [mp.mpf(0), mp.mpf(1), mp.mpf(1)/3, mp.mpf(1)/36,
-mp.mpf(1)/270, mp.mpf(1)/4320, mp.mpf(1)/17010,
-mp.mpf(139)/5443200, mp.mpf(1)/204120]
mp_assert_allclose(compute_alpha(9), alpha)
def test_g():
# Test data for the g_k. See DLMF 5.11.4.
with mp.workdps(30):
g = [mp.mpf(1), mp.mpf(1)/12, mp.mpf(1)/288,
-mp.mpf(139)/51840, -mp.mpf(571)/2488320,
mp.mpf(163879)/209018880, mp.mpf(5246819)/75246796800]
mp_assert_allclose(compute_g(7), g)
def test_d():
# Compare the d_{k, n} to the results in appendix F of [1].
#
# Sources
# -------
# [1] DiDonato and Morris, Computation of the Incomplete Gamma
# Function Ratios and their Inverse, ACM Transactions on
# Mathematical Software, 1986.
with mp.workdps(50):
dataset = [(0, 0, -mp.mpf('0.333333333333333333333333333333')),
(0, 12, mp.mpf('0.102618097842403080425739573227e-7')),
(1, 0, -mp.mpf('0.185185185185185185185185185185e-2')),
(1, 12, mp.mpf('0.119516285997781473243076536700e-7')),
(2, 0, mp.mpf('0.413359788359788359788359788360e-2')),
(2, 12, -mp.mpf('0.140925299108675210532930244154e-7')),
(3, 0, mp.mpf('0.649434156378600823045267489712e-3')),
(3, 12, -mp.mpf('0.191111684859736540606728140873e-7')),
(4, 0, -mp.mpf('0.861888290916711698604702719929e-3')),
(4, 12, mp.mpf('0.288658297427087836297341274604e-7')),
(5, 0, -mp.mpf('0.336798553366358150308767592718e-3')),
(5, 12, mp.mpf('0.482409670378941807563762631739e-7')),
(6, 0, mp.mpf('0.531307936463992223165748542978e-3')),
(6, 12, -mp.mpf('0.882860074633048352505085243179e-7')),
(7, 0, mp.mpf('0.344367606892377671254279625109e-3')),
(7, 12, -mp.mpf('0.175629733590604619378669693914e-6')),
(8, 0, -mp.mpf('0.652623918595309418922034919727e-3')),
(8, 12, mp.mpf('0.377358774161109793380344937299e-6')),
(9, 0, -mp.mpf('0.596761290192746250124390067179e-3')),
(9, 12, mp.mpf('0.870823417786464116761231237189e-6'))]
d = compute_d(10, 13)
res = [d[k][n] for k, n, std in dataset]
std = map(lambda x: x[2], dataset)
mp_assert_allclose(res, std)
def test_sin(self):
with mp.workdps(30):
sincoeffs = mp.taylor(mp.sin, 0, 10)
asincoeffs = mp.taylor(mp.asin, 0, 10)
invsincoeffs = lagrange_inversion(sincoeffs)
mp_assert_allclose(invsincoeffs, asincoeffs, atol=1e-30)
def test_log(self):
with mp.workdps(30):
logcoeffs = mp.taylor(lambda x: mp.log(1 + x), 0, 10)
expcoeffs = mp.taylor(lambda x: mp.exp(x) - 1, 0, 10)
invlogcoeffs = lagrange_inversion(logcoeffs)
mp_assert_allclose(invlogcoeffs, expcoeffs)
def test_sin(self):
with mp.workdps(30):
sincoeffs = mp.taylor(mp.sin, 0, 10)
asincoeffs = mp.taylor(mp.asin, 0, 10)
invsincoeffs = lagrange_inversion(sincoeffs)
mp_assert_allclose(invsincoeffs, asincoeffs, atol=1e-30)
def test_log(self):
with mp.workdps(30):
logcoeffs = mp.taylor(lambda x: mp.log(1 + x), 0, 10)
expcoeffs = mp.taylor(lambda x: mp.exp(x) - 1, 0, 10)
invlogcoeffs = lagrange_inversion(logcoeffs)
mp_assert_allclose(invlogcoeffs, expcoeffs)