302 lines
11 KiB
Python
302 lines
11 KiB
Python
import scipy
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import scipy.special as sc
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import sys
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import numpy as np
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import pytest
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from numpy.testing import assert_equal, assert_allclose
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def test_zeta():
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assert_allclose(sc.zeta(2,2), np.pi**2/6 - 1, rtol=1e-12)
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def test_zetac():
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# Expected values in the following were computed using Wolfram
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# Alpha's `Zeta[x] - 1`
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x = [-2.1, 0.8, 0.9999, 9, 50, 75]
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desired = [
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-0.9972705002153750,
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-5.437538415895550,
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-10000.42279161673,
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0.002008392826082214,
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8.881784210930816e-16,
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2.646977960169853e-23,
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]
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assert_allclose(sc.zetac(x), desired, rtol=1e-12)
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def test_zetac_special_cases():
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assert sc.zetac(np.inf) == 0
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assert np.isnan(sc.zetac(-np.inf))
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assert sc.zetac(0) == -1.5
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assert sc.zetac(1.0) == np.inf
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assert_equal(sc.zetac([-2, -50, -100]), -1)
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def test_riemann_zeta_special_cases():
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assert np.isnan(sc.zeta(np.nan))
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assert sc.zeta(np.inf) == 1
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assert sc.zeta(0) == -0.5
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# Riemann zeta is zero add negative even integers.
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assert_equal(sc.zeta([-2, -4, -6, -8, -10]), 0)
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assert_allclose(sc.zeta(2), np.pi**2/6, rtol=1e-12)
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assert_allclose(sc.zeta(4), np.pi**4/90, rtol=1e-12)
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def test_riemann_zeta_avoid_overflow():
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s = -260.00000000001
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desired = -5.6966307844402683127e+297 # Computed with Mpmath
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assert_allclose(sc.zeta(s), desired, atol=0, rtol=5e-14)
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@pytest.mark.parametrize(
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"z, desired, rtol",
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[
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## Test cases taken from mpmath with the script:
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# import numpy as np
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# import scipy.stats as stats
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# from mpmath import mp
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# # seed = np.random.SeedSequence().entropy
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# seed = 154689806791763421822480125722191067828
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# rng = np.random.default_rng(seed)
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# default_rtol = 1e-13
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# # A small point in each quadrant outside of the critical strip
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# cases = []
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# for x_sign, y_sign in [1, 1], [1, -1], [-1, 1], [-1, -1]:
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# x = x_sign * rng.uniform(2, 8)
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# y = y_sign * rng.uniform(2, 8)
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# z = x + y*1j
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# reference = complex(mp.zeta(z))
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# cases.append((z, reference, default_rtol))
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# # Moderately large imaginary part in each quadrant outside of critical strip
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# for x_sign, y_sign in [1, 1], [1, -1], [-1, 1], [-1, -1]:
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# x = x_sign * rng.uniform(2, 8)
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# y = y_sign * rng.uniform(50, 80)
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# z = x + y*1j
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# reference = complex(mp.zeta(z))
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# cases.append((z, reference, default_rtol))
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# # points in critical strip
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# x = rng.uniform(0.0, 1.0, size=5)
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# y = np.exp(rng.uniform(0, 5, size=5))
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# z = x + y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# cases.append((complex(t), reference, default_rtol))
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# z = x - y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# cases.append((complex(t), reference, default_rtol))
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# # Near small trivial zeros
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# x = np.array([-2, -4, -6, -8])
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# y = np.array([1e-15, -1e-15])
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# x, y = np.meshgrid(x, y)
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# x, y = x.ravel(), y.ravel()
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# z = x + y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# cases.append((complex(t), reference, 1e-7))
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# # Some other points near real axis
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# x = np.array([-0.5, 0, 0.2, 0.75])
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# y = np.array([1e-15, -1e-15])
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# x, y = np.meshgrid(x, y)
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# x, y = x.ravel(), y.ravel()
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# z = x + y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# cases.append((complex(t), reference, 1e-7))
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# # Moderately large real part
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# x = np.array([49.33915930750887, 50.55805244181687])
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# y = rng.uniform(20, 100, size=3)
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# x, y = np.meshgrid(x, y)
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# x, y = x.ravel(), y.ravel()
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# z = x + y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# cases.append((complex(t), reference, default_rtol))
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# # Very large imaginary part
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# x = np.array([0.5, 34.812847097948854, 50.55805244181687])
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# y = np.array([1e6, -1e6])
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# x, y = np.meshgrid(x, y)
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# x, y = x.ravel(), y.ravel()
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# z = x + y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# rtol = 1e-7 if t.real == 0.5 else default_rtol
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# cases.append((complex(t), reference, rtol))
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#
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# # Naive implementation of reflection formula suffers internal overflow
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# x = -rng.uniform(200, 300, 3)
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# y = np.array([rng.uniform(10, 30), -rng.uniform(10, 30)])
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# x, y = np.meshgrid(x, y)
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# x, y = x.ravel(), y.ravel()
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# z = x + y*1j
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# for t in z:
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# reference = complex(mp.zeta(t))
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# cases.append((complex(t), reference, default_rtol))
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#
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# A small point in each quadrant outside of the critical strip
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((3.12838509346655+7.111085974836645j),
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(1.0192654793474945+0.08795174413289127j),
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1e-13),
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((7.06791362314716-7.219497492626728j),
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(1.0020740683598117-0.006752725913243711j),
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1e-13),
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((-6.806227077655519+2.724411451005281j),
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(0.06312488213559667-0.061641496333765956j),
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1e-13),
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((-3.0170751511621026-6.3686522550665945j),
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(-0.10330747857150148-1.214541994832571j),
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1e-13),
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# Moderately large imaginary part in each quadrant outside of critical strip
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((6.133994402212294+60.03091448000761j),
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(0.9885701843417336+0.009636925981078128j),
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1e-13),
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((6.17268142822657-64.74883149743795j),
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(1.0080474225840865+0.012032804974965354j),
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1e-13),
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((-3.462191939791879+76.16258975567534j),
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(18672.072070850158+2908.5104826247184j),
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1e-13),
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((-6.955735216531752-74.75791554155748j),
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(-77672258.72276545+71625206.0401107j),
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1e-13),
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# Points in critical strip
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((0.4088038289823922+1.4596830498094384j),
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(0.3032837969400845-0.47272237994110344j),
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1e-13),
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((0.9673493951209633+4.918968547259143j),
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(0.7488756907431944+0.17281553371482428j),
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1e-13),
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((0.8692482679977754+66.6142398421354j),
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(0.5831942469552066-0.26848904799062334j),
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1e-13),
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((0.42771847720003764+21.783747851715468j),
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(0.4767032638444329+0.6898148744603123j),
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1e-13),
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((0.20479494678428956+33.17656449538932j),
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(-0.6983038977487848+0.18060923618150224j),
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1e-13),
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((0.4088038289823922-1.4596830498094384j),
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(0.3032837969400845+0.47272237994110344j),
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1e-13),
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((0.9673493951209633-4.918968547259143j),
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(0.7488756907431944-0.17281553371482428j),
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1e-13),
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((0.8692482679977754-66.6142398421354j),
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(0.5831942469552066+0.26848904799062334j),
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1e-13),
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((0.42771847720003764-21.783747851715468j),
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(0.4767032638444329-0.6898148744603123j),
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1e-13),
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((0.20479494678428956-33.17656449538932j),
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(-0.6983038977487848-0.18060923618150224j),
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1e-13),
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# Near small trivial zeros
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((-2+1e-15j), (3.288175809370978e-32-3.0448457058393275e-17j), 1e-07),
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((-4+1e-15j), (-2.868707923051182e-33+7.983811450268625e-18j), 1e-07),
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((-6+1e-15j), (-1.7064292323640116e-34-5.8997591435159376e-18j), 1e-07),
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((-8+1e-15j), (2.5060859548261706e-33+8.316161985602247e-18j), 1e-07),
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((-2-1e-15j), (3.288175809371319e-32+3.0448457058393275e-17j), 1e-07),
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((-4-1e-15j), (-2.8687079230520114e-33-7.983811450268625e-18j), 1e-07),
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((-6-1e-15j), (-1.70642923235801e-34+5.8997591435159376e-18j), 1e-07),
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((-8-1e-15j), (2.5060859548253293e-33-8.316161985602247e-18j), 1e-07),
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# Some other points near real axis
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((-0.5+1e-15j), (-0.20788622497735457-3.608543395999408e-16j), 1e-07),
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(1e-15j, (-0.5-9.189384239689193e-16j), 1e-07),
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((0.2+1e-15j), (-0.7339209248963406-1.4828001150329085e-15j), 1e-07),
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((0.75+1e-15j), (-3.4412853869452227-1.5924832114302393e-14j), 1e-13),
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((-0.5-1e-15j), (-0.20788622497735457+3.608543395999408e-16j), 1e-07),
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(-1e-15j, (-0.5+9.189387416062746e-16j), 1e-07),
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((0.2-1e-15j), (-0.7339209248963406+1.4828004007675122e-15j), 1e-07),
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((0.75-1e-15j), (-3.4412853869452227+1.5924831974403957e-14j), 1e-13),
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# Moderately large real part
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((49.33915930750887+53.213478698903955j),
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(1.0000000000000009+1.0212452494616078e-15j),
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1e-13),
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((50.55805244181687+53.213478698903955j),
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(1.0000000000000004+4.387394180390787e-16j),
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1e-13),
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((49.33915930750887+40.6366015728302j),
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(0.9999999999999986-1.502268709924849e-16j),
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1e-13),
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((50.55805244181687+40.6366015728302j),
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(0.9999999999999994-6.453929613571651e-17j),
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1e-13),
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((49.33915930750887+85.83555435273925j),
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(0.9999999999999987-2.7014400611995846e-16j),
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1e-13),
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((50.55805244181687+85.83555435273925j),
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(0.9999999999999994-1.160571605555322e-16j),
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1e-13),
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# Very large imaginary part
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((0.5+1e6j), (0.0760890697382271+2.805102101019299j), 1e-07),
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((34.812847097948854+1e6j),
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(1.0000000000102545+3.150848654056419e-11j),
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1e-13),
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((50.55805244181687+1e6j),
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(1.0000000000000002+5.736517078070873e-16j),
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1e-13),
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((0.5-1e6j), (0.0760890697382271-2.805102101019299j), 1e-07),
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((34.812847097948854-1e6j),
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(1.0000000000102545-3.150848654056419e-11j),
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1e-13),
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((50.55805244181687-1e6j),
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(1.0000000000000002-5.736517078070873e-16j),
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1e-13),
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((-294.86605461349745+13.992648136816397j), (-np.inf+np.inf*1j), 1e-13),
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((-294.86605461349745-16.147667799398363j), (np.inf-np.inf*1j), 1e-13),
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]
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)
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def test_riemann_zeta_complex(z, desired, rtol):
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assert_allclose(sc.zeta(z), desired, rtol=rtol)
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# Some of the test cases below fail for intel compilers
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cpp_compiler = scipy.__config__.CONFIG["Compilers"]["c++"]["name"]
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gcc_linux = cpp_compiler == "gcc" and sys.platform == "linux"
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clang_macOS = cpp_compiler == "clang" and sys.platform == "darwin"
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@pytest.mark.skipif(
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not (gcc_linux or clang_macOS),
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reason="Underflow may not be avoided on other platforms",
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)
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@pytest.mark.parametrize(
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"z, desired, rtol",
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[
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# Test cases generated as part of same script for
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# test_riemann_zeta_complex. These cases are split off because
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# they fail on some platforms.
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#
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# Naive implementation of reflection formula suffers internal overflow
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((-217.40285743524163+13.992648136816397j),
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(-6.012818500554211e+249-1.926943776932387e+250j),
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5e-13,),
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((-237.71710702931668+13.992648136816397j),
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(-8.823803086106129e+281-5.009074181335139e+281j),
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1e-13,),
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((-217.40285743524163-16.147667799398363j),
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(-5.111612904844256e+251-4.907132127666742e+250j),
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5e-13,),
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((-237.71710702931668-16.147667799398363j),
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(-1.3256112779883167e+283-2.253002003455494e+283j),
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5e-13,),
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],
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)
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def test_riemann_zeta_complex_avoid_underflow(z, desired, rtol):
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assert_allclose(sc.zeta(z), desired, rtol=rtol)
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