154 lines
5.0 KiB
Python
154 lines
5.0 KiB
Python
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#!/usr/bin/python
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# -*- coding: utf-8 -*-
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from mpmath import mp
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from mpmath import libmp
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xrange = libmp.backend.xrange
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# Attention:
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# These tests run with 15-20 decimal digits precision. For higher precision the
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# working precision must be raised.
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def test_levin_0():
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mp.dps = 17
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eps = mp.mpf(mp.eps)
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with mp.extraprec(2 * mp.prec):
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L = mp.levin(method = "levin", variant = "u")
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S, s, n = [], 0, 1
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while 1:
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s += mp.one / (n * n)
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n += 1
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S.append(s)
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v, e = L.update_psum(S)
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if e < eps:
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break
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if n > 1000: raise RuntimeError("iteration limit exceeded")
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eps = mp.exp(0.9 * mp.log(eps))
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err = abs(v - mp.pi ** 2 / 6)
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assert err < eps
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w = mp.nsum(lambda n: 1/(n * n), [1, mp.inf], method = "levin", levin_variant = "u")
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err = abs(v - w)
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assert err < eps
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def test_levin_1():
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mp.dps = 17
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eps = mp.mpf(mp.eps)
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with mp.extraprec(2 * mp.prec):
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L = mp.levin(method = "levin", variant = "v")
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A, n = [], 1
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while 1:
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s = mp.mpf(n) ** (2 + 3j)
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n += 1
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A.append(s)
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v, e = L.update(A)
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if e < eps:
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break
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if n > 1000: raise RuntimeError("iteration limit exceeded")
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eps = mp.exp(0.9 * mp.log(eps))
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err = abs(v - mp.zeta(-2-3j))
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assert err < eps
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w = mp.nsum(lambda n: n ** (2 + 3j), [1, mp.inf], method = "levin", levin_variant = "v")
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err = abs(v - w)
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assert err < eps
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def test_levin_2():
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# [2] A. Sidi - "Pratical Extrapolation Methods" p.373
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mp.dps = 17
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z=mp.mpf(10)
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eps = mp.mpf(mp.eps)
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with mp.extraprec(2 * mp.prec):
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L = mp.levin(method = "sidi", variant = "t")
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n = 0
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while 1:
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s = (-1)**n * mp.fac(n) * z ** (-n)
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v, e = L.step(s)
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n += 1
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if e < eps:
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break
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if n > 1000: raise RuntimeError("iteration limit exceeded")
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eps = mp.exp(0.9 * mp.log(eps))
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exact = mp.quad(lambda x: mp.exp(-x)/(1+x/z),[0,mp.inf])
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# there is also a symbolic expression for the integral:
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# exact = z * mp.exp(z) * mp.expint(1,z)
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err = abs(v - exact)
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assert err < eps
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w = mp.nsum(lambda n: (-1) ** n * mp.fac(n) * z ** (-n), [0, mp.inf], method = "sidi", levin_variant = "t")
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assert err < eps
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def test_levin_3():
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mp.dps = 17
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z=mp.mpf(2)
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eps = mp.mpf(mp.eps)
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with mp.extraprec(7*mp.prec): # we need copious amount of precision to sum this highly divergent series
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L = mp.levin(method = "levin", variant = "t")
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n, s = 0, 0
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while 1:
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s += (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n))
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n += 1
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v, e = L.step_psum(s)
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if e < eps:
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break
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if n > 1000: raise RuntimeError("iteration limit exceeded")
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eps = mp.exp(0.8 * mp.log(eps))
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exact = mp.quad(lambda x: mp.exp( -x * x / 2 - z * x ** 4), [0,mp.inf]) * 2 / mp.sqrt(2 * mp.pi)
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# there is also a symbolic expression for the integral:
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# exact = mp.exp(mp.one / (32 * z)) * mp.besselk(mp.one / 4, mp.one / (32 * z)) / (4 * mp.sqrt(z * mp.pi))
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err = abs(v - exact)
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assert err < eps
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w = mp.nsum(lambda n: (-z)**n * mp.fac(4 * n) / (mp.fac(n) * mp.fac(2 * n) * (4 ** n)), [0, mp.inf], method = "levin", levin_variant = "t", workprec = 8*mp.prec, steps = [2] + [1 for x in xrange(1000)])
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err = abs(v - w)
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assert err < eps
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def test_levin_nsum():
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mp.dps = 17
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with mp.extraprec(mp.prec):
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z = mp.mpf(10) ** (-10)
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a = mp.nsum(lambda n: n**(-(1+z)), [1, mp.inf], method = "l") - 1 / z
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assert abs(a - mp.euler) < 1e-10
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eps = mp.exp(0.8 * mp.log(mp.eps))
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a = mp.nsum(lambda n: (-1)**(n-1) / n, [1, mp.inf], method = "sidi")
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assert abs(a - mp.log(2)) < eps
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z = 2 + 1j
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f = lambda n: mp.rf(2 / mp.mpf(3), n) * mp.rf(4 / mp.mpf(3), n) * z**n / (mp.rf(1 / mp.mpf(3), n) * mp.fac(n))
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v = mp.nsum(f, [0, mp.inf], method = "levin", steps = [10 for x in xrange(1000)])
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exact = mp.hyp2f1(2 / mp.mpf(3), 4 / mp.mpf(3), 1 / mp.mpf(3), z)
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assert abs(exact - v) < eps
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def test_cohen_alt_0():
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mp.dps = 17
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AC = mp.cohen_alt()
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S, s, n = [], 0, 1
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while 1:
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s += -((-1) ** n) * mp.one / (n * n)
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n += 1
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S.append(s)
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v, e = AC.update_psum(S)
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if e < mp.eps:
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break
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if n > 1000: raise RuntimeError("iteration limit exceeded")
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eps = mp.exp(0.9 * mp.log(mp.eps))
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err = abs(v - mp.pi ** 2 / 12)
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assert err < eps
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def test_cohen_alt_1():
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mp.dps = 17
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A = []
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AC = mp.cohen_alt()
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n = 1
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while 1:
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A.append( mp.loggamma(1 + mp.one / (2 * n - 1)))
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A.append(-mp.loggamma(1 + mp.one / (2 * n)))
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n += 1
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v, e = AC.update(A)
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if e < mp.eps:
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break
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if n > 1000: raise RuntimeError("iteration limit exceeded")
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v = mp.exp(v)
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err = abs(v - 1.06215090557106)
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assert err < 1e-12
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