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For a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method.(more...)

def cycle_length(f, x0, nmax=None, values=False): """For a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method. >>> from sympy.ntheory.generate import cycle_length This will yield successive values of i <-- func(i): >>> def iter(func, i): ... while 1: ... ii = func(i) ... yield ii ... i = ii ... A function is defined: >>> func = lambda i: (i**2 + 1) % 51 and given a seed of 4 and the mu and lambda terms calculated: >>> next(cycle_length(func, 4)) (6, 2) We can see what is meant by looking at the output: >>> n = cycle_length(func, 4, values=True) >>> list(ni for ni in n) [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] There are 6 repeating values after the first 2. If a sequence is suspected of being longer than you might wish, ``nmax`` can be used to exit early (and mu will be returned as None): >>> next(cycle_length(func, 4, nmax = 4)) (4, None) >>> [ni for ni in cycle_length(func, 4, nmax = 4, values=True)] [17, 35, 2, 5] Code modified from: http://en.wikipedia.org/wiki/Cycle_detection. """ nmax = int(nmax or 0) # main phase: search successive powers of two power = lam = 1 tortoise, hare = x0, f(x0) # f(x0) is the element/node next to x0. i = 0 while tortoise != hare and (not nmax or i < nmax): i += 1 if power == lam: # time to start a new power of two? tortoise = hare power *= 2 lam = 0 if values: yield hare hare = f(hare) lam += 1 if nmax and i == nmax: if values: return else: yield nmax, None return if not values: # Find the position of the first repetition of length lambda mu = 0 tortoise = hare = x0 for i in range(lam): hare = f(hare) while tortoise != hare: tortoise = f(tortoise) hare = f(hare) mu += 1 if mu: mu -= 1 yield lam, mu

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trailing, divisor_count, primorial, pollard_pm1 from sympy.ntheory.factor_ import smoothness, smoothness_p from sympy.ntheory.generate import cycle_length from sympy.ntheory.primetest import _mr_safe_helper, mr from sympy.ntheory.bbp_pi import pi_hex_digits

func = lambda i: (i**2 + 1) % 51 assert next(cycle_length(func, 4)) == (6, 2) assert list(cycle_length(func, 4, values=True)) == \ [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] assert next(cycle_length(func, 4, nmax=5)) == (5, None) assert list(cycle_length(func, 4, nmax=5, values=True)) == \

src/s/y/sympy-0.7.5/sympy/ntheory/tests/test_ntheory.py

**sympy**(Download)

from sympy.ntheory.residue_ntheory import _primitive_root_prime_iter from sympy.ntheory.factor_ import smoothness, smoothness_p from sympy.ntheory.generate import cycle_length from sympy.ntheory.primetest import _mr_safe_helper, mr from sympy.ntheory.bbp_pi import pi_hex_digits

func = lambda i: (i**2 + 1) % 51 assert next(cycle_length(func, 4)) == (6, 2) assert list(cycle_length(func, 4, values=True)) == \ [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] assert next(cycle_length(func, 4, nmax=5)) == (5, None) assert list(cycle_length(func, 4, nmax=5, values=True)) == \