code: def _initialize_ith_poly(N, factor_base, i, g, B):
      """Initialization stage of ith poly. After we finish sieving 1`st polynomial
      here we quickly change to the next polynomial from which we will again
      start sieving. Suppose we generated ith sieve polynomial and now we
      want to generate (i + 1)th polynomial, where ``1 <= i <= 2**(j - 1) - 1``
      where `j` is the number of prime factors of the coefficient `a`
      then this function can be used to go to the next polynomial. If
      ``i = 2**(j - 1) - 1`` then go to _initialize_first_polynomial stage.
  
      Parameters
      ==========
  
      N : number to be factored
      factor_base : factor_base primes
      i : integer denoting ith polynomial
      g : (i - 1)th polynomial
      B : array that stores a//q_l*gamma
      """
      from sympy.functions.elementary.integers import ceiling
  
      v = 1
      j = i
      while j % 2 == 0:
          v += 1
          j //= 2
      if ceiling(i / (2**v)) % 2 == 1:
          neg_pow = -1
      else:
          neg_pow = 1
      b = g.b + 2 * neg_pow * B[v - 1]
      a = g.a
      g = SievePolynomial([a * a, 2 * a * b, b * b - N], a, b)
      for fb in factor_base:
          if a % fb.prime == 0:
              continue
          fb.soln1 = (fb.soln1 - neg_pow * fb.b_ainv[v - 1]) % fb.prime
          fb.soln2 = (fb.soln2 - neg_pow * fb.b_ainv[v - 1]) % fb.prime
  
      return g
  
imports: from sympy.core.random import _randint
  from sympy.external.gmpy import gcd, invert, sqrt
  from sympy.ntheory.residue_ntheory import _sqrt_mod_prime_power
  from sympy.ntheory import isprime
  from math import log, sqrt
label_desc: Use image node_id 3 for calling a global function with example usage: _initialize_ith_poly(N, factor_base, i, g, B) and returns: g