try: import mpmath as mp except ImportError: pass try: from sympy.abc import x except ImportError: pass def lagrange_inversion(a): """Given a series f(x) = a[1]*x + a[2]*x**2 + ... + a[n-1]*x**(n - 1), use the Lagrange inversion formula to compute a series g(x) = b[1]*x + b[2]*x**2 + ... + b[n-1]*x**(n - 1) so that f(g(x)) = g(f(x)) = x mod x**n. We must have a[0] = 0, so necessarily b[0] = 0 too. The algorithm is naive and could be improved, but speed isn't an issue here and it's easy to read. """ n = len(a) f = sum(a[i]*x**i for i in range(n)) h = (x/f).series(x, 0, n).removeO() hpower = [h**0] for k in range(n): hpower.append((hpower[-1]*h).expand()) b = [mp.mpf(0)] for k in range(1, n): b.append(hpower[k].coeff(x, k - 1)/k) b = [mp.mpf(x) for x in b] return b