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- 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
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