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