127 lines
3.7 KiB
Python
127 lines
3.7 KiB
Python
from sympy.core import (Function, Pow, sympify, Expr)
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from sympy.core.relational import Relational
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from sympy.core.singleton import S
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from sympy.polys import Poly, decompose
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from sympy.utilities.misc import func_name
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from sympy.functions.elementary.miscellaneous import Min, Max
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def decompogen(f, symbol):
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"""
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Computes General functional decomposition of ``f``.
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Given an expression ``f``, returns a list ``[f_1, f_2, ..., f_n]``,
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where::
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f = f_1 o f_2 o ... f_n = f_1(f_2(... f_n))
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Note: This is a General decomposition function. It also decomposes
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Polynomials. For only Polynomial decomposition see ``decompose`` in polys.
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Examples
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========
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>>> from sympy.abc import x
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>>> from sympy import decompogen, sqrt, sin, cos
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>>> decompogen(sin(cos(x)), x)
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[sin(x), cos(x)]
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>>> decompogen(sin(x)**2 + sin(x) + 1, x)
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[x**2 + x + 1, sin(x)]
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>>> decompogen(sqrt(6*x**2 - 5), x)
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[sqrt(x), 6*x**2 - 5]
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>>> decompogen(sin(sqrt(cos(x**2 + 1))), x)
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[sin(x), sqrt(x), cos(x), x**2 + 1]
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>>> decompogen(x**4 + 2*x**3 - x - 1, x)
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[x**2 - x - 1, x**2 + x]
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"""
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f = sympify(f)
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if not isinstance(f, Expr) or isinstance(f, Relational):
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raise TypeError('expecting Expr but got: `%s`' % func_name(f))
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if symbol not in f.free_symbols:
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return [f]
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# ===== Simple Functions ===== #
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if isinstance(f, (Function, Pow)):
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if f.is_Pow and f.base == S.Exp1:
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arg = f.exp
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else:
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arg = f.args[0]
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if arg == symbol:
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return [f]
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return [f.subs(arg, symbol)] + decompogen(arg, symbol)
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# ===== Min/Max Functions ===== #
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if isinstance(f, (Min, Max)):
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args = list(f.args)
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d0 = None
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for i, a in enumerate(args):
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if not a.has_free(symbol):
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continue
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d = decompogen(a, symbol)
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if len(d) == 1:
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d = [symbol] + d
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if d0 is None:
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d0 = d[1:]
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elif d[1:] != d0:
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# decomposition is not the same for each arg:
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# mark as having no decomposition
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d = [symbol]
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break
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args[i] = d[0]
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if d[0] == symbol:
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return [f]
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return [f.func(*args)] + d0
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# ===== Convert to Polynomial ===== #
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fp = Poly(f)
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gens = list(filter(lambda x: symbol in x.free_symbols, fp.gens))
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if len(gens) == 1 and gens[0] != symbol:
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f1 = f.subs(gens[0], symbol)
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f2 = gens[0]
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return [f1] + decompogen(f2, symbol)
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# ===== Polynomial decompose() ====== #
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try:
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return decompose(f)
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except ValueError:
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return [f]
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def compogen(g_s, symbol):
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"""
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Returns the composition of functions.
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Given a list of functions ``g_s``, returns their composition ``f``,
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where:
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f = g_1 o g_2 o .. o g_n
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Note: This is a General composition function. It also composes Polynomials.
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For only Polynomial composition see ``compose`` in polys.
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Examples
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========
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>>> from sympy.solvers.decompogen import compogen
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>>> from sympy.abc import x
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>>> from sympy import sqrt, sin, cos
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>>> compogen([sin(x), cos(x)], x)
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sin(cos(x))
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>>> compogen([x**2 + x + 1, sin(x)], x)
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sin(x)**2 + sin(x) + 1
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>>> compogen([sqrt(x), 6*x**2 - 5], x)
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sqrt(6*x**2 - 5)
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>>> compogen([sin(x), sqrt(x), cos(x), x**2 + 1], x)
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sin(sqrt(cos(x**2 + 1)))
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>>> compogen([x**2 - x - 1, x**2 + x], x)
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-x**2 - x + (x**2 + x)**2 - 1
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"""
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if len(g_s) == 1:
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return g_s[0]
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foo = g_s[0].subs(symbol, g_s[1])
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if len(g_s) == 2:
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return foo
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return compogen([foo] + g_s[2:], symbol)
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