如何使 sympy 简化为零的部首表达式

问题描述

多项式 x^3 - 3x + 1 的三个(实数)根相加为 0.但是 sympy 似乎不能简化这个根的总和:

>>>从 sympy 导入 *>>>从 sympy.abc 导入 x>>>rr = real_roots(x**3 -3*x + 1)>>>总和(rr)CRootOf(x**3 - 3*x + 1, 0) + CRootOf(x**3 - 3*x + 1, 1) + CRootOf(x**3 - 3*x + 1, 2)

函数simplify 和radsimp 不能简化这个表达式.然而，最小多项式计算正确:

>>>最小多项式(总和(rr))_X

由此我们可以得出结论，总和为0.有没有一种直接的方法可以让 sympy 简化这个根的总和?

解决方案

如果可能，以下函数计算等于代数项的有理数:

import sympy as sp# 尝试将代数项简化为有理数def try_simplify_to_rational(expr):尝试:float(expr) # 表达式计算结果为实数吗?minPoly = sp.poly(sp.minimal_polynomial(expr))print('最小多项式:', minPoly)如果 len(minPoly.monoms()) == 1: # minPoly == x返回 0如果 minPoly.degree() == 1: # minPoly == a*x + ba,b = minPoly.coeffs()返回 sp.Rational(-b, a)除了类型错误:pass # 表达式不计算为实数除了 sp.polys.polyerrors.NotAlgebraic:pass # 表达式不计算为代数数除了例外作为 exc:打印(意外异常:"，str(exc))print('化简为有理数不成功')return expr # 化简不成功

查看工作示例:

x = sp.symbols('x')rr = sp.real_roots(x**3 - 3*x + 1)# 根之和等于 (-1)* x^2 的系数，这里为 0打印(sp.simplify(sum(rr)))打印(try_simplify_to_rational(sum(rr)))# ->0

在sympy issue #19726.

The three (real) roots of the polynomial x^3 - 3x + 1 sum up to 0. But sympy does not seem to be able to simplify this sum of roots:

>>> from sympy import *
>>> from sympy.abc import x
>>> rr = real_roots(x**3 -3*x + 1)
>>> sum(rr)
CRootOf(x**3 - 3*x + 1, 0) + CRootOf(x**3 - 3*x + 1, 1) + CRootOf(x**3 - 3*x + 1, 2)

The functions simplify and radsimp cannot simplify this expression. The minimal polynomial, however, is computed correctly:

>>> minimal_polymial(sum(rr))
_x

From this we can conclude that the sum is 0. Is there a direct way of making sympy simplify this sum of roots?

解决方案

The following function computes the rational number equal to an algebraic term if possible:

import sympy as sp

# try to simplify an algebraic term to a rational number
def try_simplify_to_rational(expr):
  try:
    float(expr) # does the expression evaluate to a real number?
    minPoly = sp.poly(sp.minimal_polynomial(expr))
    print('minimal polynomial:', minPoly)
    if len(minPoly.monoms()) == 1: # minPoly == x
      return 0
    if minPoly.degree() == 1: # minPoly == a*x + b
      a,b = minPoly.coeffs()
      return sp.Rational(-b, a)
  except TypeError:
    pass # expression does not evaluate to a real number
  except sp.polys.polyerrors.NotAlgebraic:
    pass # expression does not evaluate to an algebraic number
  except Exception as exc:
    print("unexpected exception:", str(exc))
  print('simplification to rational number not successful')
  return expr # simplification not successful

See the working example:

x = sp.symbols('x')
rr = sp.real_roots(x**3 - 3*x + 1)
# sum of roots equals (-1)*coefficient of x^2, here 0
print(sp.simplify(sum(rr)))
print(try_simplify_to_rational(sum(rr))) # -> 0

A more elaborate function computing also simple radical expressions is proposed in sympy issue #19726.

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