如何用 Python 重新排列一个复杂的方程

Question

我想r使用 Python 重新排列变量的以下等式。

P = C * ((1-(1+r)**(-n)))/r + fv*(1+r)**(-n)

to

r = blabla...

我知道 sympy 与这样的重新安排任务有关。所以，我写了下面的代码。

# Solve the equation for r

import sympy

from sympy import symbols

P, C, r, n, fv = sympy.symbols('P C r n fv')

eq = sympy.Eq(P, C * ((1-1/(1+r)**n))/r + fv/(1+r)**n)

sympy.solve(eq, r)

但是，我得到了这样的错误。

NotImplementedError                       Traceback (most recent call last)

<ipython-input-47-a183add313da> in <module>

      3 P, C, r, n, fv = sympy.symbols('P C r n fv')

      4 eq = sympy.Eq(P, C * ((1-1/(1+r)**n))/r + fv/(1+r)**n)

----> 5 sympy.solve(eq, r)

~\Anaconda3\lib\site-packages\sympy\solvers\solvers.py in solve(f, *symbols, **flags)

   1169     ###########################################################################

   1170     if bare_f:

-> 1171         solution = _solve(f[0], *symbols, **flags)

   1172     else:

   1173         solution = _solve_system(f, symbols, **flags)

~\Anaconda3\lib\site-packages\sympy\solvers\solvers.py in _solve(f, *symbols, **flags)

   1740 

   1741     if result is False:

-> 1742         raise NotImplementedError('\n'.join([msg, not_impl_msg % f]))

   1743 

   1744     if flags.get('simplify', True):

NotImplementedError: multiple generators [r, (r + 1)**n]

No algorithms are implemented to solve equation -C*(1 - (r + 1)**(-n))/r + P - fv*(r + 1)**(-n)

我猜想 sympy 无法计算功率。你知道如何对方程执行这种复杂的重排吗？我正在使用 Python==3.7，sympy==1.4。

Answer 1

这不是一个简单的方程来解决。它与功率计算无关，只是方程太复杂，sympy 无法求解 r。但是，如果其他变量有特定值并且您需要求解 r（即为非平凡方程找到零），则可以使用数值求解器：nsolve# Solve the equation for rfrom sympy import var, Eq, solvevar('C, r, n, fv, P', positive = True)# this throws an error: no algorithms are implemented to solve equationequation = Eq(P, C * ((1-1/(1+r)**n))/r + fv/(1+r)**n)# a simple calculation for power works fineequation = Eq(P, (1+r)**n)solve(equation, r)

Answer 2

您要解决的方程式是：In [23]: eq&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Out[23]:&nbsp;&nbsp; &nbsp; &nbsp; ⎛&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;-n⎞&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; C⋅⎝1 - (r + 1)&nbsp; ⎠&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;-nP = ───────────────── + fv⋅(r + 1)&nbsp;&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; r&nbsp;我们可以将它重新排列成这样的多项式In [24]: eq2 = Eq(eq.lhs * (1+r)**n * r, eq.rhs * (1+r)**n * r).expand()&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;In [25]: eq2&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;Out[25]:&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;n&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; n&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;P⋅r⋅(r + 1)&nbsp; = C⋅(r + 1)&nbsp; - C + fv⋅r现在我们看到这是一个多项式，只是指数n是符号的。一般来说，这种方程不会有一个可以用封闭形式表达的解——这就是为什么 sympy 没有针对这种特殊情况的算法（它不是 sympy 本身的限制）。可以对这个方程进行数值求解，但只有当我们对每个参数都有数值时，数值求解才有效。如果我们用数字代替参数，那么nsolve可以用数字找到解决方案：In [26]: eq3 = eq.subs({P:1, C:2, fv:1, n:100})&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;In [27]: eq3&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;Out[27]:&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;⎛&nbsp; &nbsp; &nbsp; &nbsp; 1&nbsp; &nbsp; &nbsp;⎞&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;2⋅⎜1 - ──────────⎟&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;⎜&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;100⎟&nbsp; &nbsp; &nbsp; &nbsp; 1&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ⎝&nbsp; &nbsp; (r + 1)&nbsp; &nbsp;⎠1 = ────────── + ──────────────────&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;100&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;r&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp; &nbsp; (r + 1)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;In [28]: nsolve(eq3, r, 1)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;Out[28]: 2.00000000000000但请注意，此方程的解不是唯一的，例如 -2 也是这里的解：In [52]: nsolve(eq3, r, -1.9)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Out[52]: -2.00000000000000这个特殊的方程有大约 100 个根，但不一定都是实数。