Sympy ValueError：第一个变量不能是数字（在函数中）

Question

这是牛顿方程求解的简单代码，旨在找到复数根。

代码应该像这样工作：

导入库：代码导入 sympy 库来处理符号数学。

定义多项式：使用 sympy 库定义多项式方程 p(x)=x5+11x4−21x3−10x2−21x−5。

牛顿法函数：newton_method 函数实现牛顿法以在给定公差内查找函数的根。

求实根：它使用循环应用具有不同初始猜测的牛顿法来求多项式方程的实根。

减少多项式：reduce_polynomial 函数将多项式除以 (x−root) 以减少多项式的次数。

求复数根：利用获得的实根来约简多项式，并通过求解约简多项式来找到任意复数根。

输出：最后，它打印使用牛顿法和归约技术找到的实根和复根。

此代码利用牛顿法求实根，然后根据获得的实根简化多项式以求任意复数根。调整容差和初始猜测会影响不同多项式的方法的准确性和收敛性。

import sympy as sp

# Define the variable
x = sp.symbols('x')

# Define the polynomial
p = x**5 + 11*x**4 - 21*x**3 - 10*x**2 - 21*x - 5

# Find the real roots using Newton's method
tolerance = 1e-5  # Tolerance for approximation
roots = []

# Function for Newton's method
def newton_method(f, x0, tol, max_iter=100):
    x = x0
    iteration = 0
    while iteration < max_iter:
        x_next = x - f.subs(x, x) / f.diff(x).subs(x, x)
        if abs(x - x_next) < tol:
            return x_next
        x = x_next
        iteration += 1
    return None

# Find real roots
for i in range(5):
    root = newton_method(p, i, tolerance)
    if root is not None:
        roots.append(root)

print("Real roots:", roots)

# Reduce the polynomial to a lower degree
# This reduces the polynomial by dividing it by (x - root)
def reduce_polynomial(poly, root):
    return sp.simplify(poly / (x - root))

# Find complex roots
complex_roots = []
for root in roots:
    reduced_poly = reduce_polynomial(p, root)
    complex_root = sp.solve(reduced_poly, x)
    complex_roots.extend(complex_root)

print("Complex roots:", complex_roots)

**但是，我遇到了这个常见错误： **

---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-3-beb367549ec5> in <cell line: 27>()
     26 # Find real roots
     27 for i in range(5):
---> 28     root = newton_method(p, i, tolerance)
     29     if root is not None:
     30         roots.append(root)

3 frames
<ipython-input-3-beb367549ec5> in newton_method(f, x0, tol, max_iter)
     17     iteration = 0
     18     while iteration < max_iter:
---> 19         x_next = x - f.subs(x, x) / f.diff(x).subs(x, x)
     20         if abs(x - x_next) < tol:
     21             return x_next

/usr/local/lib/python3.10/dist-packages/sympy/core/expr.py in diff(self, *symbols, **assumptions)
   3584     def diff(self, *symbols, **assumptions):
   3585         assumptions.setdefault("evaluate", True)
-> 3586         return _derivative_dispatch(self, *symbols, **assumptions)
   3587 
   3588     ###########################################################################

/usr/local/lib/python3.10/dist-packages/sympy/core/function.py in _derivative_dispatch(expr, *variables, **kwargs)
   1907         from sympy.tensor.array.array_derivatives import ArrayDerivative
   1908         return ArrayDerivative(expr, *variables, **kwargs)
-> 1909     return Derivative(expr, *variables, **kwargs)
   1910 
   1911 

/usr/local/lib/python3.10/dist-packages/sympy/core/function.py in __new__(cls, expr, *variables, **kwargs)
   1294             if isinstance(v, Integer):
   1295                 if i == 0:
-> 1296                     raise ValueError("First variable cannot be a number: %i" % v)
   1297                 count = v
   1298                 prev, prevcount = variable_count[-1]

ValueError: First variable cannot be a number: 0```


I searched for value error in other post but problem unsolved.

Answer 1

x 是包含

sympy.symbol

的顶级变量，但在

newton_method

内部 x 被重新声明为包含浮点值

x = x0

的局部变量，因此它不再是符号，因此

f.subs(x, x)

无法工作，

f.diff(x).subs(x, x)

也无法工作还有。

def newton_method(f, x0, tol, max_iter=100):
    iteration = 0
    while iteration < max_iter:
        x_next = (x0 - f.subs(x, x0) / f.diff(x).subs(x, x0)).evalf()
        if abs(x0 - x_next) < tol:
            return x_next
        x0 = x_next
        iteration += 1
    return None