sympy方程的混合解

Question

我正在求解具有以下形式的时间相关方程（求 \lambda 的根）：

import sympy as sp
import matplotlib.pyplot as plt

t = sp.symbols(r't', real=True, positive=True)


eq = ...

print(repr(eq))
-\lambda**3 - 2*\lambda**2*t + 4*\lambda*t**4 - 8*\lambda*t**3 + 8*\lambda*t**2 - 8*\lambda*t + 4*\lambda + 8*t**3 - 16*t**2 + 8*t

求解方程并将根保存到列表中：

sol = sp.solve(eq)
e_list = [list(sol[i].values())[0] for i in range(len(sol))]

明确显示它们的演变：

x =e_list[0]
lam_x = sp.lambdify(t, x, modules=['numpy'])

x_vals = np.linspace(0.01, .9999, 1000, dtype=complex)
y_vals = np.around(lam_x(x_vals),decimals=5)

plt.plot(np.real(x_vals), np.real(y_vals),label='r1')


x =e_list[1]
lam_x = sp.lambdify(t, x, modules=['numpy'])

x_vals = np.linspace(0.01, .9999, 1000, dtype=complex)
y_vals = np.around(lam_x(x_vals),decimals=5)

plt.plot(np.real(x_vals), np.real(y_vals),label='r2')


x =e_list[2]
lam_x = sp.lambdify(t, x, modules=['numpy'])

x_vals = np.linspace(0.01, .9999, 1000, dtype=complex)
y_vals = np.around(lam_x(x_vals),decimals=5)

plt.plot(np.real(x_vals), np.real(y_vals),label='r3')
plt.legend()
plt.show()

但由于某些原因，两个第一根的解在 ~ .72 之前混合了。我不知道如何纠正这个，然后不要混合

Answer 1

Numpy 会出现这种行为，因为您要进行复杂的分支切割。在这种情况下，您应该使用 Mpmath 作为 lambdify 的评估模块，它以不同方式处理分支切割。例如：

import numpy as np
import matplotlib.pyplot as plt

x =e_list[0]
lam_x = lambdify(t, x, modules=['mpmath'])

x_vals = np.linspace(0.01, .9999, 1000, dtype=complex)
y_vals = []
for _x in x_vals:
    y_vals.append(complex(lam_x(_x)))

plt.figure()
plt.plot(np.real(x_vals), np.real(y_vals),label='r1')


x =e_list[1]
lam_x = lambdify(t, x, modules=['mpmath'])

x_vals = np.linspace(0.01, .9999, 1000, dtype=complex)

y_vals = []
for _x in x_vals:
    y_vals.append(complex(lam_x(_x)))

plt.plot(np.real(x_vals), np.real(y_vals),label='r2')


x =e_list[2]
lam_x = lambdify(t, x, modules=['numpy'])

x_vals = np.linspace(0.01, .9999, 1000, dtype=complex)
y_vals = np.around(lam_x(x_vals),decimals=5)

plt.plot(np.real(x_vals), np.real(y_vals),label='r3')
plt.legend()
plt.show()

sympy方程的混合解

Mixing solutions of sympy equation

sympy

solver