优化非线性方程组 - 处理复数

Question

我正在尝试求解非线性方程组。问题是解决方案很复杂，根据 Octave/Matlab，虚部非常小。我正在尝试将其移至 python，但不幸的是我不确定我应该如何优雅地处理它。

在Octave中，我可以直接使用fsolve，然后将解通过"real"函数传递，得到数字的实部。问题是，它很容易解决它而不会返回任何错误

不幸的是，在尝试求解方程式时，在 python returns 错误中使用 numpy。这是写在Python中的方程式：

import numpy as np
from scipy.optimize import fsolve
import scipy.io as spio

params = dict()
params['cbeta'] = 0.96
params['cdelta'] = 0.1
params['calpha'] = 0.33
params['cgamma'] = 1.2
params['clambda']= 1.0
params['csigma'] = 0.8
params['etau'] = 0.0

def steady_s(vars0):
    # unpacking paramters
    cbeta = params['cbeta']
    cdelta = params['cdelta']
    calpha = params['calpha']
    cgamma = params['cgamma']
    clambda= params['clambda']
    csigma = params['csigma']

    # guesses for initial values
    c = vars0[0]
    y = vars0[1]
    k = vars0[2]
    g = vars0[3]
    r = vars0[4]

    # == functions to minimize to find steady states == #
    f = np.empty((5,))

    # HH Euler
    f[0] = (1.0/c)*cbeta*(r + 1.0 - cdelta) - (1.0+g)/c

    # Goods market clearing
    f[1] = y - c - k*(1.0 + g) + k*(1.0-cdelta)

    # Capital Market clearing
    f[2] = r - (k)**(calpha-1.0)*calpha**2.0

    # production function for final good
    f[3] = y - k**calpha

    # growth rate
    pi = (calpha - 1.0) * k**calpha #small pi, this isnt actual profits
    f[4] = g - (cgamma - 1.0) * clambda * (csigma*clambda*pi)**(csigma/(1.0-csigma))

    return f

# == Initial Guesses == #
vars0 = np.ones((5,))

# == Solving for Steady State == #

xss = fsolve(steady_s, vars0)

在 Octave 中实现同样的事情给出了这个解决方案：

 Columns 1 through 3:

   0.7851388 + 0.0000000i   0.8520544 + 0.0000000i   0.6155938 + 0.0000000i

 Columns 4 and 5:

   0.0087008 - 0.0000000i   0.1507300 - 0.0000000i

我通过 Octave 中的 "real" 函数传递此解决方案，以获得我想要的结果。

特别是python连一次方程式都解不出来。特别是如果我尝试运行 f[4] 在定义了所有参数的函数之外，它 returns 一个 nan 值。

如有任何帮助，我们将不胜感激！

提前向我道歉 missed/formatted-badly。

Answer 1

的确，scipy 很难处理复数。但是，一个名为 mpmath 的项目可以解决您的问题。此处：http://mpmath.org/. It used to come with sympy (sympy.org). You can find documentation here：此解决方案适用于我：

from mpmath import findroot
import numpy as np
import scipy.io as spio

params = dict()
params['cbeta'] = 0.96
params['cdelta'] = 0.1
params['calpha'] = 0.33
params['cgamma'] = 1.2
params['clambda']= 1.0
params['csigma'] = 0.8
params['etau'] = 0.0

def steady_s(c,y,k,g,r):
    # unpacking paramters
    cbeta = params['cbeta']
    cdelta = params['cdelta']
    calpha = params['calpha']
    cgamma = params['cgamma']
    clambda= params['clambda']
    csigma = params['csigma']

    # guesses for initial values
    #c = vars0[0]
    #y = vars0[1]
    #k = vars0[2]
    #g = vars0[3]
    #r = vars0[4]

    # == functions to minimize to find steady states == #
    f = [0,0,0,0,0]

    # HH Euler
    f[0] = (1.0/c)*cbeta*(r + 1.0 - cdelta) - (1.0+g)/c

    # Goods market clearing
    f[1] = y - c - k*(1.0 + g) + k*(1.0-cdelta)

    # Capital Market clearing
    f[2] = r - (k)**(calpha-1.0)*calpha**2.0

    # production function for final good
    f[3] = y - k**calpha

    # growth rate
    pi = (calpha - 1.0) * k**calpha #small pi, this isnt actual profits
    f[4] = g - (cgamma - 1.0) * clambda * (csigma*clambda*pi)**(csigma/(1.0-csigma))

    return f

# == Initial Guesses == #
vars0 = list(np.ones((5,)))

# == Solving for Steady State == #
xss = findroot(steady_s, vars0)

优化非线性方程组 - 处理复数

Optimizing System of Non Linear Equations - Handling Complex Numbers

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numpy

complex-numbers