寻找指数求和的解

Question

我正在尝试用Python在数字上求解此方程（numpy / scipy，一切可用）

在此公式中， K 是一个常数， f 和 g 是两个依赖于E计数器的术语（这是一个离散表示）整数），其中 x 是我要查找的变量。

例如，以 E 为3个术语，即：

f（E）和 g（E）也是已知的。

我从numpy上读到有关使用“ fsolve”的信息，尽管我不明白如何自动生成作为项总和的函数。我可能会手动完成，但是要花50个学期才能用完一段时间，所以我也想学习一些新知识。

Answer 1

You can use scipy.optimize.fsolve where the function is constructed using numpy.sum:

import numpy as np
import scipy.optimize

np.random.seed(123)

f = np.random.uniform(size=50)
g = np.random.uniform(size=f.size)
K = np.sum(f * np.exp(-g*np.pi))

def func(x, f, g, K):
    return np.sum(f * np.exp(-g*x), axis=0) - K

# The argument to the function is an array itself,
# so we need to introduce extra dimensions for f, g.
res = scipy.optimize.fsolve(func, x0=1, args=(f[:, None], g[:, None], K))

Note that for your specific function you can also assist the algorithm by providing the derivative of the function:

def derivative(x, f, g, K):
    return np.sum(-g*f * np.exp(-g*x), axis=0)

res = scipy.optimize.fsolve(func, fprime=derivative,
                            x0=1, args=(f[:, None], g[:, None], K))

Finding multiple roots

You can vectorize the procedure in a sense that the function accepts N inputs (for example for each of the rows) and produces N outputs (again one for each row). Hence the inputs and outputs are independent of each other and the corresponding jacobian is a diagonal matrix. Here's some sample code:

import numpy as np
import scipy.optimize

np.random.seed(123)

image = np.random.uniform(size=(4000, 3000, 2))
f, g = image[:, :, 0], image[:, :, 1]
x_original = np.random.uniform(size=image.shape[0])  # Compute for each of the rows.
K = np.sum(f * np.exp(-g * x_original[:, None]), axis=1)

def func(x, f, g, K):
    return np.sum(f * np.exp(-g*x[:, None]), axis=1) - K

def derivative(x, f, g, K):
    return np.diag(np.sum(-g*f * np.exp(-g*x[:, None]), axis=1))

res = scipy.optimize.fsolve(func, fprime=derivative,
                            x0=0.5*np.ones(x_original.shape), args=(f, g, K))
assert np.allclose(res, x_original)