用python求解非线性超定系统

Question

我正试图找到一个用Python解决一个非线性超定系统的好方法。我在这里查看了优化工具http://docs.cipy.org/doc/scipy/reference/optimize.nonlin.html，但我不知道如何使用它们。到目前为止我所掌握的是

#overdetermined nonlinear system that I'll be using
'''
a = cos(x)*cos(y)                           
b = cos(x)*sin(y)                           
c = -sin(y)                                   
d = sin(z)*sin(y)*sin(x) + cos(z)*cos(y)    
e = cos(x)*sin(z)                           
f = cos(z)*sin(x)*cos(z) + sin(z)*sin(x)    
g = cos(z)*sin(x)*sin(y) - sin(z)*cos(y)    
h = cos(x)*cos(z)
a-h will be random int values in the range 0-10 inclusive
'''
import math
from random import randint
import scipy.optimize

def system(p):
    x, y, z = p
    return(math.cos(x)*math.cos(y)-randint(0,10),
           math.cos(x)*math.sin(y)-randint(0,10),
           -math.sin(y)-randint(0,10),
           math.sin(z)*math.sin(y)*math.sin(x)+math.cos(z)*math.cos(y)-randint(0,10),
           math.cos(x)*math.sin(z)-randint(0,10),
           math.cos(z)*math.sin(x)*math.cos(z)+math.sin(z)*math.sin(x)-randint(0,10),
           math.cos(z)*math.sin(x)*math.sin(y)-math.sin(z)*math.cos(y)-randint(0,10),
           math.cos(x)*math.cos(z)-randint(0,10))

x = scipy.optimize.broyden1(system, [1,1,1], f_tol=1e-14)

你能帮我一下吗？

Answer 1

如果我没理解错的话，你想找到非线性方程组f(x)=b的近似解，其中b是包含随机值b=[a,...,h]的向量。

为此，您首先需要从system函数中删除随机值，否则在每次迭代中，求解器将尝试求解不同的方程组。而且，我认为基本的Broyden方法只对一个未知量和方程一样多的系统起作用。或者，您可以使用scipy.optimize.leastsq。一个可能的解决方案如下所示：

# I am using numpy because it's more convenient for the generation of
# random numbers.
import numpy as np
from numpy.random import randint
import scipy.optimize

# Removed random right-hand side values and changed nomenclature a bit.
def f(x):
    x1, x2, x3 = x
    return np.asarray((math.cos(x1)*math.cos(x2),
                       math.cos(x1)*math.sin(x2),
                       -math.sin(x2),
                       math.sin(x3)*math.sin(x2)*math.sin(x1)+math.cos(x3)*math.cos(x2),
                       math.cos(x1)*math.sin(x3),
                       math.cos(x3)*math.sin(x1)*math.cos(x3)+math.sin(x3)*math.sin(x1),
                       math.cos(x3)*math.sin(x1)*math.sin(x2)-math.sin(x3)*math.cos(x2),
                       math.cos(x1)*math.cos(x3)))

# The second parameter is used to set the solution vector using the args
# argument of leastsq.
def system(x,b):
    return (f(x)-b)

b = randint(0, 10, size=8)
x = scipy.optimize.leastsq(system, np.asarray((1,1,1)), args=b)[0]

我希望这对你有帮助。但是，请注意，您找到解决方案的可能性极小，特别是当您在区间[0,10]中生成随机整数，而f的范围被限制在[-2,2]时