使用Python求解非线性超定系统

Question

我正在尝试找到一种用 python 解决非线性超定系统的好方法。我在这里研究了优化工具http://docs.scipy.org/doc/scipy/reference/optimize.nonlin.html，但我不知道如何使用它们。到目前为止我所拥有的是

#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

如果我理解正确，您想要找到非线性方程组的近似解，f(x) = b其中 b 是包含随机值 的向量b=[a,...,h]。

为此，您首先需要从函数中删除随机值system，否则在每次迭代中求解器将尝试求解不同的方程组。此外，我认为基本的布罗伊登方法仅适用于具有与方程一样多的未知数的系统。或者你可以使用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]

我希望这对您有帮助。但是，请注意，您不太可能找到解决方案，特别是当您生成区间 [0,10] 内的随机整数而 的范围f仅限于 [-2,2]时