python计算牛顿迭代法

利用python的符号计算库sympy库来计算函数的雅可比矩阵和海森矩阵

from sympy import *
import numpy as np

#计算具体值
def calculateValue(Jac_or_Hess, x, isJacMac):
    result = []
    x_dic = {}

    for data in range(1, len(x)+1):
        x_dic['x{}'.format(data)] = x[data-1]
    
    if(isJacMac):
        for fun in Jac_or_Hess:
            result.append(float(fun.evalf(subs = x_dic)))
    else:
        for fun in Jac_or_Hess:
            tmp = []
            for f in fun:
                tmp.append(float(f.evalf(subs = x_dic)))
            result.append(tmp)
    
    return np.matrix(result).T

#返回雅可比矩阵
def jacobian(Y, paraSymbos):
    grad = []

    for para in paraSymbos:
        fun = diff(Y, para)
        grad.append(fun)

    return grad

#返回海森矩阵
def hessianM(Y, paraSymbos):
    hess = []
    
    for i in jacobian(Y, paraSymbos):
        hess.append(jacobian(i, paraSymbos))
        
    return hess

def coutSolve(x):
    print('[ ', end='')
    for i in x:
        print(i, end=' ')
    print(']')


x1, x2, x3, x4 = symbols("x1 x2 x3 x4")
paraSymbos = [x1, x2, x3, x4]

Y = 100*((x1**2-x2)**2)+(x1-1)**2+(x3-1)**2+90*((x3**2-x4)**2)+10.1*((x2-1)**2+(x4-1)**2)+19.8*(x2-1)*(x4-1)
x0 = [-3, -1, -3, -1]

time = 2000#最大迭代次数
err = 1e-8#误差小于此值时停止迭代

print("\n优化函数为:fx =", Y)
print("初值为： x = ", end='')
coutSolve(x0)
print("允许的最大迭代次数为:", time)
print("精度为:{}\n".format(err))

for i in range(1, time):
    print("开始第{}轮迭代....".format(i))
    grad = calculateValue( jacobian(Y, paraSymbos), x0, True) #一阶梯度即雅可比矩阵
    grad2 = calculateValue( hessianM(Y, paraSymbos), x0, False) #二阶梯度即海森矩阵
    try:
        delta = -(grad2.I) * grad
    except(np.linalg.LinAlgError):
        print("二阶导矩阵不可逆，结束迭代。")
        print("优化失败。")
        break
    print("第{}次迭代结束。x = ".format(i),end='')
    for index in range(len(x0)):
        x0[index] += delta[index, 0]
    coutSolve(x0)

    if(delta.T * delta <err):
        print("达到定义精度，停止迭代。")
        break
    print("")