我的 function 没有迭代循环，我找不到问题

Question

I'm trying to create a Function that does the Newton-Raphson root finding for multiple variables (dx = -Jinv^-1 * F), but it won't iterate over a loop continuously (ie output won't become the next input for next iteration). Does anyone see the issue?

PS. I'm trying to code it without using any of the built in matrix functions so those solutions won't apply here.

# Multivariable Root Finding via Newton Raphson


# import necessary functions
import numpy as np

# define array function for f1 and f2
x = np.arange(-3,3,.01)

def f1(x):
    return 2*(x[0]**3) - 6*(x[0]*(x[1]**2)) - 1

def f2(x):
    return -2*(x[1]**3) + 6*(x[1]*(x[0]**2)) + 3

# define matrix F(x) 2x1
def F(x_vec): 
    return [f1(x_vec),f2(x_vec)]

#print(F([2,3]))


# Create Inverse Jacobian 2x2

# InvJacobian Upper Left
def JUL(x):
    return (6*x[0]**2 - 6*x[1]**2)/((6*x[0]**2 - 6*x[1]**2)**2 + 144*(x[0]**2)*(x[1]**2))

# InvJacobian Upper Right
def JUR(x):
    return (12*x[0]*x[1])/((6*x[0]**2 - 6*x[1]**2)**2 + 144*(x[0]**2)*(x[1]**2))

# InvJacobian Lower Left
def JLL(x):
    return (-12*x[0]*x[1])/((6*x[0]**2 - 6*x[1]**2)**2 + 144*(x[0]**2)*(x[1]**2))

# InvJacobian Lower Left
def JLR(x):
    return (6*x[0]**2 - 6*x[1]**2)/((6*x[0]**2 - 6*x[1]**2)**2 + 144*(x[0]**2)*(x[1]**2))

# Combine all Jacobian into Matrix
def Jinv(x_vec):
    return [JUL(x_vec),JUR(x_vec),JLL(x_vec),JLR(x_vec)]


# Create Newton Raphson Functon

def rf_newton2d(F_system, Jinv_system, x_vec0, tol, maxiter):
    # let ou be the starting array x_vec0
    ou = x_vec0
    i = 0
    err = 10**-4
    

    # F*Jinv should output a 2x1 matrix
    while err > tol and i <= maxiter:
        F_system = F(ou)
        Jinv_system = Jinv(ou)
        # w is the [0] position of the 2x1 output matrix
        w = F_system[0]*Jinv_system[0] + F_system[1]*Jinv_system[1]
    
        # z is the [1] position of the 2x1 output
        z = F_system[0]*Jinv_system[2] + F_system[1]*Jinv_system[3]
    
        # u is the output matrix of F*Jinv
        u = [w,z]
    
        # out is the new array involving u + ou
        out = [ou[0] + u[0],ou[1] +u[1]]
    
        #define nu as out
        nu = out
    
        # let nu be the ou for the next iteration
        ou = nu
    
        #run continuous iterations
        i += 1
    
        return nu

print(rf_newton2d(F([x_vec0]),Jinv([x_vec0]),[2,3],10**-5, 5))

Answer 1

return statements in a function immediately stop the function and continue running the main code from where the function was called. It looks to me like the return statement in the while loop is not supposed to be in the while loop.

Answer 2

In your final line you have the function F([x_vec0]) , this means within the function F the variable x_vec = [x_vec0] . You then pass this variable to the function f1 , so within f1 , x = [x_vec0] . Within x[1] can be found within f1 . The problem is that x will never have a second entry. If x_vec0 = [0,1] (or some other array object such as an np.ndarray ), then [x_vec0] = [[0,1]] , ie [x_vec0][0] = [0,1] and [x_vec0][1] does not exist.

I got all the information from the traceback (obviously I had to invent a value for x_vec0 ):

---------------------------------------------------------------------------
IndexError                                Traceback (most recent call last)
<ipython-input-4-d57a9d10dd94> in <module>
     80         return nu
     81 x_vec0 = np.array([1,2])
---> 82 print(rf_newton2d(F([x_vec0]),Jinv([x_vec0]),[2,3],10**-5, 5))

<ipython-input-4-d57a9d10dd94> in F(x_vec)
     16 # define matrix F(x) 2x1
     17 def F(x_vec):
---> 18     return [f1(x_vec),f2(x_vec)]
     19 
     20 #print(F([2,3]))

<ipython-input-4-d57a9d10dd94> in f1(x)
      9 
     10 def f1(x):
---> 11     return 2*(x[0]**3) - 6*(x[0]*(x[1]**2)) - 1
     12 
     13 def f2(x):

IndexError: list index out of range

Without knowing the maths of Newton-Raphson roots I can't suggest what the fix is.