SymPy 不评估指数 function 与潜艇

Question

I'm attempting to write an implementation of the multivariate Newton-Raphson method, in Python . To get started, I'm trying to solve the system:

10 * x + 3 * y * y - 3
x * x - exp(y) - 2

Although it is my intent to expand this to any arbitrary mxn system (assuming a solution is possible/exists, of course).

My code is as follows:

from dataclasses import dataclass
from sympy import *

x, y, z = symbols('x y z')


@dataclass
class Solve:

    @staticmethod
    def newton_raphson():

        F = Matrix([10 * x + 3 * y * y - 3, x * x - exp(y) - 2])
        v = Matrix([x, y])
        print(J := F.jacobian(v))
        print(J * -1)

        xx, yy = 0, 0
        for i in range(10):
            A = J.subs({"x": xx, "y": yy})
            b = F.subs({"x": xx, "y": yy})

            update = linsolve((A, b), [x, y])

            (xx, yy) = tuple(*update) # should be adding, not setting equal to

            print(xx, yy)




mySolver = Solve
mySolver.newton_raphson()

My issue being that, when I perform .subs() on the matrices F and J , everything but the exponential function evaluates. That is, the line print(xx, yy) prints:

3*(573 - 50*exp(3))/(20*(27 - 25*exp(3))) 5*(13 + 20*exp(3))/(4*(-27 + 25*exp(3)))

(-28650000*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 9) - 7500000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 9) - 53550000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 6) - 105598890*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6))) + 92498679*exp(5*(351 + 215*exp(3))/(4*(-1350*exp(3) + 729 + 625*exp(6)))) + 164313360*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 3) + 15235500*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 6) + 164358750*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 3))/(-81000000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 6) - 107010000*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 6) - 31492800*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6))) + 72404280*exp(5*(351 + 215*exp(3))/(4*(-1350*exp(3) + 729 + 625*exp(6)))) + 38032200*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 3) + 87480000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 3) + 9000000*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 9) + 25000000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 9)) (-16200000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 6) - 40086000*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 6) - 68751693*exp(5*(351 + 215*exp(3))/(4*(-1350*exp(3) + 729 + 625*exp(6)))) - 6298560*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6))) + 17496000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 3) + 68874390*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 3) + 10450000*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 9) + 5000000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 9))/(8*(-2025000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 6) - 2675250*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 6) - 787320*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6))) + 1810107*exp(5*(351 + 215*exp(3))/(4*(-1350*exp(3) + 729 + 625*exp(6)))) + 950805*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 3) + 2187000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 3) + 225000*exp(1755/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 1075*exp(3)/(4*(-1350*exp(3) + 729 + 625*exp(6))) + 9) + 625000*exp(625*exp(6)/(-1350*exp(3) + 729 + 625*exp(6)) + 9)))

...etc

While I need a strictly numerical value, for updating the guesses xx and yy .

Answer 1

Sympy is specialized in symbolic evaluation. exp(3) is a irrational number, and thus can not be represented numerically. Sympy knows this.

To force a numerical approximation, you can use the N function from sympy, as in

A = N(J.subs({"x": xx, "y": yy}))