Python: How calculate 2 variable nonlinear equation or plot those equation on graph in python?

Question

I hope to gain some solutions (x and y) from two nonlinear equations. So I write some code, and insert the equations, but It does not work.

As I know, The problem is generated at f2=math.acos(~~~) , that is "ValueError: math domain error" (Actually, When I erase math.acos and they show some wrong but specific solution.)

So, please I ask some help to know the way, (1) how I gain certain solution of 'f1=~', 'f2=~' as x, y. (2) how I draw some plot for 'sub_equation=~' and 'f1=~'.

I am really looking for some help. Thank you.

from scipy.optimize import fsolve
import math
import numpy as np
import matplotlib.pyplot as plt




###Input###
Angle = 120.0
length_Porpyrin =18.6
length_linker = 12.5
###parameter###
length_1 = length_Porpyrin/2.0
lenght_2 = length_linker/2.0
delta = np.pi*Angle/180.0/2.0
ramda = 30.18/180.0*np.pi
bond_angle = 2.0*np.pi/3.0
length_d = 1.35



def equations(p):
    x,y = p
    ### modified Variable ###
    atr1 = np.arctan(length_1 / x)
    atr2 = np.arctan(lenght_2 / y)
    sub_equation = ( length_d ** 2+(y/np.cos(np.arctan(lenght_2 / y))) ** 2-(x/np.cos(np.arctan(length_1 / x))) ** 2 )*np.cos(np.arctan(lenght_2 / y)) / ( 2 * length_d * y )
    ##########################
    f1 = (  (x/np.cos(np.arctan(length_1 / x))) ** 2 + (y/np.cos(np.arctan(lenght_2 / y))) ** 2 - 2 *( x/np.cos(np.arctan(length_1 / x))) * (y/np.cos(np.arctan(length_1 / x))) *  np.cos(ramda-np.arctan(length_1 / x)-np.arctan(lenght_2 / y))  ) - length_d ** 2
    f2 = math.acos(sub_equation)  -  ( bond_angle -(np.pi-np.arctan(lenght_2 / y)-delta))
    return (f1, f2)


solution = fsolve(equations, (25,25))
radius1 = solution[0]
radius2 = solution[1] 


print('[solution]')
print(solution)
print('radius1', radius1)
print('radius2', radius2)

Answer 1

I think the error might be in the fact that when you use an inverse trig function (like arccos (acos), arcsin (asin)). Some inverse trig functions have domains, and if a value that happens to be plugged in that is it out of that domain, it will result in a domain error.

Below are the domains of each inverse trig function (R = all real nums):

So the solution would be to put some kind of bounds on the parameter that can be entered into the inverse (arc) functions. Or you could try handling the exception using a try except block. Here's the Python Documentation for that: https://docs.python.org/3/tutorial/errors.html (go to section 8.3).

Answer 2

slongo has already explained what the error means: math.acos() was called with an argument either greater than 1 or smaller than -1 -- and that should never happen.

In other words: Try directly plotting the value of sub_equation -- that should always stay within [-1;1]. If does not, then there is probably something wrong, either with your definition of sub_equation or with the values of the variables you put into it.

I assume that you have an idea of what the meaning of those equations and the individual terms is, so if there is an error in the definition of sub_equation but you can't easily find it, I'd suggest looking at the individual parts of it: define and plot them separately, to see which ones are what you'd expect and which ones are not.