Solve nonlinear equation in python

Question

I am trying to find the fundamental TE mode of the dielectric waveguide. The way I try to solve it is to compute two function and try to find their intersection on graph. However, I am having trouble get the intersect point on the plot. My code:

def LHS(w):
    theta = 2*np.pi*1.455*10*10**(-6)*np.cos(np.deg2rad(w))/(900*10**(-9))
    if(theta>(np.pi/2) or theta < 0):
        y1 = 0
    else:
        y1 = np.tan(theta)
    return y1

def RHS(w):
    y = ((np.sin(np.deg2rad(w)))**2-(1.440/1.455)**2)**0.5/np.cos(np.deg2rad(w))
    return y

x = np.linspace(80, 90, 500)

LHS_vals = [LHS(v) for v in x]
RHS_vals = [RHS(v) for v in x]


# plot
fig, ax = plt.subplots(1, 1, figsize=(6,3))
ax.plot(x,LHS_vals)
ax.plot(x,RHS_vals)
ax.legend(['LHS','RHS'],loc='center left', bbox_to_anchor=(1, 0.5))

plt.ylim(0,20)
plt.xlabel('degree')
plt.ylabel('magnitude')
plt.show()

I got plot like this:

The intersect point is around 89 degree, however, I am having trouble to compute the exact value of x. I have tried fsolve, solve to find the solution but still in vain. It seems not able to print out solution if it is not the only solution. Is it possible to only find solution that x is in certain range? Could someone give me any suggestion here? Thanks!

edit: the equation is like this (m=0):

and I am trying to solve theta here by finding the intersection point

edit: One of the way I tried is as this:

from scipy.optimize import fsolve
def f(wy):
   w, y = wy
   z = np.array([y - LHS(w), y - RHS(w)])
   return z

fsolve(f,[85, 90])

However it gives me the wrong answer. I also tried something like this:

import matplotlib.pyplot as plt

x = np.arange(85, 90, 0.1)
f = LHS(x)
g = RHS(x)

plt.plot(x, f, '-')
plt.plot(x, g, '-')

idx = np.argwhere(np.diff(np.sign(f - g)) != 0).reshape(-1) + 0
plt.plot(x[idx], f[idx], 'ro')
print(x[idx])
plt.show()

But it shows: ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()

Answer 1

First, you need to make sure that the function can actually handle numpy arrays. Several options for defining piecewise functions are shown in Plot Discrete Distribution using np.linspace(). E.g.

def LHS(w):
    theta = 2*np.pi*1.455*10e-6*np.cos(np.deg2rad(w))/(900e-9)
    y1 = np.select([theta < 0, theta <= np.pi/2, theta>np.pi/2], [np.nan, np.tan(theta), np.nan])
    return y1

This already allows to use the second approach, plotting a point at the index which is closest to the minimum of the difference between the two curves.

import numpy as np
import matplotlib.pyplot as plt

def LHS(w):
    theta = 2*np.pi*1.455*10e-6*np.cos(np.deg2rad(w))/(900e-9)
    y1 = np.select([theta < 0, theta <= np.pi/2, theta>np.pi/2], [np.nan, np.tan(theta), np.nan])
    return y1

def RHS(w):
    y = ((np.sin(np.deg2rad(w)))**2-(1.440/1.455)**2)**0.5/np.cos(np.deg2rad(w))
    return y

x = np.linspace(82.1, 89.8, 5000)
f = LHS(x)
g = RHS(x)

plt.plot(x, f, '-')
plt.plot(x, g, '-')

idx = np.argwhere(np.diff(np.sign(f - g)) != 0).reshape(-1) + 0
plt.plot(x[idx], f[idx], 'ro')
plt.ylim(0,40)
plt.show()

One may then also use scipy.optimize.fsolve to get the actual solution.

idx = np.argwhere(np.diff(np.sign(f - g)) != 0)[-1]

from scipy.optimize import fsolve

h = lambda x: LHS(x)-RHS(x)
sol = fsolve(h,x[idx])

plt.plot(sol, LHS(sol), 'ro')
plt.ylim(0,40)
plt.show()

Answer 2

Something quick and (very) dirty that seems to work (at least it gave theta value of ~89 for your parameters) - add the following to your code before the figure, after RHS_vals = [RHS(v) for v in x] line:

# build a list of differences between the values of RHS and LHS
diff = [abs(r_val- l_val) for r_val, l_val in zip(RHS_vals, LHS_vals)]

# find the minimum of absolute values of the differences
# find the index of this minimum difference, then find at which angle it occured
min_diff = min(diff)
print "Minimum difference {}".format(min_diff)
print "Theta = {}".format(x[diff.index(min_diff)])

I looked in range of 85-90:

x = np.linspace(85, 90, 500)