Solve Non Linear Equation Numpy.

Edit: Everything is good :) This is a code which works with small values of t=20 and TR=([[30,20,12,23..],[...]]) but when I put higher values it is shown 'Expect x to be a 1-D so

Solution 1:

Here is a solution using an 1D interpolation to compute the inverse of the F(h) function. Because non standard root finding method is used, the error is not controlled, and the discretization grid have to be chosen with care. However, the interpolated inverse function can be directly computed over an array.

note: the definition of F is modified, the problem is now Solve h for F(h) = TR

import numpy as np
from scipy.interpolate import interp1d
import matplotlib.pylab as plt

# The function to inverse:
t = 10
alfa = 1.1*10**(-7)
k = 0.18
T1 = 20
Tpow = 100defF(h):
    A = np.exp(h**2*alfa*t/k**2)
    B = h**3*2/(3*np.sqrt(3))*(alfa*t)**(3/2)/k**3return -(Tpow-T1)*( 1 - A + B )

# Interpolation 
h_eval = np.linspace(40, 100, 50)   # critical step: define the discretization grid
F_inverse = interp1d( F(h_eval), h_eval, kind='cubic', bounds_error=True )


# Some random data:
TR = np.array([[13, 10, 13, 13, 13],
       [ 9, 11, 10, 12,  9],
       [13, 13, 10, 10, 13],
       [12, 11, 12,  9, 11],
       [11, 13, 13, 11, 13]])

# Compute the array h for a given array TR
h = F_inverse(TR)
print(h)

# Graph to verify the interpolation 
plt.plot(h_eval, F(h_eval), '.-', label='discretized F(h)');
plt.plot(h.ravel(), TR.ravel(), 'or', label='interpolated values')
plt.xlabel('h'); plt.ylabel('F(h) or TR'); plt.legend();

With the other function, the following lines are changed:

from scipy.special import erf
defF(h):
    return (Tpow-T1)*(1-np.exp((h**2*alfa*t)/k**2)*(1.0-erf(h*np.sqrt(alfa*t)/k)))

# Interpolation 
h_eval = np.linspace(15, 35, 50)   # the range is changed