###############
# Authored by Weisheng Jiang
# Book 4  |  From Basic Arithmetic to Machine Learning
# Published and copyrighted by Tsinghua University Press
# Beijing, China, 2022
###############

# Bk4_Ch17_02.py

import sympy
import numpy as np
from sympy.functions import exp

#define symbolic vars, function
x1,x2 = sympy.symbols('x1 x2')

f_x = x1*exp(-(x1**2 + x2**2))

print(f_x)

#take the gradient symbolically
grad_f = [sympy.diff(f_x,var) for var in (x1,x2)]
print(grad_f)

f_x_fcn = sympy.lambdify([x1,x2],f_x)

#turn into a bivariate lambda for numpy
grad_fcn = sympy.lambdify([x1,x2],grad_f)

import matplotlib.pyplot as plt

xx1, xx2 = np.meshgrid(np.linspace(-2,2,40),np.linspace(-2,2,40))

# coarse mesh
xx1_, xx2_ = np.meshgrid(np.linspace(-2,2,15),np.linspace(-2,2,15))
V = grad_fcn(xx1_,xx2_)


ff_x = f_x_fcn(xx1,xx2)
ff_x_ = f_x_fcn(xx1_,xx2_)

color_array = np.sqrt(V[0]**2 + V[1]**2)

# 3D visualization + vectors
ax = plt.figure().add_subplot(projection='3d')
ax.plot_wireframe(xx1, xx2, ff_x, rstride=1, 
                  cstride=1, color = [0.5,0.5,0.5],
                  linewidth = 0.2)
ax.contour3D(xx1, xx2, ff_x, 20, cmap = 'RdBu_r')

lengths = np.sqrt(V[0]**2+V[1]**2+1**2)

for x1,y1,z1,u1,v1,w1,l in zip(xx1_.flatten(),
                               xx2_.flatten(),
                               ff_x_.flatten(),
                               V[0].flatten(), 
                               V[1].flatten(), 
                               V[0].flatten()*0 - 1,
                               lengths.flatten()):
    
    ax.quiver(x1, y1, z1, u1, v1, w1, length=l*0.2,
              color = 'k')

ax.xaxis.set_ticks([])
ax.yaxis.set_ticks([])
ax.zaxis.set_ticks([])
plt.xlim(-2,2)
plt.ylim(-2,2)
ax.view_init(30, -125)
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$f(x_1,x_2)$')

plt.tight_layout()