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Book4_Ch19_Python_Codes/Bk4_Ch19_01.py

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+
+###############
+# Authored by Weisheng Jiang
+# Book 4  |  From Basic Arithmetic to Machine Learning
+# Published and copyrighted by Tsinghua University Press
+# Beijing, China, 2022
+###############
+
+
+# Bk4_Ch19_01.py
+
+import sympy
+
+#define symbolic vars, function
+x1,x2 = sympy.symbols('x1 x2')
+# f_x = 3*x1**2-5*x2**2
+f_x = x1 - x2
+
+#take the gradient symbolically
+grad_f = [sympy.diff(f_x,var) for var in (x1,x2)]
+
+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 numpy as np
+import matplotlib.pyplot as plt
+
+xx1, xx2 = np.meshgrid(np.linspace(-4,4,41),np.linspace(-4,4,41))
+
+# coarse mesh
+xx1_, xx2_ = np.meshgrid(np.linspace(-4,4,10),np.linspace(-4,4,10))
+V = grad_fcn(xx1_,xx2_)
+V_z = np.ones_like(V[1]);
+
+ff_x = f_x_fcn(xx1,xx2)
+# ff_x_ = f_x_fcn(xx1_,xx2_)
+
+color_array = np.sqrt(V[0]**2 + V[1]**2)
+l_3D_vectors = np.sqrt(V[0]**2 + V[1]**2 + V_z**2)
+
+# 3D 
+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)
+ax.contour3D(xx1, xx2, ff_x, levels = 0, colors = 'k')
+
+# plt.quiver (xx1_, xx2_, ff_x_, V[0], V[1], V_z,length = .5)
+ax.xaxis.set_ticks([])
+ax.yaxis.set_ticks([])
+ax.zaxis.set_ticks([])
+plt.xlim(-4,4)
+plt.ylim(-4,4)
+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()
+
+
+# 2D
+fig, ax = plt.subplots()
+plt.quiver (xx1_, xx2_, V[0], V[1], color_array,
+            angles='xy', scale_units='xy',scale=2,
+            edgecolor='none', facecolor= 'b')
+
+plt.contour(xx1, xx2, ff_x,20)
+plt.contour(xx1, xx2, ff_x, levels = 0, colors = 'k')
+plt.show()
+ax.set_aspect('equal')
+ax.xaxis.set_ticks([])
+ax.yaxis.set_ticks([])
+ax.set_xlabel('$x_1$')
+ax.set_ylabel('$x_2$')
+plt.tight_layout()