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Book4_Ch17_Python_Codes/Bk4_Ch17_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_Ch17_01.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(-2,2,20),np.linspace(-2,2,20))
+V = grad_fcn(xx1_,xx2_)
+
+
+ff_x = f_x_fcn(xx1,xx2)
+
+color_array = np.sqrt(V[0]**2 + V[1]**2)
+
+# 3D visualization
+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')
+
+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()
+
+
+# 2D visualization
+fig, ax = plt.subplots()
+
+plt.contourf(xx1, xx2, ff_x,20, cmap = 'RdBu_r')
+
+plt.quiver (xx1_, xx2_, V[0], V[1],
+            angles='xy', scale_units='xy',
+            edgecolor='none', facecolor= '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()

Book4_Ch17_Python_Codes/Bk4_Ch17_02.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_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()

Book4_Ch17_Python_Codes/Bk4_Ch17_03.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_Ch17_03.py
+
+import sympy
+from sympy import Matrix, Transpose
+import numpy as np
+from sympy.functions import exp
+import matplotlib.pyplot as plt
+
+
+def mesh_circ(c1, c2, r, num):
+    
+    theta = np.arange(0,2*np.pi+np.pi/num,np.pi/num)
+    r     = np.arange(0,r,r/num)
+    theta,r = np.meshgrid(theta,r)
+    xx1 = np.cos(theta)*r + c1
+    xx2 = np.sin(theta)*r + c2
+    
+    return xx1, xx2
+
+
+#define symbolic vars, function
+x1,x2,p1,p2 = sympy.symbols('x1 x2 p1 p2')
+
+f_x = -4*x1**2 -4*x2**2
+f_p = -4*p1**2 -4*p2**2
+
+print(f_x)
+
+#take the gradient symbolically
+grad_f = [sympy.diff(f_p,var) for var in (p1,p2)]
+print(grad_f)
+
+f_x_fcn = sympy.lambdify([x1,x2],f_x)
+
+t_x = Matrix(grad_f).T*Matrix([[x1 - p1], [x2 - p2]]) + Matrix([f_p])
+print(t_x)
+
+t_x_fcn = sympy.lambdify([x1,x2,p1,p2],t_x)
+
+#turn into a bivariate lambda for numpy
+grad_fcn = sympy.lambdify([x1,x2],grad_f)
+
+xx1, xx2 = mesh_circ(0, 0, 3, 20)
+
+# expansion point
+p1 = -1.5
+p2 = 0
+py = f_x_fcn(p1,p2)
+
+# coarse mesh
+xx1_, xx2_ = np.meshgrid(np.linspace(p1-1, p1+1, 10),
+                         np.linspace(p2-1, p2+1, 10))
+
+# quadratic surface
+ff_x = f_x_fcn(xx1,xx2)
+
+# tangent plane
+tt_x = t_x_fcn(xx1_, xx2_, p1, p2)
+
+# 3D visualization
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+
+ax.plot_wireframe(xx1, xx2, ff_x, rstride=1, 
+                  cstride=1, color = [0.5,0.5,0.5],
+                  linewidth = 0.2)
+
+ax.plot_wireframe(xx1_, xx2_, np.squeeze(tt_x), rstride=1, 
+                  cstride=1, color = [1,0,0],
+                  linewidth = 0.2)
+ax.plot(p1,p2,py,'x')
+
+ax.xaxis.set_ticks([])
+ax.yaxis.set_ticks([])
+ax.zaxis.set_ticks([])
+plt.xlim(-3,3)
+plt.ylim(-3,3)
+ax.view_init(30, -125)
+ax.set_xlabel('$x_1$')
+ax.set_ylabel('$x_2$')
+ax.set_zlabel('$f(x_1,x_2)$')