From fded368d87a081f014f59f119c7a0f2babe7b781 Mon Sep 17 00:00:00 2001 From: Visualize-ML <105787223+Visualize-ML@users.noreply.github.com> Date: Sun, 24 Jul 2022 10:52:35 -0400 Subject: [PATCH] Add files via upload --- Book4_Ch17_Python_Codes/Bk4_Ch17_01.py | 78 +++++++++++++++ Book4_Ch17_Python_Codes/Bk4_Ch17_02.py | 75 +++++++++++++++ Book4_Ch17_Python_Codes/Bk4_Ch17_03.py | 89 ++++++++++++++++++ ...函数微分__数学要素__从加减乘除到机器学习.pdf | Bin 0 -> 1036659 bytes 4 files changed, 242 insertions(+) create mode 100644 Book4_Ch17_Python_Codes/Bk4_Ch17_01.py create mode 100644 Book4_Ch17_Python_Codes/Bk4_Ch17_02.py create mode 100644 Book4_Ch17_Python_Codes/Bk4_Ch17_03.py create mode 100644 Book4_Ch17_多元函数微分__数学要素__从加减乘除到机器学习.pdf diff --git a/Book4_Ch17_Python_Codes/Bk4_Ch17_01.py b/Book4_Ch17_Python_Codes/Bk4_Ch17_01.py new file mode 100644 index 0000000..d48391f --- /dev/null +++ b/Book4_Ch17_Python_Codes/Bk4_Ch17_01.py @@ -0,0 +1,78 @@ + +############### +# 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() diff --git a/Book4_Ch17_Python_Codes/Bk4_Ch17_02.py b/Book4_Ch17_Python_Codes/Bk4_Ch17_02.py new file mode 100644 index 0000000..2934eac --- /dev/null +++ b/Book4_Ch17_Python_Codes/Bk4_Ch17_02.py @@ -0,0 +1,75 @@ + +############### +# 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() diff --git a/Book4_Ch17_Python_Codes/Bk4_Ch17_03.py b/Book4_Ch17_Python_Codes/Bk4_Ch17_03.py new file mode 100644 index 0000000..e9eecde --- /dev/null +++ b/Book4_Ch17_Python_Codes/Bk4_Ch17_03.py @@ -0,0 +1,89 @@ + +############### +# 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('$f(x_1,x_2)$') diff --git a/Book4_Ch17_多元函数微分__数学要素__从加减乘除到机器学习.pdf b/Book4_Ch17_多元函数微分__数学要素__从加减乘除到机器学习.pdf new file mode 100644 index 0000000000000000000000000000000000000000..03a99e940a6dbf149a660bba38cd883fedaee13e GIT binary patch literal 1036659 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