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###############
# Authored by Weisheng Jiang
# Book 3 | From Basic Arithmetic to Machine Learning
# Published and copyrighted by Tsinghua University Press
# Beijing, China, 2022
###############
# Bk3_Ch17_03
import numpy as np
from sympy import lambdify, diff, exp, latex, simplify
from sympy.abc import x, y
import numpy as np
from matplotlib import pyplot as plt
from matplotlib import cm
num = 301; # number of mesh grids
x_array = np.linspace(-1.5,1.5,num)
y_array = np.linspace(-1.5,1.5,num)
# global mesh
xx,yy = np.meshgrid(x_array,y_array)
num_stride = 10
plt.close('all')
f_xy = exp(-x**2 - y**2)
f_xy_fcn = lambdify([x,y],f_xy)
f_xy_zz = f_xy_fcn(xx,yy)
# expansion point
x_a = -0.1
y_b = -0.2
# local mesh
x_a_array = np.linspace(x_a - 0.5,x_a + 0.5,101)
y_b_array = np.linspace(y_b - 0.5,y_b + 0.5,101)
xx_local,yy_local = np.meshgrid(x_a_array,y_b_array)
f_xy_zz_local = f_xy_fcn(xx_local,yy_local)
# expansion point
f_ab = f_xy_fcn(x_a,y_b)
#%% constant approximation
f_ab = f_xy_fcn(x_a,y_b)
fig, ax = plt.subplots(subplot_kw={'projection': '3d'})
ax.plot_wireframe(xx,yy, f_xy_zz,
color = [0.5,0.5,0.5],
rstride=num_stride, cstride=num_stride,
linewidth = 0.25)
approx_zero_order = f_ab + xx_local*0
ax.plot_wireframe(xx_local,yy_local, approx_zero_order,
color = [1,0,0],
rstride=num_stride, cstride=num_stride,
linewidth = 0.25)
ax.plot(x_a,y_b,f_ab, marker = 'x', color = 'r',
markersize = 12)
ax.set_proj_type('ortho')
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$f(x_1,x_2)$')
ax.set_xlim(xx.min(), xx.max())
ax.set_ylim(yy.min(), yy.max())
ax.set_zlim(f_xy_zz.min(), 1.5)
ax.view_init(azim=-145, elev=30)
# ax.view_init(azim=-90, elev=0)
plt.tight_layout()
ax.grid(False)
plt.show()
#%% first order approximation
df_dx = f_xy.diff(x)
df_dx_fcn = lambdify([x,y],df_dx)
df_dx_a_b = df_dx_fcn(x_a,y_b)
df_dy = f_xy.diff(y)
df_dy_fcn = lambdify([x,y],df_dy)
df_dy_a_b = df_dy_fcn(x_a,y_b)
approx_first_order = approx_zero_order + df_dx_a_b*(xx_local - x_a) + df_dy_a_b*(yy_local - y_b)
fig, ax = plt.subplots(subplot_kw={'projection': '3d'})
ax.plot_wireframe(xx,yy, f_xy_zz,
color = [0.5,0.5,0.5],
rstride=num_stride, cstride=num_stride,
linewidth = 0.25)
ax.plot_wireframe(xx_local, yy_local, approx_first_order,
color = [1,0,0],
rstride=num_stride, cstride=num_stride,
linewidth = 0.25)
ax.plot(x_a,y_b,f_ab, marker = 'x', color = 'r',
markersize = 12)
ax.set_proj_type('ortho')
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$f(x_1,x_2)$')
ax.set_xlim(xx.min(), xx.max())
ax.set_ylim(yy.min(), yy.max())
ax.set_zlim(f_xy_zz.min(), 1.5)
ax.view_init(azim=-145, elev=30)
# ax.view_init(azim=-90, elev=0)
plt.tight_layout()
ax.grid(False)
plt.show()
#%% second order approximation
d2f_dxdx = f_xy.diff(x,2)
d2f_dxdx_fcn = lambdify([x,y],d2f_dxdx)
d2f_dxdx_a_b = d2f_dxdx_fcn(x_a,y_b)
d2f_dxdy = f_xy.diff(x,y)
d2f_dxdy_fcn = lambdify([x,y],d2f_dxdy)
d2f_dxdy_a_b = d2f_dxdy_fcn(x_a,y_b)
d2f_dydy = f_xy.diff(y,2)
d2f_dydy_fcn = lambdify([x,y],d2f_dydy)
d2f_dydy_a_b = d2f_dydy_fcn(x_a,y_b)
approx_second_order = approx_first_order + (d2f_dxdx_a_b*(xx_local - x_a)**2
+ 2*d2f_dxdy_a_b*(xx_local - x_a)*(yy_local - y_b)
+ d2f_dydy_a_b*(yy_local - y_b)**2)/2
fig, ax = plt.subplots(subplot_kw={'projection': '3d'})
ax.plot_wireframe(xx,yy, f_xy_zz,
color = [0.5,0.5,0.5],
rstride=num_stride, cstride=num_stride,
linewidth = 0.25)
ax.plot_wireframe(xx_local,yy_local, approx_second_order,
color = [1,0,0],
rstride=num_stride, cstride=num_stride,
linewidth = 0.25)
ax.plot(x_a,y_b,f_ab, marker = 'x', color = 'r',
markersize = 12)
ax.set_proj_type('ortho')
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$f(x_1,x_2)$')
ax.set_xlim(xx.min(), xx.max())
ax.set_ylim(yy.min(), yy.max())
ax.set_zlim(f_xy_zz.min(), 1.5)
ax.view_init(azim=-145, elev=30)
# ax.view_init(azim=-90, elev=0)
plt.tight_layout()
ax.grid(False)
plt.show()