In [10]:
from demos.setup import np, plt
from compecon.quad import qnwlogn
from compecon.tools import nodeunif
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
%matplotlib inline
Univariate Taylor approximation¶
For the function $$f(x) = (x+1)\exp(2x) $$
the first-order Taylor approximation is $$\hat{f}(x) = 1+3x $$
and the second-order approximation is $$\hat{f}(x) = 1+3x + 8x^2 $$
In [11]:
x = np.linspace(-1, 1, 100)
y = (x + 1) * np.exp(2 * x)
y1 = 1 + 3 * x
y2 = 1 + 3 * x + 8 * x ** 2
plt.figure(figsize=[8, 5])
plt.plot(x, y, 'k', linewidth=3, label='Function')
plt.plot(x, y1, 'b', linewidth=3, label='1st order approximation')
plt.plot(x, y2, 'r', linewidth=3, label='2nd order approximation')
plt.legend()
plt.xticks([-1, 0, 1])
plt.show()
Bivariate Taylor approximation¶
For the function $$g(x) = x_1^2\exp(-x_2) $$
the first-order Taylor approximation around $(1,0)$ is $$\hat{g}(x) = 2x_1 -x_2 - 1$$
and the second-order approximation is $$\hat{g}(x) = x_1^2 - 2x_1 x_2 +0.5x_2^2 + x_2$$
In [12]:
nplot = [101, 101]
a = [0, -1]
b = [2, 1]
x1, x2 = nodeunif(nplot, a, b)
x1.shape = nplot
x2.shape = nplot
y = np.exp(x2) * x1 ** 2
y1 = 2 * x1 - x2 - 1
y2 = x1 ** 2 - 2 * x1 * x2 + 0.5 * x2 ** 2 + x2
def newPlot(title, Y, k):
ax = fig.add_subplot(1, 3, k, projection='3d')
ax.plot_surface(x1, x2, Y, rstride=1, cstride=1, cmap=cm.coolwarm,
linewidth=0, antialiased=False)
ax.set_xlabel('$x_1$')
ax.set_ylabel('$x_2$')
ax.set_zlabel('$y$')
plt.title(title)
fig = plt.figure(figsize=[15, 6])
newPlot('Original function', y, 1)
newPlot('First-order approximation error', y1-y, 2)
newPlot('Second-order approximation error', y2-y, 3)
plt.show()
