Useful features

#### APPROXIMATION ####
import math
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

def f(x):
    return np.sin(x) + 0.5 * x


x = np.linspace(-2 * np.pi, 2 * np.pi, 50)

plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.plot(x, f(x), 'b')


#### REGRESSION ####
reg = np.polyfit(x, f(x), deg=1)
ry = np.polyval(reg, x)

plt.plot(x, f(x), 'b', label='f(x)')
plt.plot(x, ry, 'r.', label='regression')
plt.legend(loc=0)
plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')

reg = np.polyfit(x, f(x), deg=5)
ry = np.polyval(reg, x)
plt.plot(x, f(x), 'b', label='f(x)')
plt.plot(x, ry, 'r.', label='regression')
plt.legend(loc=0)
plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')

reg = np.polyfit(x, f(x), 7)
ry = np.polyval(reg, x)
plt.plot(x, f(x), 'b', label='f(x)')
plt.plot(x, ry, 'r.', label='regression')
plt.legend(loc=0)
plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')

np.allclose(f(x), ry) #checks results
np.sum((f(x) - ry) ** 2) / len(x) #MSE

matrix = np.zeros((3 + 1, len(x)))
matrix[3, :] = x ** 3
matrix[2, :] = x ** 2
matrix[1, :] = x
matrix[0, :] = 1

reg = np.linalg.lstsq(matrix.T, f(x))[0]

reg

ry = np.dot(reg, matrix)
plt.plot(x, f(x), 'b', label='f(x)')
plt.plot(x, ry, 'r.', label='regression')
plt.legend(loc=0)
plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')

matrix[3, :] = np.sin(x)
reg = np.linalg.lstsq(matrix.T, f(x))[0]
ry = np.dot(reg, matrix)
plt.plot(x, f(x), 'b', label='f(x)')
plt.plot(x, ry, 'r.', label='regression')
plt.legend(loc=0)
plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')

np.allclose(f(x), ry)
np.sum((f(x) - ry) ** 2) / len(x)


#### INTERPOLATION ####
import scipy.interpolate as spi
x = np.linspace(-2 * np.pi, 2 * np.pi, 25)

def f(x):
    return np.sin(x) + 0.5 * x

ipo = spi.splrep(x, f(x), k=1)
iy = spi.splev(x, ipo)

plt.plot(x, f(x), 'b', label='f(x)')
plt.plot(x, iy, 'r.', label='interpolation')
plt.legend(loc=0)
plt.grid(True)
plt.xlabel('x')
plt.ylabel('f(x)')

#### INTEGRATION ####
import scipy.integrate as sci
def f(x):
    return np.sin(x) + 0.5 * x
a = 0.5 # left integral limit
b = 9.5 # right integral limit
x = np.linspace(0, 10)
y = f(x)

from matplotlib.patches import Polygon
fig, ax = plt.subplots(figsize=(7, 5))
plt.plot(x, y, 'b', linewidth=2)
plt.ylim(ymin=0)
# area under the function
# between lower and upper limit
Ix = np.linspace(a, b)
Iy = f(Ix)
verts = [(a, 0)] + list(zip(Ix, Iy)) + [(b, 0)]
poly = Polygon(verts, facecolor='0.7', edgecolor='0.5')
ax.add_patch(poly)
# labels
plt.text(0.75 * (a + b), 1.5, r'$\int_a^b f(x)dx$', horizontalalignment='center', fontsize=20)
plt.figtext(0.9, 0.075, '$x$')
plt.figtext(0.075, 0.9, '$f(x)$')
ax.set_xticks((a, b))
ax.set_xticklabels(('$a$', '$b$'))
ax.set_yticks([f(a), f(b)])