SciPy

# needed to display the graphs
%matplotlib inline
from pylab import *

from numpy import *
from scipy.integrate import quad, dblquad, tplquad

# integration
val, abserr = quad(lambda x: exp(-x ** 2),  Inf, Inf)
val, abserr

from scipy.integrate import odeint, ode

# differential equation
def dy(y, t, zeta, w0):
    x, p = y[0], y[1]
    
    dx = p
    dp = -2 * zeta * w0 * p - w0**2 * x

    return [dx, dp]

# initial state
y0 = [1.0, 0.0]

# time coodinate to solve the ODE for
t = linspace(0, 10, 1000)
w0 = 2*pi*1.0

# solve the ODE problem for three different values of the damping ratio
y1 = odeint(dy, y0, t, args=(0.0, w0)) # undamped
y2 = odeint(dy, y0, t, args=(0.2, w0)) # under damped
y3 = odeint(dy, y0, t, args=(1.0, w0)) # critial damping
y4 = odeint(dy, y0, t, args=(5.0, w0)) # over damped

fig, ax = subplots()
ax.plot(t, y1[:,0], 'k', label="undamped", linewidth=0.25)
ax.plot(t, y2[:,0], 'r', label="under damped")
ax.plot(t, y3[:,0], 'b', label=r"critical damping")
ax.plot(t, y4[:,0], 'g', label="over damped")
ax.legend();

from scipy.fftpack import *

# fourier transform
N = len(t)
dt = t[1]-t[0]

# calculate the fast fourier transform
# y2 is the solution to the under-damped oscillator from the previous section
F = fft(y2[:,0]) 

# calculate the frequencies for the components in F
w = fftfreq(N, dt)

fig, ax = subplots(figsize=(9,3))
ax.plot(w, abs(F));

A = array([[3,1], [1,2]])
b = array([9,8])

# solve a system of linear equations
x = solve(A, b)
x

# eigenvalues and eigenvectors
A = rand(3,3)
B = rand(3,3)

evals, evecs = eig(A)

evals

evecs

svd(A)

from scipy import optimize

def f(x):
    return 4*x**3 + (x-2)**2 + x**4

fig, ax  = subplots()
x = linspace(-5, 3, 100)
ax.plot(x, f(x));

x_min = optimize.fmin_bfgs(f, -0.5)
x_min

from scipy import stats

# create a (continous) random variable with normal distribution
Y = stats.norm()

x = linspace(-5,5,100)

fig, axes = subplots(3,1, sharex=True)

# plot the probability distribution function (PDF)
axes[0].plot(x, Y.pdf(x))

# plot the commulative distributin function (CDF)
axes[1].plot(x, Y.cdf(x));

# plot histogram of 1000 random realizations of the stochastic variable Y
axes[2].hist(Y.rvs(size=1000), bins=50);

Y.mean(), Y.std(), Y.var()

# t-test example
t_statistic, p_value = stats.ttest_ind(Y.rvs(size=1000), Y.rvs(size=1000))
t_statistic, p_value