Learning scipy

Exercise 10.1: Least squares

Generate matrix A ∈ $R_{m×n}$ with m > n. Also generate some vector b ∈ $R_m$ .
Now find $x=arg \min_x||Ax-b||_2$.

Print the norm of the residual.

Code

import numpy as np
import scipy.linalg

m = 20
n = 10
A = np.random.rand(m, n)
b = np.random.rand(m).reshape(m, 1)

x = scipy.linalg.lstsq(A, b)[0]

print("x = "+str(x))
err = A.dot(x) - b
residual = scipy.linalg.norm(err,ord=2)
print("norm of the residual: "+str(residual))

Result

x = [[ 0.29354894]
 [ 0.3171796 ]
 [ 0.3076955 ]
 [ 0.16784761]
 [ 0.37531644]
 [-0.12455975]
 [ 0.24540134]
 [ 0.17275431]
 [-0.28511405]
 [-0.52271425]]
norm of the residual: 0.7740732857559364

Exercise 10.2: Optimization

Find the maximum of the function
$$ f(x) = sin^2(x-2)e^{-x^2}$$

Code

import numpy as np
import scipy.linalg
import scipy.optimize

f = lambda x: -scipy.sin(x-2)**2*(x-2)*scipy.exp(-(x*x))
# 由于只有取最小值的方法，因此要取反函数并对答案取反
res = scipy.optimize.fmin(f, scipy.rand(), full_output=True)
print("Max of the func: "+str(-res[1]))

Result

Optimization terminated successfully.
         Current function value: -0.000225
         Iterations: 20
         Function evaluations: 40
Max of the func: 0.0002253477813642845

Exercise 10.3: Pairwise distances

Let X be a matrix with n rows and m columns. How can you compute the pairwise distances between every two rows?

As an example application, consider n cities, and we are given their coordinates in two columns. Now we want a nice table that tells us for each two cities, how far they are apart.

Again, make sure you make use of Scipy’s functionality instead of writing your own routine.

Code

import numpy as np
import scipy.spatial.distance

m, n = 4, 3
# 4 cities, x,y,z to discribe a city
A = np.random.rand(m,n)
print("4 Cities: ")
print(A)
D = scipy.spatial.distance.pdist(A)
print("Distance between each 2 cities: ")
print(D)

Result

4 Cities:
[[0.42422873 0.17197828 0.82144308]
 [0.15882959 0.42992466 0.80128027]
 [0.7649197  0.21990926 0.38651754]
 [0.75747478 0.15525905 0.16286141]]
Distance between each 2 cities:
[0.37064751 0.55455201 0.73828332 0.7638585  0.91727634 0.23293162]