Scipy exercises

2018-06-01

Scipy Exercises

Exercise 10.1: Least squares Generate matrix A ∈ Rm×n with m > n. Also generate some vector b ∈ Rm. Now ﬁnd x = argmin||Ax−b||2. Print the norm of the residual.

from scipy.optimize import leastsq
import numpy as np

m = 20
n = 10

def func(p,x):
	s = np.random.rand(m)
	for i in range(m):
		tmp = 0.0
		for j in range(n):
			tmp += x[i][j]*p[j]
		s[i] = tmp
	
	return s

def err(p,A,y):
	e = (func(p,A)-y)
	return e
	
A = np.random.rand(m,n)
b = np.random.rand(m)
p = np.random.rand(n)

x = np.linalg.solve(A.T@A,A.T@b)
print(x)
rst = leastsq(err,p,args=(A,b))
x = rst[0]
print(x)

#通过比较np.linalg.solve()和scipy.optimize.leastsq(),两者所得的解基本上一样

print(np.sum((err(x,A,b)**2)/m))    #计算残差

Exercise 10.2: Optimization Find the maximum of the function f(x) = sin^2(x-2)e^{-x^2}

import math
from scipy.optimize import minimize_scalar

func = lambda x: -((math.sin(x-2))**2)*math.e**(-(x**2))
rst = minimize_scalar(func)
print(-rst['fun'])
``` 

* 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.

```python
import numpy as np
from scipy.spatial import distance

X = np.random.randn(10,8)
dm = distance.pdist(X,metric='euclidean')
print(dm)