(Week 13)Python-Scipy_exercises

Exercise 10.1: Least squares

Generate matrix $A ∈ R^{m×n}$ with $m > n$ . Also generate some vector $b ∈ R^m$ .

Now ﬁnd $x = argmin_x||Ax−b||_2$ .

Print the norm of the residual.

编写代码如下，调用lstsq函数求最小二乘解，再调用norm求范数即可（默认为2-范数）。

import scipy as sp
import scipy.linalg as lg
import numpy as np

def least_square():
    A = np.random.random((10, 20)) * 10
    b1 = np.random.random(10) * 5

    x = lg.lstsq(A, b1, cond=-1)[0]
    print(x)
    residual = A.dot(x) - b1
    print(lg.misc.norm(residual))

least_square()

Exercise 10.2: Optimization

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

编写代码如下，使用optimize的minimize_scalar函数求 $-f(x)$ 的最小值，再取负即可。

import scipy.optimize as op
import math

def get_max():
    fun = lambda x: math.sin(x - 2) ** 2 * math.exp(- x * x) * -1
    ans = op.minimize_scalar(fun, bounds=[-5, 5])
    print(ans.x, -ans.fun)  #取负

get_max()

运行结果如下，同图像一致。

D:\python\python.exe D:/pycharm/projects/Matplot/scipy_exercise.py
0.21624132858697098 0.9116854118471548

Process finished with exit code 0

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.

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编写代码如下：

import scipy.spatial.distance as ds
import numpy as np

def pairwise_distance():
    m, n =5, 2 # five cities represented by a 2-dim vector
    x = np.random.randint(0, 10, (m, n))
    dis_list = []
    table = {}
    for i in range(m):
        for j in range(i + 1, m):
            dis_list.append(str(i) + ' to ' + str(j))
    print(x)
    dis = ds.pdist(x, 'euclidean')
    for ci, di in zip(dis_list, dis):
        table[ci] = di
    print(table)

pairwise_distance()

执行结果如下：

D:\python\python.exe D:/pycharm/projects/Matplot/scipy_exercise.py
[[6 7]
 [2 1]
 [0 2]
 [6 5]
 [1 4]]
{'0 to 1': 7.211102550927978, '0 to 2': 7.810249675906654, '0 to 3': 2.0, '0 to 4': 5.830951894845301, '1 to 2': 2.23606797749979, '1 to 3': 5.656854249492381, '1 to 4': 3.1622776601683795, '2 to 3': 6.708203932499369, '2 to 4': 2.23606797749979, '3 to 4': 5.0990195135927845}

Process finished with exit code 0