(Week 13)Python-Scipy_exercises

最新推荐文章于 2023-11-17 11:45:14 发布

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本文通过三个实例介绍Scipy的应用：最小二乘法求解线性方程组、优化函数求极值及计算矩阵中行间距离。每个实例都附带了Python代码实现。

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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.

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