第十一周作业 Numpy习题

查阅numpy函数使用的博客：

Numpy线性代数：https://blog.csdn.net/pianoorrock/article/details/79223080

numpy基本方法总结：https://blog.csdn.net/blog_empire/article/details/39298557

生成矩阵A和B，A由随机高斯项组成，B为toeplitz矩阵

n,m值调整至10,15，便于测试

import numpy as np
import numpy.matlib
import scipy
import time
import math
from scipy.linalg import toeplitz


n = 10
m = 15
temp = np.matlib.rand(1,m)
A = np.matlib.randn(n, m)   #生成A
B = toeplitz(temp)          #生成B
print("Matrix A:")
print(A)
print("Matrix B:")
print(B)
print("\n")

Exercise_9_1

常规的矩阵运算

def Exercise_9_1(k):
    print("Exercise[9.1]:")

    print("A+A:")   #A+A
    print(A+A)

    print("AA^T:")  #AA^T
    print(A*A.T)

    print("A^T*A:") #A^T*A
    print(A.T*A)

    print("AB:")    #AB
    print(A*B)

    print("A(B-kI) for k = " + str(k))
    print(A * (B - k * np.matlib.identity(m)))
    print("\n")

Exercise_9_1(1)

Exercise_9_2

线性方程求解，用numpy.linalg.solve求解方程

def Exercise_9_2():
    print("Exercise[9.2]:")
    b = np.matlib.rand(m, 1)      #随机生成向量b
    print("vector b:")
    print(b)
    x = np.linalg.solve(B, b)#用numpy.linalg.solve求解方程 
    print("The x of B*x = b is :")
    print(x)
    print("\n")

Exercise_9_2()

Exercise_9_3

求A，B的范数和 B的奇异值的max和min

def Exercise_9_3(F):
    print("Exercise_9_3")
    print("||A||"+str(F)+" :")
    print(np.linalg.norm(A, F))  #用numpy.linalg.norm求范数，求A的F范数
    print("||B||∞ : ")
    print(np.linalg.norm(B, np.inf))#numpy的inf表示无穷，即求无穷范数
    print("singular value of B is :")
    u,s,v= np.linalg.svd(B,full_matrices=True)  #求奇异值，由svd函数进行svd分解可得
    print("max:")
    print(max(s))
    print("min:")
    print(min(s))

Exercise_9_3(2)

Exercise_9_4

用高斯项生成矩阵Z，n×n，用幂迭代法求Z的最大特征值和相应的特征向量，直到收敛为止需要多少次迭代？

def Exercise_9_4(n):
    start = time.clock()
    Z = np.matlib.randn(n, n)
    u = np.matlib.rand(n, 1)
    print("Matrix Z:")
    print(Z)
    t1 = 0
    t2 = 1
    count = 0
    while math.fabs(t1 - t2) > 1e-10:
        count = count + 1
        v = Z * u
        t1 = t2
        t2 = max(np.abs(v))
        u = v / t2
    end = time.clock();
    print("iterations count : " + str(count))
    print("max eigenvalue : "+ str(t2))
    print("corresponding eigenvalue : ")
    print(u)
    print("Time used : " + str(end - start))

Exercise_9_4(10)

此处测试n=10时，迭代次数为80（随机性很强，n=10时迭代次数也不定。。。。）

Exercise_9_5

numpy.random里的二项分布生成C矩阵

#求奇异值，由svd函数进行svd分解可得

def Exercise_9_5(n,p):
    print("n : " + str(n) + "   p : " + str(p))
    C = np.random.binomial(1,p,(n,n)) 
    print("Matrix C:")
    print(C)
    u, s, v = np.linalg.svd(C)
    print("max singular value of C is :")
    print(max(s) ) 
    
Exercise_9_5()

p越接近0.5，其奇异值越大。

Exercise_9_6

def Exercise_9_6(A, z):
    print("Matrix A :")
    print(A)
    print("z : " + str(z))
    print(A[0, np.argmin(np.abs(A-z))])
    print()

Exercise_9_6(A, z)

由提示可得，先取差值，得到矩阵A-z,取绝对值，元素则表示z到A矩阵该位置元素的距离，再利用numpy.argmin函数得到最小值下标，再取元素值即可

第十一周作业 Numpy习题

p越接近0.5，其奇异值越大。

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