Numpy

Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where A ∈ Rn×m and B ∈ Rm×m,for n = 200, m = 500.

import numpy as np
import scipy.linalg

mu, sigma = 0, 0.1
n = 200
m = 500
A = np.random.normal(mu, sigma, size=(n, m))

c = [i for i in range(m+1)]
B = scipy.linalg.toeplitz(c,c)

Exercise 9.1: Matrix operations

Calculate A + A, AA⊤, A⊤A and AB. Write a function that computes A(B − λI) for any λ.

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

print("A·A⊤:")
print(np.dot(A,A.T))

print("A⊤A:")
print(np.dot(A.T,A))

print("AB:")
print(np.dot(A,B))


def compute_func(A,B,λ):
	print("A(B − λI):")
	print(np.dot(A,B-λ*(np.eye(m, k=0))))
	
compute_func(A,B,2.0)

Exercise 9.2: Solving a linear system

Generate a vector b with m entries and solve Bx = b.

b = np.array([1 for i in range(m)])
x = np.linalg.solve(B, b)
print(x)

Exercise 9.3: Norms

Compute the Frobenius norm of A: ∥A∥F and the infinity norm of B: ∥B∥∞. Also find the largest andsmallest singular values of B.

print("the Frobenius norm of A:")
print(np.linalg.norm(A,'fro'))
print("the infinity norm of B:")
print(np.linalg.norm(B,np.inf))
print("the largest singular values:")
print(np.linalg.norm(A,2))
print("the smallest singular values:")
print(np.linalg.norm(A,-2))

Exercise 9.4: Power iteration

Generate a matrix Z, n × n, with Gaussian entries, and use the power iteration to find the largesteigenvalue and corresponding eigenvector of Z. How many iterations are needed till convergence?

Optional: use the time.clock() method to compare computation time when varying n.

Z = np.random.normal(mu,sigma,size=(n,n))
x = np.array([0 for i in range(n)])
y = np.array([1 for i in range(n)])
tmp = 0
tmp1 = 100
num = 0

begin = time.clock()
while(abs(tmp1 - tmp)>0.0000001):
	x = np.dot(Z,y)
	tmp1 = tmp
	tmp = np.linalg.norm(x,2)
	y = x/tmp
	num += 1

end = time.clock()
	
print(end-begin)
print("the largest eigenvalue:", tmp)  
print("the corresponding eigenvector:", y)    
print("The number of iterations:", num)

Exercise 9.5: Singular values

Generate an n × n matrix, denoted by C, where each entry is 1 with probability p and 0 otherwise. Usethe linear algebra library of Scipy to compute the singular values of C. What can you say about therelationship between n, p and the largest singular value?

x = 0
def Singular_values(n, p):
	C = []
	for i in range(n):
		C1 = []
		for j in range(n):
			if np.random.random() <= p:
				C1.append(1)
			else:
				C1.append(0)
		C.append(C1)
	C = np.mat(C)
	x = scipy.linalg.svdvals(C)
	return np.max(x)

for n in [10*(i + 1) for i in range(10)]:  
	for p in [0.1*(i + 1) for i in range(9)]:
		print( str(Singular_values(n, p))+ " ",end='')
	print("")

result:

横坐标为p，纵坐标为n。由输出结果得出，奇异值随n的增长而增长，随p的增长而增长。

Exercise 9.6: Nearest neighbor

Write a function that takes a value z and an array A and finds the element in A that is closest to z. Thefunction should return the closest value, not index.

Hint: Use the built-in functionality of Numpy rather than writing code to find this value manually. Inparticular, use brackets and argmin.

def find_closest(A, z):
	C = np.abs(A-z)
	idx = np.argmin(C)
	return A[idx // A[0].size][idx % A[0].size]

A = np.random.rand(n, n)  
print(A)  
print(find_closest(A, 1))