numpy是python中一个类似于c语言中math的函数库，可以简化许多数学方法的使用。以下是练习题和代码：
Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where A ∈Rn×m and B ∈Rm×m, for n = 200, m = 500.
Exercise 9.1: Matrix operations Calculate A + A, AA>,A>A and AB. Write a function that computes A(B−λI) for any λ. Exercise 9.2: Solving a linear system Generate a vector b with m entries and solve Bx = b.
Exercise 9.3: Norms Compute the Frobenius norm of A: kAkF and the infinity norm of B: kBk∞. Also find the largest and smallest singular values of B.
Exercise 9.4: Power iteration Generate a matrix Z, n × n, with Gaussian entries, and use the power iteration to find the largest eigenvalue 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.
Exercise 9.5: Singular values Generate an n×n matrix, denoted by C, where each entry is 1 with probability p and 0 otherwise. Use the linear algebra library of Scipy to compute the singular values of C. What can you say about the relationship between n, p and the largest singular value?
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. The function should return the closest value, not index.
Hint: Use the built-in functionality of Numpy rather than writing code to find this value manually. In particular, use brackets and argmin.
import numpy as np
from scipy.linalg import toeplitz
import time
#9.1
def ex9_1(A, B, n, m):
print("A + A:")
C = A + A
print(C)
print("AA^:")
print(np.dot(A, A.T))
print("A^A:")
print(np.dot(A.T, A))
print("AB:")
print(np.dot(A, B))
print("A(B − λI):")
C = B - lamda * (np.eye(m))
print(np.dot(A, C))
print("\n")
#9.2
def ex9_2(A, B, n, m):
b = np.ones((m, 1))
x = np.linalg.solve(B, b)
print(x)
print("\n")
#9.3
def ex9_3(A, B, n, m):
A_F = np.linalg.norm(A, 'fro')
print("the Frobenius norm:", A_F)
B_F = np.linalg.norm(B, np.inf)
print("the infinity norm:", B_F)
lar_sin = np.linalg.norm(B, 2)
smal_sin = np.linalg.norm(B, -2)
print("the largest singular:", lar_sin)
print("the smallest singular:", smal_sin)
print("\n")
#9.4
def ex9_4(A, B, n, m):
Z = np.random.standard_normal((n, n))
num = 0
u_k = np.ones(n)
v_k_norm = 0
v_k = np.zeros(n)
begin = time.clock()
while(True):
v_k = np.dot(Z, u_k)
v_k_norm_temp = v_k_norm
v_k_norm = np.linalg.norm(v_k)
u_k = v_k / v_k_norm
num += 1
if(abs(v_k_norm_temp - v_k_norm) < 0.0005):
break;
end = time.clock()
print("the largest eigenvalue:", v_k_norm)
print("the corresponding eigenvector:", u_k)
print("The number of iterations:", num)
print("computation time when varying n:", end-begin)
print("\n")
#9.5
def ex9_5(A, B, n, m):
p = 0.5
C = np.random. binomial(1, p, (n, n))
lar_sin = np.linalg.norm(C, 2)
smal_sin = np.linalg.norm(C, -2)
print("the smallest singular:", smal_sin)
print("the largest singular:", lar_sin)
print("n * p:", n*p)
print("the largest singular is closed with n * p \nso that we can say they are equal!")
print("\n")
#9.6
def ex9_6(A, B, n, m):
z = -5
B, C = A[A>z], A[A<=z]
ceil, floor = 0, 0
if(len(B)):
ceil = np.argmin(B)
else:
return C[np.argmax(C)]
if(len(C)):
floor = np.argmax(C)
else:
return B[ceil]
if(abs(B[ceil]-z) < abs(C[floor]-z)):
return B[ceil]
else:
return C[floor]
print("the closest value:", closest)
print("\n")
#赋初值
mu, sigma = 0, 1.0
n, m = 200, 500
A = np.random.normal(loc=mu, scale=sigma, size=(n, m))
c = [a for a in range(1, m+1)]
B = toeplitz(c, c)
ex_9_1(A, B, n, m)
ex_9_2(A, B, n, m)
ex_9_3(A, B, n, m)
ex_9_4(A, B, n, m)
ex_9_5(A, B, n, m)
ex_9_6(A, B, n, m)
2018/5/22