Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where A∈ R^n x m, and B ∈ R^m x m, for n = 200, m = 500
首先使用numpy.random.mormal生成A
函數語法：
numpy.random.normal(loc, scale, size)
loc : 概率分佈的均值，對應着分佈中心
scale : 概率分佈的標準差，對應分佈的寬度（scale越大越矮胖， 越小越高瘦）
size : 輸出的空間大小
使用scipy.linalg.toeplitz生成B
import numpy
from scipy.linalg import toeplitz
import time
A = numpy.random.normal(size = (200, 500))
print('A = ')
print(A)
B = toeplitz(range(1, 501))
print('\nB = ')
print(B)
9-1
Exercise 9.1: Matrix operations
Calculate A + A, A AT ,AT A and A B. Write a function that computes A(B − λI) for any λ.
print('A + A = ')
print(A + A)
print('\nAA^T = ')
print(A.dot(A.T))
print('\nA^TA = ')
print(A.T.dot(A))
print('\nAB = ')
print(A.dot(B))
def calcul(A, B, linda):
input()
tmp = B - linda * numpy.eye(500)
return numpy.dot(A, tmp)
9-2
Exercise 9.2: Solving a linear system
Generate a vector b with m entries and solve Bx = b.
b = numpy.array(range(1, 501))
print('b = ')
print(b)
x = numpy.linalg.solve(B, b)
print('\nx = ')
print(x)
9-3
Exercise 9.3: Norms
Compute the Frobenius norm of A: kAk F and the infinity norm of B: kBk ∞ . Also find the largest and smallest singular values of B.
from numpy import linalg
print('|A|f = ', linalg.norm(A, ord=2))
print('|B| = ', linalg.norm(B, ord=numpy.inf), '\n')
sigma = linalg.eigvals(B)
print('largest = ', numpy.max(sigma), '\nsmallest:', numpy.min(sigma))
9-4
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.
print('\n----------------------9.4----------------------')
def power_iter(n, Z):
esp = 1e-5
begin = time.clock()
iter = 0
u = v = numpy.ones([n , 1]) #ones()返回一個全1數組
lam = iter_counter = 0
while(1):
v = numpy.matmul(Z, u)
pre = lam
lam = v.max()
u = v / lam
if(abs(pre - lam) < esp):
break
iter_counter += 1
end = time.clock()
time_used = end - begin
return lam, iter_counter, time_used
n = [100, 150, 200]
for ni in n:
Z = numpy.random.normal(size = (ni, ni))
lam, iter_counter, time_used = power_iter(ni, Z)
print('when n = ', ni, ': ')
print('Z = ', Z)
print('lambda = ', lam)
print('iter = ', iter_counter)
print('time = ', time_used, '\n')
9-5
Exercise 9.5: Singular valuesGenerate 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?
print('\n----------------------9.5----------------------')
def singular_value(n , p):
C = numpy.random.binomial(1, p, (n, n))
u, sigma, v = linalg.svd(C)
return sigma
for n in [10, 20]:
for p in [0.2, 0.3, 0.5]:
sigma = singular_value(n, p)
print('n = ', n, ', p = ', p)
print('The largest singular value is: \n', sigma, '\n')
9-6
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.
def nearest_neighbor(A, z):
B = numpy.abs(A - z)
index = numpy.argmin(B)
return A[index]
A = numpy.array(range(1, 100))
print('The closest value is: ', nearest_neighbor(A, 2))