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.
Exercise 9.1: Matrix operations

　　Calculate A + A, AA^T,A^TA 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: ||A||_F and the infinity norm of B: ||B||_∞. 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 orresponding 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 Arr and finds the element in Arr 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
import random as rd
from scipy.linalg import toeplitz

def E_9_1_1(A):
    return A + A

def E_9_1_2(A):
    return np.dot(A, A.T)

def E_9_1_3(A):
    return np.dot(A.T, A)

def E_9_1_4(A, B):
    return np.dot(A, B)

def E_9_1_5(A, B, l):
    return np.dot(A, (B - l * np.eye(len(B))))

def E_9_2(B, b):
    return np.dot(np.linalg.inv(B), b)

def E_9_3_1(A):
    return np.linalg.norm(A)

def E_9_3_2(B):
    return np.linalg.norm(B, np.inf)

def E_9_3_3(B):
    s = np.linalg.svd(B)[1]
    return (s[0], s[len(s) - 1])

def E_9_4(Z):
    e = np.linalg.eig(Z)
    temp = np.max(e[0])
    for i in range(len(e[0])):
        if (e[0][i] == temp):
            return (e[0][i], tuple(e[1][i]))
    return None

def E_9_5_1(n, p):
    return np.random.binomial(1, p, size = (n, n))

def E_9_5_2(C):
    return np.linalg.svd(C)[1]

def E_9_6(Arr, z):
    return Arr[np.argmin(np.abs(Arr - z))]

#n = 2
#m = 5
n = 200
m = 500
A = np.random.normal(size = (n, m))
B_ = np.random.randn(m * 2)
B = toeplitz(B_[: m], B_[: 1] + B_[m : m * 2])
l = rd.normalvariate(0, 1)
b = np.random.normal(size = (m, 1))
Z = np.random.normal(size = (n, n))
p = rd.random()
z = rd.normalvariate(0, 1)
Arr = np.random.randn(m)

print("A =", A)
print("B =", B)
print("λ =", l)
print("b =", b)
print("Z =", Z)
print("p =", p)
print("z =", z)
print("Arr =", Arr)

print("\nExercise 9.1: Matrix operations")
print("A + A =", E_9_1_1(A))
print("AAᵀ =", E_9_1_2(A))
print("AᵀA =", E_9_1_3(A))
print("AB =", E_9_1_4(A, B))
print("A(B−λI) =", E_9_1_5(A, B, l))

print("\nExercise 9.2: Solving a linear system")
print("x =", E_9_2(B, b))

print("\nExercise 9.3: Norms")
print("||A||F =", E_9_3_1(A))
print("||A||∞ =", E_9_3_2(B))
print("(max{singular values of B}, min{singular values of B}) =", E_9_3_3(B))

print("\nExercise 9.4: Power iteration")
print("(max{λ(Z)}, orresponding eigenvector of Z) =", E_9_4(Z))

print("\nExercise 9.5: Singular values")
C = E_9_5_1(n, p)
print("C =", C)
print("singular values of C =", E_9_5_2(C))

print("\nExercise 9.6: Nearest neighbor")
print("the closest value of Arr to z =", E_9_6(Arr, z))