numpy exercises

Numpy exercises

Generate matrices A, with random Gaussian entries, B, a Toeplitz matrix, where A ∈R^(n×m) and B ∈R^(m×m), for n = 200, m = 500.

• Exercise 9.1: Matrix operations Calculate A + A, AA,AAT and AB. Write a function that computes A(B−λI) for any λ.

  import numpy as np
  from scipy.linalg import toeplitz
  
  A = np.random.normal(0.,1.,(200,500))
  B = toeplitz(np.array(list(range(1,501))))
  sumA = A+A
  productA = A.dot(A.T)
  productAT = A.T.dot(A)
  productAB = A.dot(B)
  print("sum A+A:")
  print(sumA)
  print("product AA:")
  print(productA)
  print('product AAT:')
  print(productAT)
  print('product AB:')
  print(productAB)
  def compute(A,B,linda):
      temp = B-linda*np.eye(500)
      return  A@B

• Exercise 9.2: Solving a linear system Generate a vector b with m entries and solve Bx = b.

b=np.array(list(range(0,500)))
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 and smallest singular values of B.

form numpy import linalg as LA

print(LA.norm(A,ord=2))
print(LA.norm(B,ord=np.inf))
U,sigma,VT = LA.svd(B)
Max = max(sigma)
Min = min(sigma)
print(Max,Min)

• 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.

import time
n=200
def PowerIter(Z,esp):
    u = np.ones(n)
    u = u.reshape(n,1)
    max_u = np.max(u)
    sub = np.inf
    iteration = 0
    startTime = time.clock()
    while sub > esp:
        u = Z@u
        temp = np.max(u)
        sub = np.fabs(temp-max_u)
        max_u = temp
        u = u / max_u
        iteration += 1
    endTime = time.clock()
    return max_u, iteration, endTime-startTime, u

esp = 1e-5
Z = np.random.normal(0,1,(n,n))
print("Maxeig, iterations, time/s, eigVec")
print(PowerIter(Z,esp))

• 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?

n=200
for i in range(1,9):
    p = i / 8
    C = np.random.choice(2,(n,n),p=[p,1-p])
    sigma = LA.svd(C,1,0)[1]
    print("n, p , sigma: ",end='')
    print(n, p, sigma)

• 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 NearNb(A,z):
    B = A - z
    re=np.where(A==np.min(A))
    print(B[re[1][0]-1,re[0][0]-1])

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