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.
#9.1 def computes(A, B, arg): return np.dot(A, B - np.dot(arg, np.eye(np.size(B,0),np.size(B,1)))) A = np.random.normal(0, 1, (200, 500)) B = toeplitz(np.random.random(500)) AaddA = A + A np.savetxt('AaddA.txt', AaddA) AAT = np.dot(A, A.T) np.savetxt('AAT.txt', AaddA) ATA = np.dot(A.T, A) np.savetxt('ATA.txt', AaddA) AB = np.dot(A, B) np.savetxt('AB.txt', AaddA) print("Solving A(B−λI)") arg = float(input('Input the λ:')) exp = computes(A, B, arg) np.savetxt('exp.txt', exp)
Exercise 9.2: Solving a linear system Generate a vector b with m entries and solve Bx = b.
#9.2 b = np.random.random(500) x = np.linalg.solve(B, b) print("Solving B * x = b") print("Residual: ", np.linalg.norm((np.dot(B, x) - b), ord = 2))
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.
#9.3 print("Frobenius norm of A:", np.linalg.norm(A, ord = 'fro')) print("infinity norm of B:", np.linalg.norm(B, ord = np.inf)) print("The smallest singular of B:", min(np.linalg.svd(B, True, False)))
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.
#9.4 import time def power_iteration(A): u = np.random.rand(A.shape[1]) num_simulations = 0; while True: old = u[:] v = np.dot(A, u) arg = max(v) if max(v) + min(v) < 0 else min(v) u = v / arg num_simulations += 1 if np.linalg.norm(u - old, ord = np.inf) < 10 ** (-6) or num_simulations >= 10 ** 4: break return u, arg, num_simulations n = 100 Z = np.random.normal(0, 1, (n, n)) t1 = time.clock() v, arg, num_simulations = power_iteration(Z) t2 = time.clock() print("simulations:",num_simulations, "Clock:", t2-t1, "Lambda:", arg)
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?
#9.5 ''' 计算后发现最大的奇异值接近n*p ''' from scipy.linalg import svd for p in range(100): n = 100 p = p / 100 C = np.random.binomial(1, p, (n,n)) U, s, Vh = svd(C) print(max(s))
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.
#9.6 def closest(z, A): return A[np.fabs(A - z).argmin()] z = 3 A = np.random.random(10) * 10 print (closest(z, A))