Shifted Power Method Step

Problem

Perform one shifted power iteration step on matrix A with shift mu and nonzero vector v.

Compute w = (A - mu*I)*v, the Rayleigh estimate lam = mu + (w . v) / (v . v), and return [lam, w / ||w||_2].

Hint

The Rayleigh quotient is w . v over v . v before adding the shift back.

case 1

in: [[[2,0],[0,3]],0,[1,1]]

expected: [2.5,[0.5547001962252291,0.8320502943378437]]

case 2

in: [[[2,0],[0,3]],1,[1,1]]

expected: [2.5,[0.4472135954999579,0.8944271909999159]]

case 3

in: [[[4,1],[2,3]],0,[1,1]]

expected: [5,[0.7071067811865475,0.7071067811865475]]

case 4

in: [[[4,1],[2,3]],2,[1,0]]

expected: [4,[0.7071067811865475,0.7071067811865475]]

Starter codePython

def shifted_power_step(A, mu, v):
    # Your code here
    pass

4 test cases · run your code against them in the editor