Lanczos Tridiagonal Step

Problem

Perform one Lanczos iteration step for symmetric A and unit vector v.

Compute alpha = v^T * A * v, the residual w = A*v - alpha*v, and beta = ||w||_2. Return [alpha, beta].

Hint

The alpha and beta values form the diagonal and off-diagonal of the tridiagonal Lanczos matrix.

case 1

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

expected: [2,0]

case 2

in: [[[2,0],[0,3]],[0.7071067811865475,0.7071067811865475]]

expected: [2.5,0.5]

case 3

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

expected: [2,1]

case 4

in: [[[1,0,0],[0,2,0],[0,0,3]],[0.5773502691896258,0.5773502691896258,0.5773502691896258]]

expected: [2,0.816496580927726]

Starter codePython

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

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