Arnoldi Hessenberg Step

Problem

Perform one Arnoldi iteration step for matrix A and an orthonormal basis Q (list of orthonormal vectors).

Let q be the last vector of Q, compute w = A*q, subtract the projections h[i] = Q[i].w from w, and return [h, beta] where beta = ||w||_2 after orthogonalization.

Hint

The coefficients h fill one column of the upper Hessenberg matrix produced by Arnoldi.

Test cases

case 1

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

expected: [[2],0]

case 2

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

expected: [[1],3]

case 3

in: [[[1,2],[3,4]],[[0.7071067811865475,0.7071067811865475]]]

expected: [[5],2]

case 4

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

expected: [[1,2],0]

Starter codePython

def arnoldi_step(A, Q):
    # Your code here
    pass

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