projection_error()

projection_error(states, basis)

Calculate the absolute and relative projection errors induced by projecting states to a low dimensional basis, i.e.,

absolute_error = ||Q - Vr Vr^T Q||_F, relative_error = ||Q - Vr Vr^T Q||_F / ||Q||_F

where Q = states and Vr = basis. Note that Vr Vr^T is the orthogonal projector onto subspace of R^n defined by the basis.

Parameters:

states(n, k) or (k,) ndarray

Matrix of k snapshots where each column is a single snapshot, or a single 1D snapshot. If 2D, use the Frobenius norm; if 1D, the l2 norm.

Vr(n, r) ndarray

Low-dimensional basis of rank r. Each column is one basis vector.

Returns:

absolute_errorfloat

Absolute projection error ||Q - Vr Vr^T Q||_F.

relative_errorfloat

Relative projection error ||Q - Vr Vr^T Q||_F / ||Q||_F.