Diagonal (Jacobi) preconditioner application

Divides by an approximation to the diagonal of the system, which makes the iterative solver converge faster.

Usage

M_inv(v, kdiag_full, obs_idx, noise_var)

Arguments

v

Numeric vector to precondition (length \(m\)).

kdiag_full

Vector \(\mathrm{diag}(K)\) of length \(N\) (typically from `as.vector(kronecker(diag(space), diag(time)))`).

obs_idx

Integer indices of observed entries in the full vector.

noise_var

Scalar or length-\(m\) numeric nugget to add to the diagonal.

Value

A numeric vector of length \(m\): elementwise `v / (diagA + 1e-12)`.

Details

Technically: applies \(M^{-1} v \approx v / \mathrm{diag}(A)\), where \(A = S K S^\top + \mathrm{diag}(\text{noise})\) and \(\mathrm{diag}(K) = \mathrm{diag}(space) \otimes \mathrm{diag}(time)\).