Preconditioned Conjugate Gradient (PCG) solver for the observed system

In plain terms: solves the big linear system that gives the GP weights using only matrix–vector products—no huge matrices, no explicit inverse.

Usage

pcg(
  b,
  obs_idx,
  N,
  space_mat,
  time_mat,
  noise_var,
  kdiag_full,
  tol = 1e-08,
  maxit = 10000
)

Arguments

b

Right-hand side vector (observed length \(m\)).

obs_idx

Integer indices of observed entries in the full vector.

N

Total length of the full vector.

space_mat

Spatial kernel matrix.

time_mat

Temporal kernel matrix.

noise_var

Scalar or length-\(m\) nugget on the observed scale.

kdiag_full

Vector \(\mathrm{diag}(K)\) of length \(N\).

tol

Relative residual tolerance for convergence (default `1e-8`).

maxit

Maximum number of iterations (default `10000`).

Value

Numeric solution vector `x` of length \(m\).

Details

Technically: solves \((S K S^\top + \mathrm{diag}(\text{noise}))\,x = b\) by PCG, using `Amv` for matrix–vector products and `M_inv` as a Jacobi preconditioner. Stops when the relative residual falls below `tol` or after `maxit` iterations (issues a warning on `maxit`).