statespace_sqrt

The square root filter and smoother (DK §6.3).

The filter propagates a square root \(\tilde P_t\) of \(P_t\), so \(P_t\) stays positive semi-definite by construction. Each step triangularizes the pre-array

\[\begin{split}\begin{bmatrix} Z_o \tilde P & \tilde H_{oo} & 0 \\ T \tilde P & 0 & R \tilde Q \end{bmatrix} \;\to\; \begin{bmatrix} \tilde F & 0 & 0 \\ \tilde K & \tilde P_{t+1} & 0 \end{bmatrix}\end{split}\]

by an orthogonal transformation (an LQ factorization by Householder QR, where DK use Givens rotations). Then \(F = \tilde F \tilde F'\), \(K = \tilde K \tilde F^{-1}\) and \(P_{t+1} = \tilde P_{t+1} \tilde P_{t+1}'\). The smoother does the same for \(N_{t-1}\).

The output uses the types of kalman_filter and state_smoother, so the rest of the library works on it. Diffuse states are not supported; use the exact initial filter or an approximate diffuse initialization.

sqrt_kalman_filter

subroutine sqrt_kalman_filter(rep, res, info, Pchol)
  type(ssm_rep_t), intent(in) :: rep
  type(filter_result_t), intent(out) :: res
  integer, intent(out) :: info
  real(dp), allocatable, intent(out), optional :: Pchol(:, :, :)   ! (m, m, n+1)

Pchol receives the square roots \(\tilde P_t\).

sqrt_state_smoother

subroutine sqrt_state_smoother(rep, fres, sres, info)
  type(filter_result_t), intent(in) :: fres
  type(smoother_result_t), intent(out) :: sres

\(r_t\) and \(\hat\alpha_t\) as in state_smoother, with \(N_{t-1}\) propagated through its square root and \(V_t = P_t - (P_t \tilde N_{t-1})(P_t \tilde N_{t-1})'\).