These algorithms require the inverse of the covariance matrix of the received samples.
The evaluation of the likelihood functions requires the inverse of the covariance matrix, which cannot be done if some of the eigenvalues are zero.
Unlike disturbance observers, WNN_AFEE does not require the inverse of the Jacobian transpose for computing the force, thus, it has no computational problem near singular points.
As the algorithm requires the inverse of the Cholesky factor of the Hessian matrix at each iteration a procedure is presented to directly obtain a matrix that multiplied by its transpose gives the BFGS update of the Hessian.
At first, a Gaussian mixture approach, as initially introduced in [15], may seem impractical since the embedded Kalman filtering requires the inverse of matrices of size the length of the observation vector, which is typically very large in TBD.
It is important to emphasise that the de-mixing filters avoid the inverse of the deterministic mixing matrix but still require the inverse of filters h11, h22 and h33 to estimate the de-mixing filters that also need to be realisable (for details, see the inverse of non-minimum phase systems, pp. 348 349 in [9]).
The estimation of gametic effects via marker-assisted BLUP requires the inverse of the conditional gametic relationship matrix G.
The quasi-steady-state equation, Eq. (24), can be solved explicitly since the solution only requires the inverse of the non-singular matrix and the multiplication with the matrix and a vector.
Similarly, the uncle−nephew covariance (individuals 5 and 7) is 2 λ 3, lower than the covariance between half sibs (individuals 4 and 5) which is only λ 2. The mixed model implementation of the covariance between relatives requires the inverse of the T matrix to construct the mixed model equations, in a similar manner as the numerator relationship matrix in the standard model (Henderson 1984).
The backward filter requires the inverse dynamics of the forward filter.
