We currently compute the Kálmán update step using the canonical (i.e. mentioned by Frühwirth) equation $C_k = (I - KH)C_{k-1}$, but we know that this is not correct if $K$ is non-optimal or if $C_{k-1}$ is not true, and both of these conditions fail in traccc. The correct equation, therefore, is $C_k = (I - KH)C_{k-1}(I - KH)^\intercal + KVK^\intercal$ which holds for all (even non-optimal) $K$.
Now, we compute the $\chi^2$ value similarly, using $R = (I - HK)V$. Notice that this is eerily similar to the invalid covariance update, and we are not certain that this holds for arbitrary $K$. Is there a $\chi^2$ computation that holds for all $K$?
@andiwand suggested $R = (I - HK)V(I - HK)^\intercal + HCH^\intercal$.
We should verify that this is correct and, if so, migrate to it.
We currently compute the Kálmán update step using the canonical (i.e. mentioned by Frühwirth) equation$C_k = (I - KH)C_{k-1}$ , but we know that this is not correct if $K$ is non-optimal or if $C_{k-1}$ is not true, and both of these conditions fail in traccc. The correct equation, therefore, is $C_k = (I - KH)C_{k-1}(I - KH)^\intercal + KVK^\intercal$ which holds for all (even non-optimal) $K$ .
Now, we compute the$\chi^2$ value similarly, using $R = (I - HK)V$ . Notice that this is eerily similar to the invalid covariance update, and we are not certain that this holds for arbitrary $K$ . Is there a $\chi^2$ computation that holds for all $K$ ?
@andiwand suggested$R = (I - HK)V(I - HK)^\intercal + HCH^\intercal$ .
We should verify that this is correct and, if so, migrate to it.