Editing Kalman filter (section)

=== Simplification of the ''posteriori'' error covariance formula ===
The formula used to calculate the ''a posteriori'' error covariance can be simplified when the Kalman gain equals the optimal value derived above. Multiplying both sides of our Kalman gain formula on the right by '''S'''<sub>''k''</sub>'''K'''<sub>''k''</sub><sup>T</sup>, it follows that
:<math>\mathbf{K}_k \mathbf{S}_k \mathbf{K}_k^\textsf{T} = \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T}</math>

Referring back to our expanded formula for the ''a posteriori'' error covariance,
:<math> \mathbf{P}_{k\mid k} = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1}  - \mathbf{P}_{k\mid k-1} \mathbf{H}_k^\textsf{T} \mathbf{K}_k^\textsf{T} + \mathbf{K}_k \mathbf{S}_k \mathbf{K}_k^\textsf{T}</math>

we find the last two terms cancel out, giving
:<math> \mathbf{P}_{k\mid k} = \mathbf{P}_{k\mid k-1} - \mathbf{K}_k \mathbf{H}_k \mathbf{P}_{k\mid k-1} = (\mathbf{I} - \mathbf{K}_k \mathbf{H}_k) \mathbf{P}_{k\mid k-1}</math>

This formula is computationally cheaper and thus nearly always used in practice, but is only correct for the optimal gain. If arithmetic precision is unusually low causing problems with [[numerical stability]], or if a non-optimal Kalman gain is deliberately used, this simplification cannot be applied; the ''a posteriori'' error covariance formula as derived above (Joseph form) must be used.