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By Randall L. Eubank

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2 into one unified recursion that also computes the M (t) matrices for use in the backward covariance recursion. 4 This algorithm computes S(t|t), R(t), S(t|t−1), M (t), t = 1, . , n and σXε (t, j),t = 1, . , n, j = 1, . , t. 3. This two-stage approach can be perfectly satisfactory and we will see this reflected in some of the forward and backward recursions for computing signal and state vector estimators in Chapters 4 and 5. However, there are also cases where it would be more convenient to have the entire matrix ΣXε in hand after a single forward pass.

N. Consequently, if we want to compute the S(t|t − 1) and R(t) in tandem with evaluation of ΣXε we need a slightly more subtle strategy. Now, in general, for the tth row block the above diagonal blocks appear like σXε (t, j) = S(t|t − 1)M © 2006 by Taylor & Francis Group, LLC T (t) · · · M T (j − 1)H T (j) A Kalman Filter Primer 42 for j = t+ 1, . , n. So, computations above the diagonal can be carried out by storing and updating matrices of the form A(t, j) = S(t|t − 1)M T (t) · · · M T (j − 1).

T − 1). 4. Efficient order n2 (for p, q small relative to n) recursions for L can now be obtained in several ways. One © 2006 by Taylor & Francis Group, LLC A Kalman Filter Primer 56 approach is to build L row by row. 1 to see that the first column for L is I H(2)F (1)S(1|0)H T (1)R−1 (1) H(3)F (2)F (1)S(1|0)H T (1)R−1 (1) H(4)F (3)F (2)F (1)S(1|0)H T (1)R−1 (1) H(5)F (4) · · · F (1)S(1|0)H T (1)R−1 (1) . . H(n − 1)F (n − 2) · · · F (1)S(1|0)H T (1)R−1 (1) H(n)F (n − 1) · · · F (1)S(1|0)H T (1)R−1 (1) while the second has the form 0 I H(3)F (2)S(2|1)H T (2)R−1 (2) H(4)F (3)F (2)S(2|1)H T (2)R−1 (2) H(5)F (4)F (3)F (2)S(2|1)H T (2)R−1 (2) .

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