Abstract
It is now clear that the only timeconsuming operation in the Kalman filtering process is the computation of the Kalman gain matrices.
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Exercises
Exercises

7.1.
Give a proof of Lemma 7.1.

7.2.
Find the lower triangular matrix L that satisfies:

(a)
\(LL^{\top }=\left[ \begin{array}{c@{\quad }c@{\quad }c} 1 &{} 2 &{} 3\\ 2 &{} 8 &{} 2\\ 3 &{} 2 &{} 14 \end{array}\right] .\)

(b)
\(LL^{\top }=\left[ \begin{array}{c@{\quad }c@{\quad }c} 1 &{} 1 &{} 1\\ 1 &{} 3 &{} 2\\ 1 &{} 2 &{} 4 \end{array}\right] .\)

(a)

7.3.

(a)
Derive a formula to find the inverse of the matrix
$$\begin{aligned} L=\left[ \begin{array}{c@{\quad }c@{\quad }c} \ell _{11} &{} 0 &{} 0\\ \ell _{21} &{} \ell _{22} &{} 0\\ \ell _{31} &{} \ell _{32} &{} \ell _{33} \end{array}\right] , \end{aligned}$$where \(\ell _{11},\ \ell _{22}\), and \(\ell _{33}\) are nonzero.

(b)
Formulate the inverse of
$$\begin{aligned} L=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \ell _{11} &{} 0 &{} 0 &{} \cdots &{} 0\\ \ell _{21} &{} \ell _{22} &{} 0 &{} \cdots &{} 0\\ \vdots &{} \vdots &{} \ddots &{} \ddots &{} \vdots \\ \vdots &{} \vdots &{} &{} \ddots &{} 0\\ \ell _{n1} &{} \ell _{n2} &{} \cdots &{} \cdots &{} \ell _{nn} \end{array}\right] , \end{aligned}$$where \(\ell _{11},\ \cdots ,\ \ell _{nn}\) are nonzero.

(a)

7.4.
Consider the following computer simulation of the Kalman filtering process. Let \(\epsilon \ll 1\) be a small positive number such that
$$\begin{aligned}\begin{gathered} 1\epsilon \not \simeq 1 \\ 1\epsilon ^{2}\simeq 1 \end{gathered}\end{aligned}$$where “\(\simeq \)” denotes equality after rounding in the computer. Suppose that we have
$$\begin{aligned} P_{k, k}=\left[ \begin{array}{c@{\quad }c} \frac{\epsilon ^{2}}{1\epsilon ^{2}} &{} 0\\ 0 &{} 1\end{array}\right] . \end{aligned}$$Compare the standard Kalman filter with the squareroot filter for this example. Note that this example illustrates the improved numerical characteristics of the squareroot filter.

7.5.
Prove that to any positive definite symmetric matrix A, there is a unique upper triangular matrix \(A^{u}\) such that \(A=A^{u}(A^{u})^{\top }\).

7.6.
Using the upper triangular decompositions instead of the lower triangular ones, derive a new squareroot Kalman filter.

7.7.
Combine the sequential algorithm and the squareroot scheme with upper triangular decompositions to derive a new filtering algorithm.
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Chui, C.K., Chen, G. (2017). Sequential and SquareRoot Algorithms. In: Kalman Filtering. Springer, Cham. https://doi.org/10.1007/9783319476124_7
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DOI: https://doi.org/10.1007/9783319476124_7
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