# Sequential and Square-Root Algorithms

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## Abstract

It is now clear that the only time-consuming operation in the Kalman filtering process is the computation of the Kalman gain matrices.

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## Author information

Authors

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Correspondence to Charles K. Chui .

## Exercises

### Exercises

1. 7.1.

Give a proof of Lemma 7.1.

2. 7.2.

Find the lower triangular matrix L that satisfies:

1. (a)

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

2. (b)

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

3. 7.3.
1. (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.

2. (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.

4. 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 square-root filter for this example. Note that this example illustrates the improved numerical characteristics of the square-root filter.

5. 7.5.

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

6. 7.6.

Using the upper triangular decompositions instead of the lower triangular ones, derive a new square-root Kalman filter.

7. 7.7.

Combine the sequential algorithm and the square-root scheme with upper triangular decompositions to derive a new filtering algorithm.

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