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COMPUTATIONAL ISSUES 287

8.3.4 Square root filtering

The square root of a square matrix P is a matrix A such that P ¼ AA.
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Sometimes the matrix B that satisfies P ¼ B B is also called a square
root, but strictly speaking such a B is a Cholesky factor, and not a square
root. Anyway, square roots, Cholesky factors and other factorizations
are useful matrix decomposition methods that enable stable implemen-
tations of the Kalman filter. The principal idea in square root filtering is
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to decompose a covariance matrix P as P ¼ B B (or likewise), and to use
B as a representation of P. This effectively doubles the precision of the
number representation. The various factorization methods lead to
various forms of square root filtering.
This section describes one particular implementation of a square root
filter, the Potter implementation (Potter and Stern, 1963). It uses:

. Triangular Cholesky factorization of the error covariance matrices.
. Sequentially processing of the measurements using symmetric ele-
mentary matrices.
. QR factorization for the prediction.

The update in Potter’s square root filter

We will represent the error covariance matrix C(ijj) by an upper triangu-
lar matrix B(ijj) where:

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CðijjÞ¼ B ðijjÞBðijjÞ ð8:39Þ

An upper triangular matrix is a matrix with all elements below the
diagonal equal to zero, e.g.

2 3
b 00 b 01 b 02
0
6 b 11 b 12 7
6 7
0 0 ð8:40Þ
B ¼ 6 b 22 7
.
4 5
0 0 0 . .
The update formula, as expressed in (8.27):

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CðijiÞ¼ Cðiji 1Þ Cðiji 1ÞH T HCðiji 1ÞH þ C v 1 HCðiji 1Þ