Starting with the prearray

$[[\backslash$lambda $^((1)/(2))$K$^(-($H$)/(2))($n$-1),\backslash$lambda $^((1)/(2))$u$($n$)],[$hat$($x$)^($H$)($n$|$Y$\_($n$-1))$K$^(-($H$)/(2))($n$-1),$y$^(\text{'})($n$)],[0^($T$),1],[\backslash$lambda $^(-(1)/(2))$K$^((1)/(2))($n$-1),0]]$

which is the expanded version of the prearray in Eq$.(15\mathrm{.}38),$ show that the extended

square$-$root information filtering algorithm is described by

$[[\backslash$lambda $^((1)/(2))$K$^(-($H$)/(2))($n$-1),\backslash$lambda $^((1)/(2))$u$($n$)],[$hat$($x$)^($H$)($n$|$Y$\_($n$-1))$K$^(-($H$)/(2))($n$-1),$y$^(\text{'})($n$)],[0^($T$),1],[\backslash$lambda $^(-(1)/(2))$K$^((1)/(2))($n$-1),0]]\backslash$Theta $($n$)=[[$K$^(-($H$)/(2))($n$),0],[$hat$($x$)^($H$)($n$+1|$Y$\_($n$))$K$^(-($H$)/(2))($n$),$r$^(-(1)/(2))($n$)\backslash$alpha $^(*)($n$)],[\backslash$lambda $^((1)/(2))$u$^($H$)($n$)$K$^((1)/(2))($n$),$r$^(-(1)/(2))($n$)],[$K$^((1)/(2))($n$),-$g$($n$)$r$^((1)/(2))($n$)]]$

Hence, formulate an expression for the updated state estimate hat$($x$)($n$+1|$Y$\_($n$)).$