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[D, A] = ckbs_sumsq_hes(dg, dh, qinv, rinv)
\[
\begin{array}{rcl}
S ( x_1 , \ldots , x_N ) & = & \sum_{k=1}^N S_k ( x_k , x_{k-1} ) \\
S_k ( x_k , x_{k-1} ) & = &
\frac{1}{2}
( z_k - h_k - H_k * x_k )^\R{T} * R_k^{-1} * ( z_k - h_k - H_k * x_k )
\\
& + &
\frac{1}{2}
( x_k - g_k - G_k * x_{k-1} )^\R{T} * Q_k^{-1} * ( x_k - g_k - G_k * x_{k-1} )
\end{array}
\]
where the matrices
R_k
and
Q_k
are
symmetric positive definite and
x_0
is the constant zero.
Q_{N+1}
to be the
n \times n
identity
matrix and
G_{N+1}
to be zero,
the Hessian of the
affine Kalman-Bucy smoother residual sum of squares is
\[
\begin{array}{rcl}
S^{(2)} ( x_1 , \ldots , x_N ) & = &
\left( \begin{array}{cccc}
D_1 & A_2^\R{T} & 0 & \\
A_2 & D_2 & A_3^\R{T} & 0 \\
0 & \ddots & \ddots & \ddots \\
& 0 & A_N & D_N
\end{array} \right)
\\
D_k & = & H_k^\R{T} * R_k^{-1} * H_k + Q_k^{-1}
+ G_{k+1}^\R{T} * Q_{k+1}^{-1} * G_{k+1}
\\
A_k & = & - Q_k^{-1} * G_k
\end{array}
\]
dg
is a three dimensional array,
for
k = 1 , \ldots , N
\[
G_k = dg(:,:,k)
\]
and
dg
has size
n \times n \times N
.
dh
is a three dimensional array,
for
k = 1 , \ldots , N
\[
H_k = dh(:,:,k)
\]
and
dh
has size
m \times n \times N
.
qinv
is a three dimensional array,
for
k = 1 , \ldots , N
\[
Q_k^{-1} = qinv(:,:,k)
\]
and
qinv
has size
n \times n \times N
.
rinv
is a three dimensional array,
for
k = 1 , \ldots , N
\[
R_k^{-1} = rinv(:,:,k)
\]
and
rinv
has size
m \times m \times N
.
D
is a three dimensional array,
for
k = 1 , \ldots , N
\[
D_k = D(:,:,k)
\]
and
D
has size
n \times n \times N
.
A
is a three dimensional array,
for
k = 2 , \ldots , N
\[
A_k = A(:,:,k)
\]
and
A
has size
n \times n \times N
.
ckbs_sumsq_hes.
It returns true if ckbs_sumsq_hes passes the test
and false otherwise.