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Affine Residual Sum of Squares Hessian

Syntax
[DA] = ckbs_sumsq_hes(dgdhqinvrinv)

Purpose
This routine returns the diagonal and off diagonal blocks corresponding to the Hessian of the affine Kalman-Bucy smoother residual sum of squares.

Notation
The affine Kalman-Bucy smoother residual sum of squares is defined by  \[
\begin{array}{rcl}
S ( x_1 , \ldots , x_N ) & = & \sum_{k=1}^N S_k ( x_k , x_{k-1} ) \\
S_k ( x_k , x_{k-1} )    & = & 
\frac{1}{2} 
( z_k - h_k - H_k * x_k )^\T * R_k^{-1} * ( z_k - h_k - H_k * x_k )
\\
& + &
\frac{1}{2} 
( x_k - g_k - G_k * x_{k-1} )^\T * Q_k^{-1} * ( x_k - g_k - G_k * x_{k-1} )
\end{array}
\] 
where the matrices  R_k and  Q_k are symmetric positive definite and  x_0 is the constant zero.

Hessian
If we define  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  \[
\begin{array}{rcl}
S^{(2)} ( x_1 , \ldots , x_N ) & = &
\left( \begin{array}{cccc}
D_1 & A_2^\T & 0      &           \\
A_2 & D_2    & A_3^\T & 0         \\
0   & \ddots & \ddots & \ddots    \\
    & 0      & A_N    & D_N 
\end{array} \right)
\\
D_k & = & H_k^\T * R_k^{-1} * H_k + Q_k^{-1} 
      + G_{k+1}^\T * Q_{k+1}^{-1} * G_{k+1} 
\\
A_k & = & - Q_k^{-1} * G_k
\end{array}
\] 


dg
The argument 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
The argument 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
The argument 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
The argument 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
The result 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
The result A is a three dimensional array, for  k = 2 , \ldots , N  \[
     A_k = A(:,:,k)
\]
and A has size  n \times n \times N .

Example
The file sumsq_hes_ok.m contains an example and test of ckbs_sumsq_hes. It returns true if ckbs_sumsq_hes passes the test and false otherwise.
Input File: src/ckbs_sumsq_hes.m