\[
\begin{eqnarray}
M(i,j) &=& \max
\left[ \begin{array}{l}
M(i-1,j-1) + s(a_i,b_j), \\
X(i-1,j-1) + s(a_i,b_j),\\
Y(i-1,j-1) + s(a_i,b_j),
\end{array} \right. \hspace{0.5cm}
X(i,j) &=& \max
\left[ \begin{array}{l}
M(i-1,j) – u – v, \\
X(i-1,j) – u,\\
\end{array} \right. \hspace{0.5cm}
Y(i,j) &=& \max
\left[ \begin{array}{l}
M(i,j-1) – u – v, \\
Y(i,j-1) – u.
\end{array} \right.
\end{eqnarray} \tag{1}
\]
Let’s define the difference matrices for differential DP
\[
\begin{eqnarray}
\Delta H(i,j) &=& M(i,j) – M(i-1,j) \\
\Delta V(i,j) &=& M(i,j) – M(i,j-1) \\
\Delta D(i,j) &=& M(i,j) – M(i-1,j-1) \\
\Delta X(i,j) &=& X(i,j) – M(i,j) \\
\Delta Y(i,j) &=& Y(i,j) – M(i,j)
\end{eqnarray}
\]
Then the following recursive equations are derived from (1)
\[
\begin{eqnarray}
\Delta D(i,j) &=& \max
\left[ \begin{array}{l}
s(a_i,b_j), \\
\Delta X(i-1,j-1) + s(a_i,b_j)\\
\Delta Y(i-1,j-1) + s(a_i,b_j)
\end{array} \right.\\
\Delta H(i,j) &=& \Delta D(i,j) – \Delta V(i-1,j)\\
\Delta V(i,j) &=& \Delta D(i,j) – \Delta H(i,j-1)\\
\Delta X(i,j) &=& \max
\left[ \begin{array}{l}
-\Delta H(i,j) – u – v, \\
\Delta X(i-1,j) -\Delta H(i,j) – u.
\end{array} \right.\\
\Delta X(i,j) &=& \max
\left[ \begin{array}{l}
-\Delta V(i,j) -u – v, \\
\Delta Y(i,j-1) -\Delta V(i,j) – u.
\end{array} \right.
\end{eqnarray} \tag{2}
\]