2 days ago I asked a simplified version of the question below, see here. However, I simplified the question too much and I am now stuck with the following problem: \begin{align} y^{k-1} - 2y^{k} + y^{k+1} = A y^{k} + B(y^{k-1} - y^{k+1}), \quad y^{i} \in \mathbb{R}^{d}, k = 1, \dots, T-1 \end{align} where $A$ is symmetric positive definite and $B$ is antisymmetric. Furthermore, the values for $y^{0}$ and $y^{T}$ are given. I haven't found any source in google on how to solve such a problem.
Question: How does one solve such a problem? Numerical solutions are also fine.
Thanks in advance for tips and help.
Hint.
With $y_0,y_4$ given, we can solve this as a linear system
$$ \left( \begin{array}{ccc} -(A+2 I) & B+I & 0 \\ I-B & -(A+2 I) & B+I \\ 0 & I-B & -(A+2 I)\\ \end{array} \right)\left(\matrix{y_1\\ y_2\\ y_3}\right) = \left(\matrix{-(I-B)y_0\\ 0\\ -(B+I)y_4}\right) $$