If we look at matrix А1, then it becomes clear, that it is quite easy to solve this problem.
The value u0 = u(0) we know from the boundary condition. From the first equation, we find u1. After that, when we know u0 and u1, we can find u2 from the second equation. Continuing this way, we find, using direct substitution, all coordinates of the vector u = (uj)j=0,…,n.
Thus, in one-dimensional case the problem has О(n) complexity, and it seems that no faster way to solve the problem is possible
However, even in 1-dimensional case we can pay attention on one drawback of the way we got the solution mentioned above:
The process is principally sequential - we calculate elements of the vector u = (uj)j=0,…,n one by one. There is no option for parallelization of this process.
On the other hand, it is the mass-parallelization that is the key feature of our Marlin Solver. This feature is not essential in 1D case, but it is crucial in case of bigger dimensions.