  We develop an algorithm to solve tridiagonal systems of linear equations,
which appear in implicit finite-difference schemes of partial differential
equations (PDEs), being the time-dependent Schr\"{o}dinger equation (TDSE) an
ideal candidate to benefit from it. Our N-shaped partition method optimizes the
implementation of the numerical calculation on parallel architectures, without
memory size constraints. Specifically, we discuss the realization of our method
on graphics processing units (GPUs) and the Message Passing Interface (MPI). In
GPU implementations, our scheme is particularly advantageous for systems whose
size exceeds the global memory of a single processor. Moreover, because of its
lack of memory constraints and the generality of the algorithm, it is
well-suited for mixed architectures, typically available in large high
performance computing (HPC) centers. We also provide an analytical estimation
of the optimal parameters to implement our algorithm, and test numerically the
suitability of our formula in a GPU implementation. Our method will be helpful
to tackle problems which require large spatial grids for which ab-initio
studies might be otherwise prohibitive both because of large shared-memory
requirements and computation times.
