An initial-boundary value problem for the n -dimensional ($n\geq 2$) time-dependent Schrödinger equation in a semi-infinite parallelepiped is considered. Starting from the Numerov–Crank–Nicolson finite-difference scheme, we first construct higher order scheme with splitting space averages having much better spectral properties for $n\geq 3$. Next we apply the Strang-type splitting with respect to the potential and, third, construct discrete transparent boundary conditions (TBC). For the resulting double-splitting method, the uniqueness of solution and the uniform in time L2-stability are proved and an error estimate is stated. Owing to the splitting, an effective direct algorithm using FFT (in the coordinate directions perpendicular to the leading axis of the parallelepiped) is applied to implement the scheme for general potential.
We describe a simple implementation of the Takagi factorization of symmetric matrices $A = U\Lambda U^T$ with unitary $U$ and diagonal $\Lambda$ in terms of the square root of an auxiliary unitary matrix and the singular value decomposition of $A$. The method is based on an algebraically exact expression. For parameterized family $A_\epsilon = A +\epsilon R = U_\epsilon \Lambda_\epsilon U^T_\epsilon $, $\epsilon >0$ with distinct singular values, the unitary matrices $U_\epsilon $ are discontinuous at the point $\epsilon = 0$, if the singular values of $A$ are multiple, but the composition $U_\epsilon \Lambda_\epsilon U^T_\epsilon $ remains numerically stable and converges to $A$. The factorization is represented as a fast and compact algorithm. Its demo version for Wolfram Mathematica and interactive numerical tests are available on Internet.