expqEi

ExpqEi is a computational method in numerical linear algebra and applied mathematics that blends exponential integrators with eigenvalue-aware projections. The name reflects its focus on exponential matrix functions, quantum-inspired heuristics for eigenvalue indexing, and their use in integrating time-dependent problems. In broad terms, expqEi aims to efficiently approximate the action of matrix functions, such as exp(tA), on vectors for large, sparse matrices, while simultaneously providing good estimation of dominant eigenpairs.

Method and approach: The technique relies on a hybrid strategy that combines spectral information with time-stepping. A small subspace is formed from a set of approximate dominant eigenvalues and eigenvectors of the matrix A. Within this subspace, a tailored exponential-like expansion is used to advance the solution in time, while the remainder of the spectrum is treated with a complementary, typically time-stepping-based method. This separation can improve stability for stiff problems and reduce computational cost when the matrix has a favorable spectrum or structure.

Applications: ExpqEi is discussed in the context of large-scale linear and semi-linear dynamical systems, where computing actions of matrix functions is common. Potential use cases include simulations of physical processes governed by stiff PDEs, quantum-inspired models, and large network dynamics where rapid, repeated evaluations of exp(tA) v are needed.

Limitations and status: ExpqEi is an area of active research with varying performance across problem classes. Its effectiveness depends on spectral properties, the chosen subspace size, and implementation details. It is not yet universally adopted and is often considered an experimental or hybrid approach alongside established exponential integrators and Krylov methods.

See also: exponential integrators, Krylov subspace methods, matrix functions, eigenvalue problems.