Abstract. Polynomial eigenvalue problems are often found in scientific computing applications. When the coefficient matrices of the polynomial are large and sparse, usually only a few eigenpairs are required and projection methods are the best choice. We focus on Krylov methods that operate on the companion linearization of the polynomial, but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. The problem may appear in the form of a low-degree polynomial (quartic or quintic, say) expressed in the monomial basis, or a high-degree polynomial (coming from interpolation of a nonlinear eigenproblem) expressed in a non-monomial basis. We have implemented a parallel solver in SLEPc covering both cases, that is able to compute exterior as well as interior eigenvalues via spectral transformation. We discuss important issues such as scaling and restart, and illustrate the robustness and performance of the solver with some numerical experiments.

In the context of symmetric-definite generalized eigenvalue problems, it is often required to compute all eigenvalues contained in a prescribed interval. For large-scale problems, the method of choice is the so-called spectrum slicing technique: a shift-and-invert Lanczos method combined with a dynamic shift selection that sweeps the interval in a smart way. This kind of strategies were proposed initially in the context of unrestarted Lanczos methods, back in the 1990's. We propose variations that try to incorporate recent developments in the field of Krylov methods, including thick restarting in the Lanczos solver and a rational Krylov update when moving from one shift to the next. We discuss a parallel implementation in the SLEPc library and provide performance results.

Routine applications of electronic structure theory to molecules and periodic systems need to compute the electron density from given Hamiltonian and, in case of non-orthogonal basis sets, overlap matrices. System sizes can range from few to thousands or, in some examples, millions of atoms. Different discretization schemes (basis sets) and different system geometries (finite non-periodic vs. infinite periodic boundary conditions) yield matrices with

Integration of Shift-and-Invert Parallel Spectral Transformation (SIPs) eigensolver (as implemented in the SLEPc library) into an ab initio molecular dynamics package, SIESTA, is described. The effectiveness of the code is demonstrated on applications to polyethylene chains, boron nitride sheets, and bulk water clusters. For problems with the same number of orbitals, the performance of the SLEPc eigensolver depends on the sparsity of the matrices involved, favoring reduced dimensional systems such as polyethylene or boron nitride sheets in comparison to bulk systems like water clusters. For all problems investigated, performance of SIESTA-SIPs exceeds the performance of SIESTA with default solver (ScaLAPACK) at the larger number of cores and the larger number of orbitals. A method that improves the load-balance with each iteration in the self-consistency cycle by exploiting the emerging knowledge of the eigenvalue spectrum is demonstrated. © 2018 Wiley Periodicals, Inc.

We investigate how to adapt the Q-Arnoldi method for the case of symmetric quadratic eigenvalue problems, that is, we are interested in computing a few eigenpairs (λ , x) of (λ 2 M + λC + K)x = 0 with M,C, K symmetric n × n matrices. This problem has no particular structure, in the sense that eigenvalues can be complex or even defective. Still, symmetry of the matrices can be exploited to some extent. For this, we perform a symmetric linearization Ay = λ By, where A, B are symmetric 2n × 2n matrices but the pair (A, B) is indefinite and hence standard Lanczos methods are not applicable. We implement a symmetric-indefinite Lanczos method and enrich it with a thick-restart technique. This method uses pseudo inner products induced by matrix B for the orthogonalization of vectors (indefinite Gram-Schmidt). The projected problem is also an indefinite matrix pair. The next step is to write a specialized, memory-efficient version that exploits the block structure of A and B, referring only to the original problem matrices M,C, K as in the Q-Arnoldi method.This results in what we have called the Q-Lanczos method. Furthermore, we define a stabilized variant analog of the TOAR method. We show results obtained with parallel implementations in SLEPc.

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations illustrate that the pre-asymptotic phase is significantly reduced compared to standard uniform h and p strategies. Moreover, the efficiency grows with the refractive index contrast, which makes the new strategy highly attractive for metal-dielectric structures. The hp-refinement strategy together with the efficient parallel code result in highly accurate approximations and short runtimes on multi processor platforms.

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In the quadratic eigenvalue problem with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos-based method for computing all real eigenvalues contained in a given interval of large scale symmetric quadratic eigenvalue problems. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general non-hyperbolic case. Our implementation is memory-efficient by representing the computed pseudo-Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.