In this paper, we propose a Parallel Hybrid Direct-Iterative Eigensolver for Hermitian Eigenvalue Problems without tridiagonalization, denoted byPHIDE, which combines direct and iterative methods.PHIDEfirst reduces a Hermitian matrix to banded form, then applies a spectrum slicing algorithm to the banded matrix, and finally computes the eigenvectors of the original matrix via backtransformation. Compared with conventional direct eigensolvers,PHIDEavoids tridiagonalization, which involves many memory-bound operations. InPHIDE, the banded eigenvalue problem is solved using the contour integral method implemented in FEAST, which may yield slightly lower accuracy than tridiagonalization-based approaches. For sequences of correlated Hermitian eigenvalue problems arising in density functional theory (DFT),PHIDEachieves an average speedup of$1.22\times$over the state-of-the-art direct solver in ELPA when using 1024 processes. Numerical experiments are conducted on dense Hermitian matrices from real applications as well as large sparse matrices from the SuiteSparse and ELSES collections.

This short article describes the main new features added to SLEPc, the Scalable Library for Eigenvalue Problem Computations, in the past two and a half years, corresponding to five release versions. The main novelty is the extension of the SVD module with new problem types, such as the generalized SVD or the hyperbolic SVD. Additionally, many improvements have been incorporated in different parts of the library, including contour integral eigensolvers, preconditioning, and GPU support.

SLEPc is a parallel library for the solution of various types of large-scale eigenvalue problems. Over the past few years, we have been developing a module within SLEPc, called NEP, that is intended for solving nonlinear eigenvalue problems. These problems can be defined by means of a matrix-valued function that depends nonlinearly on a single scalar parameter. We do not consider the particular case of polynomial eigenvalue problems (which are implemented in a different module in SLEPc) and focus here on rational eigenvalue problems and other general nonlinear eigenproblems involving square roots or any other nonlinear function. The article discusses how the NEP module has been designed to fit the needs of applications and provides a description of the available solvers, including some implementation details such as parallelization. Several test problems coming from real applications are used to evaluate the performance and reliability of the solvers.

In this article, a parallel structured divide-and-conquer (PSDC) eigensolver is proposed for symmetric tridiagonal matrices based on ScaLAPACK and a parallel structured matrix multiplication algorithm, called PSMMA. Computing the eigenvectors via matrix-matrix multiplications is the most computationally expensive part of the divide-and-conquer algorithm, and one of the matrices involved in such multiplications is a rank-structured Cauchy-like matrix. By exploiting this particular property, PSMMA constructs the local matrices by using generators of Cauchy-like matrices without any communication, and further reduces the computation costs by using a structured low-rank approximation algorithm. Thus, both the communication and computation costs are reduced. Experimental results show that both PSMMA and PSDC are highly scalable and scale to 4096 processes at least. PSDC has better scalability than PHDC that was proposed in [16] and only scaled to 300 processes for the same matrices. Comparing with PDSTEDC in ScaLAPACK, PSDC is always faster and achieves 1.4x-1.6x speedup for some matrices with few deflations. PSDC is also comparable with ELPA, with PSDC being faster than ELPA when using few processes and a little slower when using many processes.

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

We present an ongoing effort to provide Java bindings in Open MPI [4]. We advocate including the bindings in the MPI distribution, rather than using a standalone, pure Java implementation.

SUMMARY In the numerical solution of large‐scale eigenvalue problems, Davidson‐type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson‐type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi–Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non‐symmetric problems. We present a parallel implementation that has been included in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with a Krylov–Schur eigensolver. Copyright © 2011 John Wiley & Sons, Ltd.

The Scalable Library for Eigenvalue Problem Computations (SLEPc) is a software library for computing a few eigenvalues and associated eigenvectors of a large sparse matrix or matrix pencil. It has been developed on top of PETSc and enforces the same programming paradigm.The emphasis of the software is on methods and techniques appropriate for problems in which the associated matrices are sparse, for example, those arising after the discretization of partial differential equations. Therefore, most of the methods offered by the library are projection methods such as Arnoldi or Lanczos, or other methods with similar properties. SLEPc provides basic methods as well as more sophisticated algorithms. It also provides built-in support for spectral transformations such as the shift-and-invert technique. SLEPc is a general library in the sense that it covers standard and generalized eigenvalue problems, both Hermitian and non-Hermitian, with either real or complex arithmetic.SLEPc can be easily applied to real world problems. To illustrate this, several case studies arising from real applications are presented and solved with SLEPc with little programming effort. The addressed problems include a matrix-free standard problem, a complex generalized problem, and a singular value decomposition. The implemented codes exhibit good properties regarding flexibility as well as parallel performance.

This paper presents an e-commerce tool that extends the conventional online store with a new section called room planner, a web application which is embedded in the virtual store. It allows the specification of the geometry of the room, placement of the objects and selection of the point of view. Then a realistic picture of the scene can be obtained. This functionality is very suitable for the furniture and ceramics sectors. In order to generate the images a parallel radiosity illumination algorithm has been implemented, which can be used in low-cost platforms such as a cluster of PCs, so that these technologies are affordable also for SMEs.