Dissecting the FEAST algorithm for generalized eigenproblems
TL;DR: Several critical issues that influence convergence and accuracy of the solver are identified: the choice of the starting vector space, the stopping criterion, how the inner linear systems impact the quality of the solution, and the use of FEAST for computing eigenpairs from multiple intervals.
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About: This article is published in Journal of Computational and Applied Mathematics. The article was published on 01 May 2013. and is currently open access. The article focuses on the topics: Solver & Eigenvalues and eigenvectors.
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Citations
Efficient estimation of eigenvalue counts in an interval
TL;DR: In this paper, the number of eigenvalues located in a given interval of a large sparse Hermitian matrix is estimated using polynomial and rational approximation filtering combined with a stochastic procedure.
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Zolotarev quadrature rules and load balancing for the FEAST eigensolver
TL;DR: This work proposes improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximation based on the work of Zolotarev, and improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.
118
High-performance implementation of Chebyshev filter diagonalization for interior eigenvalue computations
Andreas Pieper,Moritz Kreutzer,Andreas Alvermann,Martin Galgon,Holger Fehske,Georg Hager,Bruno Lang,Gerhard Wellein +7 more
TL;DR: The conceptual foundations of Chebyshev filter diagonalization are discussed and the impact of the choice of the damping kernel, search space size, and filter polynomial degree on the computational accuracy and effort are analyzed.
56
Nonlinear eigenvalue problems and contour integrals
Marc Van Barel,Peter Kravanja +1 more
TL;DR: Beyn's algorithm for solving nonlinear eigenvalue problems is given a new interpretation and a variant is designed in which the required information is extracted via the canonical polyadic decomposition of a Hankel tensor.
56
Parallel eigenvalue calculation based on multiple shift-invert Lanczos and contour integral based spectral projection method
Hasan Metin Aktulga,Lin Lin,Christopher Haine,Esmond G. Ng,Chao Yang +4 more
- 01 Jul 2014
TL;DR: The possibility of using multiple shift-invert Lanczos and contour integral based spectral projection method to compute a relatively large number of eigenvalues of a large sparse and symmetric matrix on distributed memory parallel computers is discussed.
40
References
•Book
Iterative Methods for Sparse Linear Systems
Yousef Saad
- 01 Apr 2003
TL;DR: This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.
•Book
The Symmetric Eigenvalue Problem
Beresford N. Parlett
- 01 Jan 1980
TL;DR: Parlett as discussed by the authors presents mathematical knowledge that is needed in order to understand the art of computing eigenvalues of real symmetric matrices, either all of them or only a few.
3.7K
Density-Matrix-Based Algorithm for Solving Eigenvalue Problems
TL;DR: A new numerical algorithm for solving the symmetric eigenvalue problem is presented, which takes its inspiration from the contour integration and density matrix representation in quantum mechanics.
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