[Paper Review] FEAST Eigenvalue Solver v3.0 User Guide

Eric Polizzi, James Kestyn|arXiv (Cornell University)|Mar 19, 2012

Matrix Theory and Algorithms1 references5 citations

TL;DR

FEAST v3.0 is a high-performance numerical library for solving large-scale eigenvalue problems, supporting Hermitian, non-Hermitian, and complex-symmetric matrices via advanced contour integration and multiple quadrature rules. It introduces support for user-defined contours, stochastic eigenvalue counting, and enhanced load balancing, significantly improving accuracy and scalability for both symmetric and non-symmetric problems.

ABSTRACT

The FEAST library package represents an unified framework for solving various family of eigenvalue problems and achieving accuracy, robustness, high-performance and scalability on parallel architectures. Its originality lies with a new transformative numerical approach to the traditional eigenvalue algorithm design - the FEAST algorithm. The algorithm gathers key elements from complex analysis, numerical linear algebra and approximation theory, to construct an optimal subspace iteration technique using approximate spectral projectors. FEAST can be used for solving both standard and generalized forms of the Hermitian or non-Hermitian problems (linear or non-linear), and it belongs to the family of contour integration eigensolvers. FEAST's main computational task consists of a numerical quadrature computation that involves solving independent linear systems along a complex contour, each with multiple right hand sides. In v4.0, FEAST has been reimplemented using an inverse residual iteration algorithm which enables the linear systems to be solved with very low accuracy (in single precision) with no impact on the FEAST double precision convergence rate. As a result, v4.0 is on average 3-4 times faster than v2.1 and v3.0 using new default optimization parameters (v2.1 has been featured as Intel-MKL's principal HPC eigensolver since 2013). v4.0 also implements new important features such as IFEAST (using Inexact Iterative solver), Non-linear polynomial FEAST, and PFEAST with its 3-MPI levels of parallelism. FEAST is both a comprehensive library package, and an easy to use software. It includes flexible reverse communication interfaces and ready to use driver interfaces for dense, banded and sparse systems.

Motivation & Objective

• To extend the FEAST eigenvalue solver to support non-Hermitian matrices, enabling solution of complex-symmetric and general-complex eigenvalue problems.
• To improve numerical efficiency and scalability through optimized quadrature rules, including Zolotarev, Gauss, and Trapezoidal, and load balancing techniques.
• To provide users with flexible contour design capabilities, including elliptical and user-defined contours, for targeted eigenvalue extraction.
• To enhance usability and performance via expert-mode routines, stochastic estimation of eigenvalue counts, and support for custom integration nodes and weights.
• To ensure backward compatibility while introducing new features such as full contour integration and improved convergence criteria in the FEAST framework.

Proposed method

• The FEAST algorithm uses contour integration in the complex plane to project the eigenvalue problem onto a subspace, isolating eigenvalues within a specified interval.
• For non-Hermitian problems, dual subspaces (Q and Q̂) are employed to compute right and left eigenvectors, respectively, enabling accurate solution of complex eigenvalue problems.
• Multiple quadrature rules—Gauss, Trapezoidal, and Zolotarev—are implemented to optimize integration accuracy and computational load balancing.
• User-defined contours are supported via the expert-mode routines, allowing specification of segmented contours using nodes, weights, and geometric parameters.
• Stochastic estimation routines are integrated to predict the number of eigenvalues in a given interval, aiding in setting the initial search subspace size M₀.
• The library provides both driver-level and expert-level interfaces, with fpm parameters controlling integration type, contour shape, and convergence behavior.

Experimental results

Research questions

• RQ1 How can the FEAST algorithm be extended to solve non-Hermitian eigenvalue problems with complex eigenvalues and eigenvectors?
• RQ2 What quadrature rules and contour shapes optimize accuracy and load balancing in large-scale eigenvalue computations?
• RQ3 How can stochastic estimation improve the robustness and efficiency of eigenvalue count prediction in the FEAST framework?
• RQ4 What is the impact of user-defined contours on targeting specific eigenvalue clusters in non-Hermitian problems?
• RQ5 How do different integration schemes and parameter settings affect convergence and performance in FEAST v3.0?

Key findings

• FEAST v3.0 successfully supports non-Hermitian eigenvalue problems through dual subspace formulation and complex contour integration, enabling accurate computation of complex eigenvalues and eigenvectors.
• The inclusion of Zolotarev quadrature for Hermitian problems significantly improves integration accuracy and load balancing, especially for clustered eigenvalues.
• Stochastic eigenvalue counting estimates reduce the need for trial-and-error in setting the initial subspace dimension M₀, improving usability and convergence robustness.
• User-defined contours allow precise targeting of eigenvalue clusters in the complex plane, particularly beneficial for non-Hermitian problems with non-symmetric spectral distributions.
• The expert-mode interface with customizable nodes and weights enables fine-grained control over integration, enhancing performance and adaptability across diverse problem types.
• Performance improvements are observed, especially in dense matrix solvers, when factorizations are stored during integration, as enabled by the fpm(10) parameter.