scholarly journals A Wilkinson-like multishift QR algorithm for symmetric eigenvalue problems and its global convergence

2012 ◽  
Vol 236 (15) ◽  
pp. 3556-3560 ◽  
Author(s):  
Kensuke Aishima ◽  
Takayasu Matsuo ◽  
Kazuo Murota ◽  
Masaaki Sugihara
Keyword(s):  
Global Convergence ◽  
Eigenvalue Problems ◽  
Qr Algorithm ◽  
Symmetric Eigenvalue Problems

scholarly journals Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation

2022 ◽  
Vol 48 (1) ◽  
pp. 1-36
Author(s):  
Mirko Myllykoski
Keyword(s):  
Eigenvalue Problems ◽  
Critical Path ◽  
Hessenberg Matrix ◽  
Single Node ◽  
Standard Case ◽  
Qr Algorithm ◽  
Schur Form ◽  
Tightly Coupled ◽  
Qz Algorithm ◽  
New Ideas

The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED) , and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.

A Filtered-Davidson Method for Large Symmetric Eigenvalue Problems

2017 ◽  
Vol 7 (1) ◽  
pp. 21-37 ◽  
Author(s):  
Cun-Qiang Miao
Keyword(s):  
Convergence Rate ◽  
Chebyshev Polynomials ◽  
Numerical Experiments ◽  
Eigenvalue Problems ◽  
Cubic Convergence ◽  
Davidson Method ◽  
Filtering Technique ◽  
Matrix Vector ◽  
Three Term Recurrence Relation ◽  
Symmetric Eigenvalue Problems

AbstractFor symmetric eigenvalue problems, we constructed a three-term recurrence polynomial filter by means of Chebyshev polynomials. The new filtering technique does not need to solve linear systems and only needs matrix-vector products. It is a memory conserving filtering technique for its three-term recurrence relation. As an application, we use this filtering strategy to the Davidson method and propose the filtered-Davidson method. Through choosing suitable shifts, this method can gain cubic convergence rate locally. Theory and numerical experiments show the efficiency of the new filtering technique.

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