scholarly journals Numerical methods for palindromic eigenvalue problems: Computing the anti-triangular Schur form

2009 ◽  
Vol 16 (1) ◽  
pp. 63-86 ◽  
Author(s):  
D. Steven Mackey ◽  
Niloufer Mackey ◽  
Christian Mehl ◽  
Volker Mehrmann
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problems ◽  
Schur Form

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 chart of numerical methods for structured eigenvalue problems

1991 ◽  
pp. 547-547 ◽  
Author(s):  
Angelika Bunse-Gerstner ◽  
Volker Mehrmann ◽  
Ralph Byers
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problems

Numerical methods for eigenvalue problems

2003 ◽  
pp. 355-391
Author(s):  
Robert Plato
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problems

Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Bénard layer eigenvalue problems

1990 ◽  
Vol 431 (1883) ◽  
pp. 433-450 ◽  
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Eigenvalue Problems ◽  
Boundary Value ◽  
Second Order ◽  
Sixth Order ◽  
Asymptotic Estimates ◽  
Eighth Order ◽  
The Family ◽  
Convergent Method

A family of numerical methods is developed for the solution of special nonlinear sixth-order boundary-value problems. Methods with second-, fourth-, sixth- and eighth-order convergence are contained in the family. The problem is also solved by writing the sixth-order differential equation as a system of three second-order differential equations. A family of second- and fourth-order convergent methods is then used to obtain the solution. A second-order convergent method is discussed for the numerical solution of general nonlinear sixth-order boundary-value problems. This method, with modifications where necessary, is applied to the sixth-order eigenvalue problems associated with the onset of instability in a Bénard layer. Numerical results are compared with asymptotic estimates appearing in the literature.

Numerical methods for nonlinear two-parameter eigenvalue problems

2015 ◽  
Vol 56 (1) ◽  
pp. 241-262 ◽  
Author(s):  
Bor Plestenjak
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problems ◽  
Two Parameter

scholarly journals On Numerical Methods for Elliptic Transmission Eigenvalue Problems

2011 ◽  
Vol 4 ◽  
pp. 32-50
Author(s):  
Anirban Roy ◽  
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problems ◽  
Transmission Eigenvalue

Numerical Methods for Eigenvalue Problems

2020 ◽  
pp. 287-298
Author(s):  
Federico Milano ◽  
Ioannis Dassios ◽  
Muyang Liu ◽  
Georgios Tzounas
Keyword(s):  
Numerical Methods ◽  
Eigenvalue Problems
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