## A block J-Lanczos method for Hamiltonian matrices

Atika Archid, Abdeslem Hafid Bentbib, and Said Agoujil

### Abstract

This work aims to present a structure-preserving block Lanczos-like method. The Lanczos-like algorithm is an effective way to solve large sparse Hamiltonian eigenvalue problems. It can also be used to approximate $\exp(A)V$ for a given large square matrix $A$ and a tall-and-skinny matrix $V$ such that the geometric property of $V$ is preserved, which interests us in this paper. This approximation is important for solving systems of ordinary differential equations (ODEs) or time-dependent partial differential equations (PDEs). Our approach is based on a block $J$-tridiagonalization procedure of a Hamiltonian and skew-symmetric matrix using symplectic similarity transformations.

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### Key words

block $J$-Lanczos method, Hamiltonian matrix, skew-Hamiltonian matrix, symplectic matrix, symplectic reflector, block $J$-tridiagonal form, block $J$-Hessenberg form

### AMS subject classifications

65F15, 65F30, 65F50

### Links to the cited ETNA articles

 [14] Vol. 33 (2008-2009), pp. 207-220 L. Elbouyahyaoui, A. Messaoudi, and H. Sadok: Algebraic properties of the block GMRES and block Arnoldi methods

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