Preserving geometric properties of the exponential matrix by block Krylov subspace methods
type: Item
instance of: scholarly article
title: Preserving geometric properties of the exponential matrix by block Krylov subspace methods
zbMATH Open document ID: 1107.65039
author: N. E. Zubov
published in: BIT
publication date: 2007-01-05T00:00:00Z
review text: Given a large square real matrix \(A\) and a rectangular tall matrix \(Q,\) many application problems require the approximation of the operation \(\exp (A)Q\). The authors show that an appropiate use of the block Lanczos method allows one to obtain a structure preserving approximation to \(\exp (A)Q\) when \(A\) is skew-symmetric or skew-symmetric and Hamiltonan. Moreover, for \(A\) Hamiltonian the authors derive a new variant of the block Lanczos method that again preserves the geometric properties of the exact scheme. Numerical results are also provided.
reviewed by: Răzvan Răducanu
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 5081857
zbMATH Keywords: matrix exponential
zbMATH Keywords: Krylov subspace method
zbMATH Keywords: exponentials integrators
zbMATH Keywords: structure preservation
zbMATH Keywords: Hamiltonian matrix
zbMATH Keywords: skew-symmetric matrix
zbMATH Keywords: block Lanczos method
zbMATH Keywords: numerical results
describes a project that uses: Matlab
describes a project that uses: QMRPACK
describes a project that uses: JDQR
describes a project that uses: ABLE
describes a project that uses: JDQZ
MaRDI profile type: Publication
full work available at URL: s10543-006-0096-6
OpenAlex ID: W2051572273
cites work: A Lanczos-type method for multiple starting vectors
cites work: ABLE: An Adaptive Block Lanczos Method for Non-Hermitian Eigenvalue Problems
cites work: Templates for the Solution of Algebraic Eigenvalue Problems
cites work: A symmetric band Lanczos process based on coupled recurrences and some applications
cites work: Q2761411
cites work: An implicitly restarted symplectic Lanczos method for the Hamiltonian eigenvalue problem
cites work: An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem
cites work: Computing Lyapunov exponents on a Stiefel manifold
cites work: Approximating the exponential from a Lie algebra to a Lie group
cites work: Methods for the approximation of the matrix exponential in a Lie-algebraic setting
cites work: A Krylov projection method for systems of ODEs
cites work: A Class of Intrinsic Schemes for Orthogonal Integration
cites work: On the implementation of Lie group methods on the Stiefel manifold
cites work: Computation of the Exponential of Large Sparse Skew-Symmetric Matrices
cites work: Computation of a few Lyapunov exponents for continuous and discrete dynamical systems
cites work: The Cayley transform in the numerical solution of unitary differential systems
cites work: Krylov subspace approximation of eigenpairs and matrix functions in exact and computer arithmetic
cites work: The Geometry of Algorithms with Orthogonality Constraints
cites work: A theory for learning by weight flow on Stiefel-Grassman manifold
cites work: Software for simplified Lanczos and QMR algorithms
cites work: Efficient Solution of Parabolic Equations by Krylov Approximation Methods
cites work: A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems
cites work: Q4531870
cites work: On Krylov Subspace Approximations to the Matrix Exponential Operator
cites work: Exponential integrators for quantum-classical molecular dynamics
cites work: Exponential Integrators for Large Systems of Differential Equations
cites work: On Cayley-transform methods for the discretization of Lie-group equations
cites work: Efficient Computation of the Matrix Exponential by Generalized Polar Decompositions
cites work: Orthosymplectic integration of linear Hamiltonian systems
cites work: Analysis of Projection Methods for Rational Function Approximation to the Matrix Exponential
cites work: Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later
cites work: Q2712884
cites work: Magnus-factorized method for numerical solving the time-dependent Schrödinger equation
cites work: Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator
cites work: Q4414854
cites work: On theoretical and numerical aspects of symplectic Gram-Schmidt-like algorithms
cites work: On Hamiltonian and symplectic Lanczos processes
cites work: Generalized polar decompositions for the approximation of the matrix exponential
DOI: 10.1007/S10543-006-0096-6
Recommended article: On Krylov Subspace Approximations to the Matrix Exponential Operator
Recommended article: Analysis of inexact Krylov subspace methods for approximating the matrix exponential
Recommended article: Analysis of Some Krylov Subspace Approximations to the Matrix Exponential Operator
Recommended article: Block Krylov subspace methods for functions of matrices
Recommended article: Approximation of the matrix exponential operator by a structure-preserving block Arnoldi-type method
Recommended article: Error Bounds for the Krylov Subspace Methods for Computations of Matrix Exponentials
Recommended article: Geometric aspects of the theory of Krylov subspace methods
Recommended article: Block Krylov Subspace Methods for Functions of Matrices II: Modified Block FOM
Recommended article: Block Krylov subspace methods for approximating the linear combination of \(\varphi\)-functions arising in exponential integrators
Recommended article: Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices
description: scientific article; zbMATH DE number 5081857