On application of the Lanczos method to solution of some partial differential equations
type: Item
instance of: scholarly article
title: On application of the Lanczos method to solution of some partial differential equations
zbMATH Open document ID: 0807.65041
DOI: 10.1016/0377-0427(94)90305-0
author: V. Druskin
author: Leonid Knizhnerman
published in: Journal of Computational and Applied Mathematics
publication date: 1994-09-22T00:00:00Z
review text: The authors consider the problem of computing the vector (1) \(u= f(A)\phi\in \mathbb{R}^ n\), where \(\phi\in \mathbb{R}^ n\) is a given vector and \(f(.)\) denotes a given function defined on the spectral interval of the symmetric \(n\times n\) matrix \(A\). Problems of the form (1) appear in solving the systems of linear algebraic equations \((f(A)= A^{-1})\) as well as in solving semidiscrete approximation to parabolic \((f(A)= \exp(- tA))\), hyperbolic \((f(A)= \cos(+tA^{1/2}))\) and elliptic \((f(A)= \exp(- tA^{1/2}))\) partial differential equations. The authors propose to take the Lanczos method in order to construct the approximation \(u_ m= \|\phi\| Qf(H)e_ 1\in \mathbb{R}^ n\) to the solution \(u\) of (1), where \(e_ 1= (1,0,\dots,0)^ T\in \mathbb{R}^ m\). The tridiagonal symmetric \(m\times m\) matrix \(H\) and the \(n\times m\) matrix \(Q\) of the \(m\) Lanczos vectors are constructed by the Lanczos process. Estimates of the error \(\| u- u_ m\|\) as well as numerical results are given.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65N22
zbMATH DE Number: 643759
zbMATH Keywords: spectral Lanczos decomposition method
zbMATH Keywords: error estimates
zbMATH Keywords: linear algebraic equations
zbMATH Keywords: semidiscrete approximation
reviewed by: Ulrich Langer
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90305-0
OpenAlex ID: W2075900061
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description: scientific article; zbMATH DE number 643759