On application of the Lanczos method to solution of some partial differential equations
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.
scientific article; zbMATH DE number 643759