Name
Abstract | ||
|---|---|---|
The Arnoldi method for standard eigenvalue problems possesses several attractive properties making it robust, reliable and efficient for many problems. The first result of this paper is a characterization of the solutions to an arbitrary (analytic) nonlinear eigenvalue problem (NEP) as the reciprocal eigenvalues of an infinite dimensional operator denoted $${\mathcal {B}}$$. We consider the Arnoldi method for the operator $${\mathcal {B}}$$ and show that with a particular choice of starting function and a particular choice of scalar product, the structure of the operator can be exploited in a very effective way. The structure of the operator is such that when the Arnoldi method is started with a constant function, the iterates will be polynomials. For a large class of NEPs, we show that we can carry out the infinite dimensional Arnoldi algorithm for the operator $${\mathcal {B}}$$ in arithmetic based on standard linear algebra operations on vectors and matrices of finite size. This is achieved by representing the polynomials by vector coefficients. The resulting algorithm is by construction such that it is completely equivalent to the standard Arnoldi method and also inherits many of its attractive properties, which are illustrated with examples. |
Year | DOI | Venue |
|---|---|---|
2012 | 10.1007/s00211-012-0453-0 | Numerische Mathematik |
Keywords | Field | DocType |
linear eigenvalue algorithm,infinite dimensional operator,standard eigenvalue problem,particular choice,standard arnoldi method,resulting algorithm,constant function,attractive property,nonlinear eigenvalue problem,standard linear algebra operation,arnoldi method,computer and information science | Linear algebra,Mathematical optimization,Polynomial,Arnoldi iteration,Eigenvalue algorithm,Mathematical analysis,Constant function,Dimensional operator,Operator (computer programming),Mathematics,Eigenvalues and eigenvectors | Journal |
Volume | Issue | ISSN |
122 | 1 | 0945-3245 |
Citations | PageRank | References |
16 | 0.90 | 14 |
Authors | ||
3 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| Jarlebring Elias | 1 | 84 | 11.48 |
| Wim Michiels | 2 | 513 | 77.24 |
| Karl Meerbergen | 3 | 443 | 55.04 |