# An algorithm for the complete solution of the quartic eigenvalue problem

@article{Drmavc2019AnAF, title={An algorithm for the complete solution of the quartic eigenvalue problem}, author={Zlatko Drmavc and Ivana vSain Glibi'c}, journal={arXiv: Numerical Analysis}, year={2019} }

Quartic eigenvalue problem $(\lambda^4 A + \lambda^3 B + \lambda^2C + \lambda D + E)x = \mathbf{0}$ naturally arises e.g. when solving the Orr-Sommerfeld equation in the analysis of the stability of the {Poiseuille} flow, in theoretical analysis and experimental design of locally resonant phononic plates, modeling a robot with electric motors in the joints, calibration of catadioptric vision system, or e.g. computation of the guided and leaky modes of a planar waveguide. This paper proposes a… Expand

#### Figures and Tables from this paper

#### 3 Citations

New Numerical Algorithm for Deflation of Infinite and Zero Eigenvalues and Full Solution of Quadratic Eigenvalue Problems

- Mathematics, Computer Science
- ACM Trans. Math. Softw.
- 2020

A new method for computing all eigenvalues and eigenvectors of quadratic matrix pencil Q(λ)=λ2 M + λ C + K so that careful preprocessing allows scaling invariant/component-wise backward error and thus a better condition number. Expand

New Numerical Algorithm for Deflation of Infinite and Zero Eigenvalues and Full Solution of Quadratic Eigenvalue Problems

- Mathematics
- 2020

This article presents a new method for computing all eigenvalues and eigenvectors of quadratic matrix pencil Q(λ)=λ2 M + λ C + K. It is an upgrade of the quadeig algorithm by Hammarlinget al., whic...

Nonlinear eigenvalue problems for coupled Helmholtz equations modeling gradient-index graphene waveguides

- Physics, Computer Science
- J. Comput. Phys.
- 2020

An improved quality factor can be obtained for a family of gradient-index host materials with internal conducting interfaces and it is demonstrated how this result lays the groundwork for solving related shape optimization problems. Expand

#### References

SHOWING 1-10 OF 25 REFERENCES

Backward Error of Polynomial Eigenproblems Solved by Linearization

- Mathematics, Computer Science
- SIAM J. Matrix Anal. Appl.
- 2007

The results are shown to be entirely consistent with those of Higham, Mackey, and Tisseur on the conditioning of linearizations of $P and to derive backward error bounds depending only on the norms of the $A_i$ for the companion pencils and for the vector space of pencils recently identified. Expand

New Numerical Algorithm for Deflation of Infinite and Zero Eigenvalues and Full Solution of Quadratic Eigenvalue Problems

- Mathematics, Computer Science
- ACM Trans. Math. Softw.
- 2020

A new method for computing all eigenvalues and eigenvectors of quadratic matrix pencil Q(λ)=λ2 M + λ C + K so that careful preprocessing allows scaling invariant/component-wise backward error and thus a better condition number. Expand

FEAST Eigensolver for Nonlinear Eigenvalue Problems

- Mathematics, Physics
- J. Comput. Sci.
- 2018

The nonlinear FEAST algorithm can be used to solve nonlinear eigenvalue problems for the eigenpairs whose eigenvalues lie in a user-defined region in the complex plane, thereby allowing for the calculation of large numbers of eigenPairs in parallel. Expand

An algorithm for the complete solution of quadratic eigenvalue problems

- Mathematics, Computer Science
- TOMS
- 2013

A new algorithm for the computation of all the eigenvalues and optionally the right and left eigenvectors of dense quadratic matrix polynomials is developed that outperforms the MATLAB function polyeig in terms of both stability and efficiency. Expand

Parallel Krylov Solvers for the Polynomial Eigenvalue Problem in SLEPc

- Computer Science, Mathematics
- SIAM J. Sci. Comput.
- 2016

This work focuses on Krylov methods that operate on the companion linearization of the polynomial but exploit the block structure with the aim of being memory-efficient in the representation of the Krylov subspace basis. Expand

Backward Error and Condition of Polynomial Eigenvalue Problems

- 1999

We develop normwise backward errors and condition numbers for the polynomial eigenvalue problem. The standard way of dealing with this problem is to reformulate it as a generalized eigenvalue problem… Expand

A projection method for nonlinear eigenvalue problems using contour integrals

- Computer Science, Mathematics
- JSIAM Lett.
- 2013

In this paper, we indicate that the Sakurai-Sugiura method with Rayleigh-Ritz projection technique, a numerical method for generalized eigenvalue problems, can be extended to nonlinear eigenvalue… Expand

Optimal Scaling of Generalized and Polynomial Eigenvalue Problems

- Computer Science, Mathematics
- SIAM J. Matrix Anal. Appl.
- 2008

This paper investigates scaling for generalized and polynomial eigenvalue problems (PEPs) of arbitrary degree and introduces a generalization of the diagonal scaling by Lemonnier and Van Dooren to PEPs that is especially effective if some information about the magnitude of the wanted eigenvalues is available. Expand

Variational Formulation for Guided and Leaky Modes in Multilayer Dielectric Waveguides

- Mathematics
- 2009

The guided and leaky modes of a planar dielectric waveguide are eigensolutions of a singular Sturm-Liouville problem. The modes are the roots of a characteristic function which can be found with… Expand

POLYNOMIAL EIGENVALUE PROBLEMS WITH HAMILTONIAN STRUCTURE

- Mathematics
- 2002

We discuss the numerical solution of eigenvalue problems for matrix polynomials, where the coefficient matrices are alternating symmetric and skew symmetric or Hamiltonian and skew Hamiltonian. We… Expand