# Algorithms for sparse, symmetric, definite quadratic lambda-matrix eigenproblems

## Abstract

Methods are presented for computing eigenpairs of the quadratic lambda-matrix, M lambda/sup 2/ + C lambda + K, where M, C, and K are large and sparse, and have special symmetry-type properties. These properties are sufficient to insure that all the eigenvalues are real and that theory analogous to the standard symmetric eigenproblem exists. The methods employ some standard techniques such as partial tri-diagonalization via the Lanczos Method and subsequent eigenpair calculation, shift-and- invert strategy and subspace iteration. The methods also employ some new techniques such as Rayleigh-Ritz quadratic roots and the inertia of symmetric, definite, quadratic lambda-matrices.

- Authors:

- Publication Date:

- Research Org.:
- Union Carbide Corp., Oak Ridge, TN (USA). Computer Sciences Div.

- OSTI Identifier:
- 6404126

- Report Number(s):
- CONF-810251-1

TRN: 81-010989

- DOE Contract Number:
- W-7405-ENG-26

- Resource Type:
- Conference

- Resource Relation:
- Conference: 1981 army numerical analysis and computers conference, Huntsville, AL, USA, 26 Feb 1981

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENFUNCTIONS; ALGORITHMS; FACTORIZATION; EIGENVALUES; EQUATIONS; EQUATIONS OF MOTION; ITERATIVE METHODS; MATRICES; SCALARS; DIFFERENTIAL EQUATIONS; FUNCTIONS; MATHEMATICAL LOGIC; PARTIAL DIFFERENTIAL EQUATIONS; 658000* - Mathematical Physics- (-1987)

### Citation Formats

```
Scott, D.S., and Ward, R.C.
```*Algorithms for sparse, symmetric, definite quadratic lambda-matrix eigenproblems*. United States: N. p., 1981.
Web.

```
Scott, D.S., & Ward, R.C.
```*Algorithms for sparse, symmetric, definite quadratic lambda-matrix eigenproblems*. United States.

```
Scott, D.S., and Ward, R.C. Thu .
"Algorithms for sparse, symmetric, definite quadratic lambda-matrix eigenproblems". United States. https://www.osti.gov/servlets/purl/6404126.
```

```
@article{osti_6404126,
```

title = {Algorithms for sparse, symmetric, definite quadratic lambda-matrix eigenproblems},

author = {Scott, D.S. and Ward, R.C.},

abstractNote = {Methods are presented for computing eigenpairs of the quadratic lambda-matrix, M lambda/sup 2/ + C lambda + K, where M, C, and K are large and sparse, and have special symmetry-type properties. These properties are sufficient to insure that all the eigenvalues are real and that theory analogous to the standard symmetric eigenproblem exists. The methods employ some standard techniques such as partial tri-diagonalization via the Lanczos Method and subsequent eigenpair calculation, shift-and- invert strategy and subspace iteration. The methods also employ some new techniques such as Rayleigh-Ritz quadratic roots and the inertia of symmetric, definite, quadratic lambda-matrices.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1981},

month = {1}

}

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