# Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices

@article{Morgan1986GeneralizationsOD, title={Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices}, author={Ronald B. Morgan and David S. Scott}, journal={Siam Journal on Scientific and Statistical Computing}, year={1986}, volume={7}, pages={817-825} }

This paper analyzes Davidson’s method for computing a few eigenpairs of large sparse symmetric matrices. An explanation is given for why Davidson’s method often performs well but occasionally performs very badly. Davidson’s method is then generalized to a method which offers a powerful way of applying preconditioning techniques developed for solving systems of linear equations to solving eigenvalue problems.

## 194 Citations

### Generalizations of davidson's method for computing eigenvalues of large nonsymmetric matrices

- Computer Science
- 1992

### Preconditioning the Lanczos Algorithm for Sparse Symmetric Eigenvalue Problems

- Computer ScienceSIAM J. Sci. Comput.
- 1993

A method for computing a few eigenpairs of sparse symmetric matrices is presented and analyzed that combines the power of preconditioning techniques with the efficiency of the Lanczos algorithm. The…

### A Jacobi-Davidson Iteration Method for Linear Eigenvalue Problems

- MathematicsSIAM Rev.
- 1996

A new method for the iterative computation of a few of the extremal eigenvalues of a symmetric matrix and their associated eigenvectors is proposed that has improved convergence properties and that may be used for general matrices.

### Numerical Solution of Linear Eigenvalue Problems

- Computer Science
- 2016

The Lanczos and Arnoldi methods are developed and described within the context of Krylov subspace eigensolvers and the idea of the Jacobi–Davidson method is presented.

### Iterative methods for the computation of a few eigenvalues of a large symmetric matrix

- Computer Science
- 1996

New iterative methods designed for the determination of a few extreme or non-extreme eigenvalues and associated eigenvectors based on the recursion formulas of the Implicitly Restarted Lanczos method, but differ from previous applications of these formulas in the selection of accelerating polynomial.

### Inexact newton preconditioning techniques for large symmetric eigenvalue problems

- Computer Science
- 1998

This paper studies a number of Newton methods and uses them to define new secondary linear systems of equations for the Davidson eigenvalue method to avoid some common pitfalls of the existing ones.

### Improved Algorithms for the Lowest Few Eigenvalues and Associated Eigenvectors of Large Matrices

- Physics
- 1992

### Block-Arnoldi and Davidson methods for unsymmetric large eigenvalue problems

- Computer Science
- 1993

The block-Arnoldi method and an adaptation of the Davidson method to unsymmetric matrices are presented, which give some theoretical results concerning the convergence and implementation aspects of the two methods.

### Iterative methods for calculations of extreme eigenvalues of large symmetric matrices

- Computer Science
- 2014

It is shown that the iterative methods for the calculation of the extreme eigenvalues and corresponding eigenvectors of the generalized symmetric matrix eigenvalue problem can be divided into two…

## References

SHOWING 1-10 OF 12 REFERENCES

### An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix

- Computer Science
- 1977

A particular class of regular splittings of not necessarily symmetric M-matrices is proposed. If the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a…

### Davidson's algorithm with and without perturbation corrections

- Computer Science
- 1980

The Davidson-Lanczos method as used here is capable, unlike the usual versions of the Lanczos method, of direct application to the generalised eigenproblem Ax= lambda Bx.

### Solution of Sparse Indefinite Systems of Linear Equations

- Computer Science
- 1975

The method of conjugate gradients for solving systems of linear equations with a symmetric positive definite matrix A is given as a logical development of the Lanczos algorithm for tridiagonalizing...

### The Symmetric Eigenvalue Problem

- Art
- 1980

According to Parlett, 'Vibrations are everywhere, and so too are the eigenvalues associated with them. As mathematical models invade more and more disciplines, we can anticipate a demand for…

### ITERATIVE SOLUTION OF IMPLICIT APPROXIMATIONS OF MULTIDIMENSIONAL PARTIAL DIFFERENTIAL EQUATIONS

- Mathematics
- 1968

Summary. A new iterative method has been developed for solving the large sets of algebraic equations that arise in the approximate solution of multidimensional partial differential equations by…

### The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices

- Mathematics
- 1975

### A two-level iterative methodfor large sparse generalized eigenvalue calculations

- Ph.D. thesis, Courant Institute, New York Univ., New York, October
- 1983

### A two-level iterative method for large sparse generalized eigenvalue calculations

- A two-level iterative method for large sparse generalized eigenvalue calculations
- 1983

### Iterative solution ofimplicit approximations ofmultidimensionalpartial differential equations

- SIAM J. Numer. Anal., 5
- 1968