# On the Convergence of an Implicitly Restarted Arnoldi Method

## Abstract

We show that Sorensen's [35] implicitly restarted Arnoldi method (including its block extension) is simultaneous iteration with an implicit projection step to accelerate convergence to the invariant subspace of interest. By using the geometric convergence theory for simultaneous iteration due to Watkins and Elsner [43], we prove that an implicitly restarted Arnoldi method can achieve a super-linear rate of convergence to the dominant invariant subspace of a matrix. Moreover, we show how an IRAM computes a nested sequence of approximations for the partial Schur decomposition associated with the dominant invariant subspace of a matrix.

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Labs., Albuquerque, NM (US); Sandia National Labs., Livermore, CA (US)

- Sponsoring Org.:
- US Department of Energy (US)

- OSTI Identifier:
- 9021

- Report Number(s):
- SAND99-1756J

TRN: AH200122%%138

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article

- Journal Name:
- SIAM Journal on Matrix Analysis and Its Applications

- Additional Journal Information:
- Other Information: Submitted to SIAM Journal on Matrix Analysis and Its Applications; PBD: 12 Jul 1999

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; CONVERGENCE; EIGENVALUES; ITERATIVE METHODS; MATRIX ELEMENTS; SIMULTANEOUS ITERATION; ARNOLDI REDUCTION; SCHUR DECOMPOSITION; RESTARTING; EIGEN-VALUES

### Citation Formats

```
Lehoucq, Richard B.
```*On the Convergence of an Implicitly Restarted Arnoldi Method*. United States: N. p., 1999.
Web.

```
Lehoucq, Richard B.
```*On the Convergence of an Implicitly Restarted Arnoldi Method*. United States.

```
Lehoucq, Richard B. Mon .
"On the Convergence of an Implicitly Restarted Arnoldi Method". United States. https://www.osti.gov/servlets/purl/9021.
```

```
@article{osti_9021,
```

title = {On the Convergence of an Implicitly Restarted Arnoldi Method},

author = {Lehoucq, Richard B},

abstractNote = {We show that Sorensen's [35] implicitly restarted Arnoldi method (including its block extension) is simultaneous iteration with an implicit projection step to accelerate convergence to the invariant subspace of interest. By using the geometric convergence theory for simultaneous iteration due to Watkins and Elsner [43], we prove that an implicitly restarted Arnoldi method can achieve a super-linear rate of convergence to the dominant invariant subspace of a matrix. Moreover, we show how an IRAM computes a nested sequence of approximations for the partial Schur decomposition associated with the dominant invariant subspace of a matrix.},

doi = {},

journal = {SIAM Journal on Matrix Analysis and Its Applications},

number = ,

volume = ,

place = {United States},

year = {1999},

month = {7}

}

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