Polynomial Approximation to the Inverse of a Large Matrix
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912245603106816 |
|---|---|
| author | Embree, Mark Henningsen, Joel A. Jackson, Jordan Morgan, Ronald B. |
| author_facet | Embree, Mark Henningsen, Joel A. Jackson, Jordan Morgan, Ronald B. |
| contents | The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its approximation to the inverse often seems to track the accuracy of the GMRES iteration. We investigate the quality of this approximation through theory and experiment, noting the practical need to add copies of some polynomial terms to improve stability. To mitigate storage and orthogonalization costs, other approaches have appeal, such as polynomial preconditioned GMRES and deflation of problematic eigenvalues. Applications of such polynomial approximations include solving systems of linear equations with multiple right-hand sides (where the solutions to subsequent problems come simply by multiplying the polynomial against the new right-hand sides) and variance reduction in multilevel Monte Carlo methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polynomial Approximation to the Inverse of a Large Matrix Embree, Mark Henningsen, Joel A. Jackson, Jordan Morgan, Ronald B. Numerical Analysis 65F10, 15A09 The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its approximation to the inverse often seems to track the accuracy of the GMRES iteration. We investigate the quality of this approximation through theory and experiment, noting the practical need to add copies of some polynomial terms to improve stability. To mitigate storage and orthogonalization costs, other approaches have appeal, such as polynomial preconditioned GMRES and deflation of problematic eigenvalues. Applications of such polynomial approximations include solving systems of linear equations with multiple right-hand sides (where the solutions to subsequent problems come simply by multiplying the polynomial against the new right-hand sides) and variance reduction in multilevel Monte Carlo methods. |
| title | Polynomial Approximation to the Inverse of a Large Matrix |
| topic | Numerical Analysis 65F10, 15A09 |
| url | https://arxiv.org/abs/2502.18317 |