Stability of the Lanczos Method for Matrix Function Approximation
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2017
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866910701106233344 |
|---|---|
| author | Musco, Cameron Musco, Christopher Sidford, Aaron |
| author_facet | Musco, Cameron Musco, Christopher Sidford, Aaron |
| contents | The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[λ_{min}(A), λ_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision.
We resolve this problem, proving that when $\max_{x \in [λ_{min}, λ_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$.
We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89].
For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $κ$ where exact arithmetic Lanczos converges in $polylog(κ)$ iterations, but Greenbaum's bound predicts $Ω(κ^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(κ^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(κ)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_07788 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Stability of the Lanczos Method for Matrix Function Approximation Musco, Cameron Musco, Christopher Sidford, Aaron Data Structures and Algorithms Numerical Analysis The ubiquitous Lanczos method can approximate $f(A)x$ for any symmetric $n \times n$ matrix $A$, vector $x$, and function $f$. In exact arithmetic, the method's error after $k$ iterations is bounded by the error of the best degree-$k$ polynomial uniformly approximating $f(x)$ on the range $[λ_{min}(A), λ_{max}(A)]$. However, despite decades of work, it has been unclear if this powerful guarantee holds in finite precision. We resolve this problem, proving that when $\max_{x \in [λ_{min}, λ_{max}]}|f(x)| \le C$, Lanczos essentially matches the exact arithmetic guarantee if computations use roughly $\log(nC\|A\|)$ bits of precision. Our proof extends work of Druskin and Knizhnerman [DK91], leveraging the stability of the classic Chebyshev recurrence to bound the stability of any polynomial approximating $f(x)$. We also study the special case of $f(A) = A^{-1}$, where stronger guarantees hold. In exact arithmetic Lanczos performs as well as the best polynomial approximating $1/x$ at each of $A$'s eigenvalues, rather than on the full eigenvalue range. In seminal work, Greenbaum gives an approach to extending this bound to finite precision: she proves that finite precision Lanczos and the related CG method match any polynomial approximating $1/x$ in a tiny range around each eigenvalue [Gre89]. For $A^{-1}$, this bound appears stronger than ours. However, we exhibit matrices with condition number $κ$ where exact arithmetic Lanczos converges in $polylog(κ)$ iterations, but Greenbaum's bound predicts $Ω(κ^{1/5})$ iterations. It thus cannot offer significant improvement over the $O(κ^{1/2})$ bound achievable via our result. Our analysis raises the question of if convergence in less than $poly(κ)$ iterations can be expected in finite precision, even for matrices with clustered, skewed, or otherwise favorable eigenvalue distributions. |
| title | Stability of the Lanczos Method for Matrix Function Approximation |
| topic | Data Structures and Algorithms Numerical Analysis |
| url | https://arxiv.org/abs/1708.07788 |