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Strictly Balancing Matrices in Polynomial Time Using Osborne's Iteration
| العنوان: | Strictly Balancing Matrices in Polynomial Time Using Osborne's Iteration |
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| المؤلفون: | Ostrovsky, Rafail, Rabani, Yuval, Yousefi, Arman |
| سنة النشر: | 2017 |
| المجموعة: | Computer Science |
| مصطلحات موضوعية: | Computer Science - Data Structures and Algorithms |
| الوصف: | Osborne's iteration is a method for balancing $n\times n$ matrices which is widely used in linear algebra packages, as balancing preserves eigenvalues and stabilizes their numeral computation. The iteration can be implemented in any norm over $\mathbb{R}^n$, but it is normally used in the $L_2$ norm. The choice of norm not only affects the desired balance condition, but also defines the iterated balancing step itself. In this paper we focus on Osborne's iteration in any $L_p$ norm, where $p < \infty$. We design a specific implementation of Osborne's iteration in any $L_p$ norm that converges to a strictly $\epsilon$-balanced matrix in $\tilde{O}(\epsilon^{-2}n^{9} K)$ iterations, where $K$ measures, roughly, the {\em number of bits} required to represent the entries of the input matrix. This is the first result that proves that Osborne's iteration in the $L_2$ norm (or any $L_p$ norm, $p < \infty$) strictly balances matrices in polynomial time. This is a substantial improvement over our recent result (in SODA 2017) that showed weak balancing in $L_p$ norms. Previously, Schulman and Sinclair (STOC 2015) showed strong balancing of Osborne's iteration in the $L_\infty$ norm. Their result does not imply any bounds on strict balancing in other norms. Comment: 12 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/1704.07406 |
| رقم الانضمام: | edsarx.1704.07406 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |