Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time $O (m^131)$
<|reference_start|>Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time $O (m^131)$: We present a linear-system solver that, given an $n$-by-$n$ symmetric positive semi-definite, diagonally dominant matrix $A$ with $m$ non-zero entries and an $n$-vector $\bb $, produces a vector $\xxt$ within relative distance $\epsilon$ of the solution to $A \xx = \bb$ in time $O (m^{1.31} \log (n \kappa_{f} (A)/\epsilon)^{O (1)})$, where $\kappa_{f} (A)$ is the log of the ratio of the largest to smallest non-zero eigenvalue of $A$. In particular, $\log (\kappa_{f} (A)) = O (b \log n)$, where $b$ is the logarithm of the ratio of the largest to smallest non-zero entry of $A$. If the graph of $A$ has genus $m^{2\theta}$ or does not have a $K_{m^{\theta}} $ minor, then the exponent of $m$ can be improved to the minimum of $1 + 5 \theta $ and $(9/8) (1+\theta)$. The key contribution of our work is an extension of Vaidya's techniques for constructing and analyzing combinatorial preconditioners.<|reference_end|>
@article{spielman2003solving,
    title={Solving Sparse, Symmetric, Diagonally-Dominant Linear Systems in Time $O
  (m^{1.31})$},
    author={Daniel A. Spielman and Shang-Hua Teng},
    journal={arXiv preprint arXiv:cs/0310036},
    year={2003},
    archivePrefix={arXiv},
    eprint={cs/0310036},
    primaryClass={cs.DS cs.NA}
}