## Abstract

In physics, it is sometimes desirable to compute the so-called density of states (DOS), also known as the spectral density, of a real symmetric matrix A. The spectral density can be viewed as a probability density distribution that measures the likelihood of finding eigenvalues near some point on the real line. The most straightforward way to obtain this density is to compute all eigenvalues of A, but this approach is generally costly and wasteful, especially for matrices of large dimension. There exist alternative methods that allow us to estimate the spectral density function at much lower cost. The major computational cost of these methods is in multiplying A with a number of vectors, which makes them appealing for large-scale problems where products of the matrix A with arbitrary vectors are relatively inexpensive. This article defines the problem of estimating the spectral density carefully and discusses how to measure the accuracy of an approximate spectral density. It then surveys a few known methods for estimating the spectral density and considers variations of existing methods. All methods are discussed from a numerical linear algebra point of view.

Original language | English (US) |
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Pages (from-to) | 34-65 |

Number of pages | 32 |

Journal | SIAM Review |

Volume | 58 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

### Bibliographical note

Funding Information:The work of the first and third authors was partially supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy contract DE-AC02-05CH11231. The work of the second and third authors was partially supported by the Scientific Discovery through the Advanced Computing (SciDAC) program funded by the U.S. Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences through grant DE-SC0008877.

## Keywords

- Approximation of distribution
- Density of states
- Large scale sparse matrix
- Quantum mechanics
- Spectral density