## Abstract

We discuss various applications of trace estimation techniques for evaluating functions of the form tr(f(A)) where f is certain function. The first problem we consider that can be cast in this form is that of approximating the Spectral density or Density of States (DOS) of a matrix. The DOS is a probability density distribution that measures the likelihood of finding eigenvalues of the matrix at a given point on the real line, and it is an important function in solid state physics. We also present a few non-standard applications of spectral densities. Other trace estimation problems we discuss include estimating the trace of a matrix inverse tr(A^{-1}), the problem of counting eigenvalues and estimating the rank, and approximating the log-determinant (trace of log function). We also discuss a few similar computations that arise in machine learning applications. We review two computationally inexpensive methods to compute traces of matrix functions, namely, the Chebyshev expansion and the Lanczos Quadrature methods. A few numerical examples are presented to illustrate the performances of these methods in different applications.

Original language | English (US) |
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Title of host publication | High Performance Computing in Science and Engineering - 3rd International Conference, HPCSE 2017, Revised Selected Papers |

Editors | Jakub Sistek, Petr Tichy, Tomas Kozubek, Martin Cermak, Dalibor Lukas, Jiri Jaros, Radim Blaheta |

Publisher | Springer Verlag |

Pages | 19-33 |

Number of pages | 15 |

ISBN (Print) | 9783319971353 |

DOIs | |

State | Published - 2018 |

Event | 3rd International Conference on High Performance Computing in Science and Engineering, HPCSE 2017 - Karolinka, Czech Republic Duration: May 22 2017 → May 25 2017 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 11087 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 3rd International Conference on High Performance Computing in Science and Engineering, HPCSE 2017 |
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Country | Czech Republic |

City | Karolinka |

Period | 5/22/17 → 5/25/17 |

### Bibliographical note

Funding Information:This work was supported byNSF under grant CCF-1318597.