### Abstract

The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time - i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components - the sample average and the sample variance - the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms aswell as a model of neural computation and decision-making. A link to relevant software is provided for readers who would like to do computations of their own.

Original language | English (US) |
---|---|

Pages (from-to) | 14973-14978 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 111 |

Issue number | 42 |

DOIs | |

State | Published - Oct 21 2014 |

### Fingerprint

### Keywords

- Decision times
- Numerical analysis
- Random matrix theory

### ASJC Scopus subject areas

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*111*(42), 14973-14978. https://doi.org/10.1073/pnas.1413446111

**Universality in numerical computations with random data.** / Deift, Percy; Menon, Govind; Olver, Sheehan; Trogdon, Thomas.

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 111, no. 42, pp. 14973-14978. https://doi.org/10.1073/pnas.1413446111

}

TY - JOUR

T1 - Universality in numerical computations with random data

AU - Deift, Percy

AU - Menon, Govind

AU - Olver, Sheehan

AU - Trogdon, Thomas

PY - 2014/10/21

Y1 - 2014/10/21

N2 - The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time - i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components - the sample average and the sample variance - the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms aswell as a model of neural computation and decision-making. A link to relevant software is provided for readers who would like to do computations of their own.

AB - The authors present evidence for universality in numerical computations with random data. Given a (possibly stochastic) numerical algorithm with random input data, the time (or number of iterations) to convergence (within a given tolerance) is a random variable, called the halting time. Two-component universality is observed for the fluctuations of the halting time - i.e., the histogram for the halting times, centered by the sample average and scaled by the sample variance, collapses to a universal curve, independent of the input data distribution, as the dimension increases. Thus, up to two components - the sample average and the sample variance - the statistics for the halting time are universally prescribed. The case studies include six standard numerical algorithms aswell as a model of neural computation and decision-making. A link to relevant software is provided for readers who would like to do computations of their own.

KW - Decision times

KW - Numerical analysis

KW - Random matrix theory

UR - http://www.scopus.com/inward/record.url?scp=84908072802&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908072802&partnerID=8YFLogxK

U2 - 10.1073/pnas.1413446111

DO - 10.1073/pnas.1413446111

M3 - Article

VL - 111

SP - 14973

EP - 14978

JO - Proceedings of the National Academy of Sciences of the United States of America

JF - Proceedings of the National Academy of Sciences of the United States of America

SN - 0027-8424

IS - 42

ER -