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Percy A Deift1, Govind Menon2, Sheehan Olver3

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Area of Science:

  • Computational mathematics
  • Numerical analysis
  • Statistical modeling

Background:

  • Numerical algorithms often involve random data, leading to variable convergence times.
  • Understanding the statistical properties of these convergence times is crucial for algorithm efficiency and reliability.

Purpose of the Study:

  • To investigate and demonstrate universality in the fluctuations of halting times for numerical computations with random data.
  • To establish that these fluctuations follow a predictable, universal statistical distribution.

Main Methods:

  • Analysis of halting time distributions for six standard numerical algorithms.
  • Inclusion of a neural computation and decision-making model to test universality.
  • Statistical analysis focusing on sample average and sample variance for scaling.

Main Results:

  • Observed two-component universality in halting time fluctuations.
  • Demonstrated that scaled histograms of halting times collapse to a universal curve.
  • Confirmed independence of this universal curve from the input data distribution with increasing dimension.

Conclusions:

  • The statistics of halting time are universally prescribed, depending only on the sample average and variance.
  • Universality simplifies the analysis of random data computations across diverse algorithms.
  • Findings have implications for algorithm design, performance prediction, and understanding complex systems.