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Neuberger's double-pass algorithm.

Ting-Wai Chiu1, Tung-Han Hsieh

  • 1Department of Physics, National Taiwan University Taipei, Taiwan 106, Taiwan.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 3, 2004
PubMed
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Neuberger's double-pass algorithm for matrix-vector multiplication is efficient, with computational cost independent of polynomial degree n. This enables high-precision approximations of matrix-vector products like (H)^(-1/2)Y without significant extra computational expense.

Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Operator Theory

Background:

  • Matrix-vector multiplication is a fundamental operation in scientific computing.
  • Rational polynomial approximations are used to approximate functions of operators.
  • Neuberger's double-pass algorithm offers a method for such approximations.

Purpose of the Study:

  • To analyze the computational cost of Neuberger's double-pass algorithm.
  • To determine the conditions under which the algorithm provides high-precision approximations.
  • To compare the efficiency of the double-pass versus single-pass algorithms.

Main Methods:

  • Analysis of floating-point operations in the double-pass algorithm.
  • Investigating the relationship between polynomial degree (n) and computational cost.

Related Experiment Videos

  • Comparing computational efficiency with varying numbers of sites and conjugate gradient iterations.

    • Main Results:

    • The number of floating-point operations is independent of the polynomial degree (n).
    • High-precision approximation of (H)^(-1/2)Y is achievable with large n without substantial cost.
    • A crossover threshold (n(T) ≈ 12-25) exists where the double-pass algorithm outperforms the single-pass algorithm.

    Conclusions:

    • Neuberger's double-pass algorithm offers a computationally efficient method for approximating matrix-vector products involving rational functions of operators.
    • The algorithm's cost-effectiveness is maintained even for high-degree polynomials, facilitating precise calculations.
    • The double-pass algorithm presents a performance advantage over single-pass methods beyond a specific threshold.