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Fast algorithm for finding the eigenvalue distribution of very large matrices

Hams1, De Raedt H

  • 1Institute for Theoretical Physics and Materials Science Centre, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|November 23, 2000
PubMed
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This study analyzes the equation of motion method for large matrices in quantum physics. The method shows linear scaling and increased efficiency with matrix size, validated by spin-1/2 model calculations.

Area of Science:

  • Quantum Physics
  • Computational Physics
  • Materials Science

Background:

  • The eigenvalue distribution (density of states) is crucial for understanding material properties.
  • Calculating this for very large matrices is computationally intensive.
  • Existing methods often require significant memory and CPU resources.

Purpose of the Study:

  • To provide a theoretical analysis of the equation of motion method for computing the density of states.
  • To rigorously estimate the statistical error associated with this method.
  • To evaluate the method's efficiency and accuracy for quantum physics applications.

Main Methods:

  • Theoretical analysis of the equation of motion method.
  • Derivation of a rigorous statistical error estimate.

Related Experiment Videos

  • Application of the method and its imaginary-time variant to exactly solvable spin-1/2 models.

    • Main Results:

    • The equation of motion method exhibits linear scaling of memory and CPU requirements with matrix dimension.
    • Computational efficiency increases with matrix size.
    • Statistical errors were studied in relation to sample and matrix size for specific models.

    Conclusions:

    • The equation of motion method is an efficient approach for determining the density of states for large matrices in quantum physics.
    • The method's performance and error characteristics are well-defined and scale favorably with problem size.
    • This provides a valuable computational tool for condensed matter physics and quantum mechanics.