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Benjamin Krakoff1, Susan M Mniszewski2, Christian F A Negre3

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This study introduces a novel algorithm for computing extremal eigenvalues and eigenvectors of symmetric matrices by solving quadratic binary optimization problems. The method offers high precision and adaptability for generalized eigenvalue problems, demonstrating robust performance across various matrix types.

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Area of Science:

  • Numerical Analysis
  • Linear Algebra
  • Optimization

Background:

  • Eigenvalue problems are fundamental in numerous scientific and engineering disciplines.
  • Existing methods for computing extremal eigenvalues can be computationally intensive or lack precision.

Purpose of the Study:

  • To present a new algorithm for computing extremal eigenvalues and eigenvectors of symmetric matrices.
  • To demonstrate the algorithm's robustness, precision, and applicability to generalized eigenvalue problems.

Main Methods:

  • The core of the algorithm involves solving a sequence of Quadratic Binary Optimization (QBO) problems.
  • The approach is designed to achieve arbitrary precision for eigenvalue/eigenvector pairs.

Main Results:

  • The algorithm successfully computes extremal eigenvalues and eigenvectors for various symmetric matrices.
  • It demonstrates robustness across different matrix classes and can be adapted for generalized eigenvalue problems.
  • Performance analysis on random and practical matrices validates the method's effectiveness.

Conclusions:

  • The proposed QBO-based algorithm provides an effective and precise method for solving eigenvalue problems.
  • This approach offers a valuable alternative for computing extremal eigenvalues and eigenvectors, especially for challenging matrix types.