article · Results in Applied Mathematics · 2025

Lanczos algorithm explained in statistics

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Abstract

The Lanczos algorithm is a well-known three-term recurrence that can be used to generate an orthogonal basis for a Krylov subspace derived by a symmetric matrix. In the paper, we present a statistical interpretation of the entries of the tridiagonal matrix generated by the Lanczos process with a diagonal matrix X and an initial vector e . We show that the entries on the main diagonal line can be interpreted as weighted mean and the entries on the super-diagonal line can be understood as weighted sum of variance . Besides, a recurrence for producing the entries on the off-diagonal entries of the tridiagonal matrix is discovered, which leads to a new implementation of the Lanczos process. Finally, numerical examples are provided to investigate the preservation of orthogonality and efficiency in data fitting.

Cite this paper

APA 7

Niu, Q., Chen, M. L., & Wu, J. (2025). Lanczos algorithm explained in statistics. Results in Applied Mathematics, 28, 100666. https://doi.org/10.1016/j.rinam.2025.100666

MLA 9

Niu, Qiang, et al. “Lanczos algorithm explained in statistics.” Results in Applied Mathematics, vol. 28, 2025, p. 100666, https://doi.org/10.1016/j.rinam.2025.100666.

Chicago

Niu, Qiang, M. L. Chen, and Joseph Wu. 2025. “Lanczos algorithm explained in statistics.” Results in Applied Mathematics 28: 100666. https://doi.org/10.1016/j.rinam.2025.100666.

Harvard

Niu, Q., Chen, M.L. and Wu, J. (2025) ‘Lanczos algorithm explained in statistics’, Results in Applied Mathematics, 28, p. 100666. Available at: https://doi.org/10.1016/j.rinam.2025.100666.

BibTeX

@article{niu2025lanczos,
  title = {{Lanczos algorithm explained in statistics}},
  author = {Niu, Qiang and Chen, M. L. and Wu, Joseph},
  journal = {Results in Applied Mathematics},
  volume = {28},
  pages = {100666},
  year = {2025},
  doi = {10.1016/j.rinam.2025.100666}
}

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