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PreprintCC BY 4.0arXiv

Improving the efficiency of a Hyperpower method for approximating inverse matrices

  • Alicia Cordero
  • Juan R. Torregrosa
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Abstract

In this manuscript, scalar norm-type accelerators are used for first time in order to increase the order of convergence of known iterative methods for solving the matrix equation X−1−A=0, while decreasing the number of matrix-matrix products. Specifically, our proposed scheme upgrades the order of convergence in a 50%, achieving the sixth-order of convergence with the computational cost of fourth-order Hyperpower method, in terms of matrix-matrix multiplications. In the main results, not only the convergence and its order is proven, but also its efficiency and stability. This new technique provides good numerical results, even compared with existing fourth-, sixth- and eighth-order schemes in large matrices of up to 106 entries, but also promises the opening of a new kind of procedures with good performance and scalability. Finally, an application to digital image restoration is made to show the behavior of the method, compared with existing ones.

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Cite

BibTeXRIS
BibTeX
@misc{cordero2026improving,
  title = {{Improving the efficiency of a Hyperpower method for approximating inverse matrices}},
  author = {Alicia Cordero and Juan R. Torregrosa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.04433},
  eprint = {2609.04433},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.04433},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Improving the efficiency of a Hyperpower method for approximating inverse matrices
AU  - Alicia Cordero
AU  - Juan R. Torregrosa
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2609.04433
UR  - https://arxiv.org/abs/2609.04433
AB  - In this manuscript, scalar norm-type accelerators are used for first time in order to increase the order of convergence of known iterative methods for solving the matrix equation X^−1−A=0, while decreasing the number of matrix-matrix products. Specifically, our proposed scheme upgrades the order of convergence in a 50%, achieving the sixth-order of convergence with the computational cost of fourth-order Hyperpower method, in terms of matrix-matrix multiplications. In the main results, not only the convergence and its order is proven, but also its efficiency and stability. This new technique provides good numerical results, even compared with existing fourth-, sixth- and eighth-order schemes in large matrices of up to 10^6 entries, but also promises the opening of a new kind of procedures with good performance and scalability. Finally, an application to digital image restoration is made to show the behavior of the method, compared with existing ones.
ER  -