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Some Hyperbolic Iterative Methods for Linear Systems

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  • K. Niazi Asil
  • M. Ghasemi Kamalvand

Abstract

The indefinite inner product defined by J = diag(j1, …, jn), jk ∈ {−1, +1}, arises frequently in some applications, such as the theory of relativity and the research of the polarized light. This indefinite scalar product is referred to as hyperbolic inner product. In this paper, we introduce three indefinite iterative methods: indefinite Arnoldi’s method, indefinite Lanczos method (ILM), and indefinite full orthogonalization method (IFOM). The indefinite Arnoldi’s method is introduced as a process that constructs a J‐orthonormal basis for the nondegenerated Krylov subspace. The ILM method is introduced as a special case of the indefinite Arnoldi’s method for J‐Hermitian matrices. IFOM is mentioned as a process for solving linear systems of equations with J‐Hermitian coefficient matrices. Finally, by providing numerical examples, the FOM, IFOM, and ILM processes have been compared with each other in terms of the required time for solving linear systems and also from the point of the number of iterations.

Suggested Citation

  • K. Niazi Asil & M. Ghasemi Kamalvand, 2020. "Some Hyperbolic Iterative Methods for Linear Systems," Journal of Applied Mathematics, John Wiley & Sons, vol. 2020(1).
    • Handle: RePEc:wly:jnljam:v:2020:y:2020:i:1:n:9874162
    DOI: 10.1155/2020/9874162
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    1. M. A. Krasnosel’skii & G. M. Vainikko & P. P. Zabreiko & Ya. B. Rutitskii & V. Ya. Stetsenko, 1972. "Approximate Solution of Operator Equations," Springer Books, Springer, number 978-94-010-2715-1, March.
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    Cited by:

    1. M. Ghasemi Kamalvand & K. Niazi Asil, 2020. "Indefinite Ruhe’s Variant of the Block Lanczos Method for Solving the Systems of Linear Equations," Advances in Mathematical Physics, John Wiley & Sons, vol. 2020(1).

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