On non strictly diagonally dominant pentadiagonal matrices
DOI:
https://doi.org/10.35819/remat2024v10i2id7012Keywords:
Crout's method, pentadiagonal matrix, non strictly diagonally dominant matricesAbstract
Based on Crout's method, we will present, in this work, new non singularity criteria and sufficient conditions for existence of the LU factorization, for non strictly diagonally dominant pentadiagonal matrices. Crout's method is a recursive process of n stages that obtains the factorization A = LU of a pentadiagonal matrix of order n. In this recursive process of obtaining both the lower triangular matrix L and the upper triangular matrix U, the parameters alpha_i, 1 <= i <= n, must be non-zero to ensure that det(A) neq 0 and A = LU. Crout's recursive method is replaced by the analysis of sufficient conditions that can be verified simultaneously with low computational cost.
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ALMEIDA, C. G. de; REMIGIO, S. A. E. Non singularity criteria for non strictly diagonally dominant pentadiagonal matrices. In: CONGRESSO NACIONAL DE MATEMÁTICA APLICADA E COMPUTACIONAL, LXIII, 2023, Universidade Federal de Mato Grosso do Sul, Bonito/MS. Proceeding Series of the Brazilian Society of Computational and Applied Mathematics, v. 10, n. 1, São Paulo: SBMAC, p. 010074-1 - 010074-7, 2023. DOI: https://doi.org/10.5540/03.2023.010.01.0074.
ALMEIDA, C. G. de; REMIGIO, S. A. E. Sufficient Conditions for Existence of the LU Factorization of Toeplitz Symmetric Tridiagonal Matrices. Trends in Computational and Applied Mathematics, São Carlos, SP, v. 24, n. 1, p. 177-190, Mar. 2023. DOI: https://doi.org/10.5540/tcam.2022.024.01.00177.
BANK, Randolph E.; ROSE, Donald J. Marching algorithms for elliptic boundary value problems. I: The constant coefficient case. SIAM Journal on Numerical Analysis, v. 14, n. 5, p. 792-829, 1977. DOI: https://doi.org/10.1137/0714055.
EL-MIKKAWY, Moawwad E. A. On the inverse of a general tridiagonal matrix. Applied Mathematics and Computation, v. 150, n. 3, p. 669-679, 2004. DOI: https://doi.org/10.1016/S0096-3003(03)00298-4.
FISCHER, Charlotte F.; USMANI, Riaz A. Properties of some tridiagonal matrices and their application to boundary value problems. SIAM Journal on Numerical Analysis, v. 6, n. 1, p. 127-142, 1969. DOI: https://doi.org/10.1137/0706014.
JOHNSON, Charles Royal; MARIJUÁN, Carlos; PISONERO, Miriam. Diagonal dominance and invertibility of matrices. Special Matrices, v. 11, n. 1, p. 20220181, 2023. DOI: https://doi.org/10.1515/spma-2022-0181.
KOLOTILINA, Liliya Yurievna. Nonsingularity/singularity criteria for nonstrictly block diagonally dominant matrices. Linear Algebra and its Applications, v. 359, n. 1-3, p. 133-159, 2003. DOI: https://doi.org/10.1016/S0024-3795(02)00422-6.
MEURANT, Gérard. A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM Journal on Matrix Analysis and Applications, v. 13, n. 3, p. 707-728, 1992. DOI: https://doi.org/10.1137/0613045.
ZHAO, Xi-Le; HUANG, Ting-Zhu. On the inverse of a general pentadiagonal matrix. Applied Mathematics and Computation, v. 202, n. 2, p. 639-646, 2008. DOI: https://doi.org/10.1016/j.amc.2008.03.004.
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Copyright (c) 2024 César Guilherme de Almeida, Santos Alberto Enriquez Remigio
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