Green Matrices, Minors and Hadamard Products

Delgado, Jorge; Peña, Guillermo; Peña, J. M.

doi:10.3390/axioms12080774

Cited by 2 publications

(1 citation statement)

References 37 publications

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“…The application of linear algebraic equations extends across a wide array of fields, demonstrating their versatility in addressing complex problems and providing essential tools for understanding, modeling, and solving diverse systems and phenomena. Here are some broader perspectives on the applications of LAEs [28][29][30][31][32].…”

Section: Introductionmentioning

confidence: 99%

Cutting-Edge Monte Carlo Framework: Novel “Walk on Equations” Algorithm for Linear Algebraic Systems

Todorov,

Dimov

2024

Axioms

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In this paper, we introduce the “Walk on Equations” (WE) Monte Carlo algorithm, a novel approach for solving linear algebraic systems. This algorithm shares similarities with the recently developed WE MC method by Ivan Dimov, Sylvain Maire, and Jean Michel Sellier. This method is particularly effective for large matrices, both real- and complex-valued, and shows significant improvements over traditional methods. Our comprehensive comparison with the Gauss–Seidel method highlights the WE algorithm’s superior performance, especially in reducing relative errors within fewer iterations. We also introduce a unique dominancy number, which plays a crucial role in the algorithm’s efficiency. A pivotal outcome of our research is the convergence theorem we established for the WE algorithm, demonstrating its optimized performance through a balanced iteration matrix. Furthermore, we incorporated a sequential Monte Carlo method, enhancing the algorithm’s efficacy. The most-notable application of our algorithm is in solving a large system derived from a finite-element approximation in constructive mechanics, specifically for a beam structure problem. Our findings reveal that the proposed WE Monte Carlo algorithm, especially when combined with sequential MC, converges significantly faster than well-known deterministic iterative methods such as the Jacobi method. This enhanced convergence is more pronounced in larger matrices. Additionally, our comparative analysis with the preconditioned conjugate gradient (PCG) method shows that the WE MC method can outperform traditional methods for certain matrices. The introduction of a new random variable as an unbiased estimator of the solution vector and the analysis of the relative stochastic error structure further illustrate the potential of our novel algorithm in computational mathematics.

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Section: Introductionmentioning

confidence: 99%

Cutting-Edge Monte Carlo Framework: Novel “Walk on Equations” Algorithm for Linear Algebraic Systems

Todorov,

Dimov

2024

Axioms

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Accurate Computations with Generalized Green Matrices

Delgado,

Peña,

Peña

2024

Symmetry

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We consider generalized Green matrices that, in contrast to Green matrices, are not necessarily symmetric. In spite of the loss of symmetry, we show that they can preserve some nice properties of Green matrices. In particular, they admit a bidiagonal decomposition. Moreover, for convenient parameters, the bidiagonal decomposition can be obtained efficiently and with high relative accuracy and it can also be used to compute all eigenvalues, all singular values, the inverse, and the solution of some linear system of equations with high relative accuracy. Numerical examples illustrate the high accuracy of the performed computations using the bidiagonal decompositions. Finally, nonsingular and totally positive generalized Green matrices are characterized.

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Green Matrices, Minors and Hadamard Products

Cited by 2 publications

References 37 publications

Cutting-Edge Monte Carlo Framework: Novel “Walk on Equations” Algorithm for Linear Algebraic Systems

Cutting-Edge Monte Carlo Framework: Novel “Walk on Equations” Algorithm for Linear Algebraic Systems

Accurate Computations with Generalized Green Matrices

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