A Laplacian decomposition algorithm for square matrices

• Alberto Conejero, J. ^[1] ; Antonio Falcó ^[2] ; María Mora-Jiménez ^[1]
  1. [1] Universidad Politécnica de Valencia

     Universidad Politécnica de Valencia

     Valencia, España

     
  2. [2] Universidad CEU Cardenal Herrera

     Universidad CEU Cardenal Herrera

     Valencia, España

     
• Localización: EACA 2022: XVII Encuentro de Álgebra Computacional y Aplicaciones / coord. por Carlos Galindo Pastor, Philippe Giménez, Fernando Javier Hernando Carrillo, F. Monserrat Delpalillo, Julio José Moyano Fernández, 2023, ISBN 978-84-19647-46-7, págs. 63-66
• Idioma: inglés
• Enlaces
  • Texto Completo Libro
• Resumen

  • Many of today’s problems can be posed as a system of equations with the form Ax = b. When A has the form A =d i=1idn1 ⊗ . . . idni−1 ⊗ Ai ⊗ idni+1 ⊗ · · · ⊗ idnd , we say that A is a Laplacian matrix. For example, these matrices appear when discretizing some PDEs, such as the Poisson equation, and are an interesting object of study since the Proper Generalized Decomposition algorithm converges very quickly with the solution of the associated linear system. This made us wonder if a generic square matrix M ∈ GL(RN ) could be decomposed in some way so that the study of the associated linear problem Mx = b would be simpler. In the main theorem of this work, we present a decomposition of the space RN×N that will help us to determine the matrix decomposition we are looking for. In addition, we will show the procedure to be carried out for it in the form of an algorithm.