The aim of the project is to study the Darboux factorization for banded matrices. This kind of factorization provides an important tool in Approximation Theory and has interesting applications that arise frequently in several problems of science and engineering. These applications include linear banded systems, quadrature formulas, random walks, or certain integrable systems such as the Toda lattice or the Volterra lattice, among others.

The following tasks will be addressed:

1. Analysis of different factorizations for banded matrices.

2. Darboux factorization for matrices. Discrete Darboux transform and circular matrix transformations for banded matrices.

3. Implementation of an algorithm for Darboux factorization.

4. Applications to banded systems and integrable systems.

The proposed research aims to broaden the connections between evolutionary computation, artificial neural networks, and function approximation theory. The interest focuses on two directions: the development or application of bioinspired computational techniques in approximation theory and the discovery of new algorithms and properties for obtaining bioinspired intelligent systems. The previous work of the supervisors supports this research proposal on which we intend to provide new insights. [RM20] provides a new population initialization algorithm for grammar-guided genetic programming that directly applies to the construction of intelligent systems. [BD18] shows how to calculate the coefficients of certain difference equations using artificial intelligence techniques. It is well-known that there is classical interest in algorithms for computing these coefficients, especially applicable to numerical integration issues. However, the disadvantages of current methods, such as the lack of convergence guarantee or the inherent computational difficulties, led to this innovative approach. 

[BD18] Barrios Rolanía, D., Delgado Martínez, G., Manrique, D., Multilayered neural
architectures evolution for computing sequences of orthogonal polynomials, Ann. Math.
Artif. Intell., 84 (3-4) (2018), 161-184.

[RM20] Ramos Criado, P., Barrios Rolanía, D., Manrique, D. Serrano, E., Grammatically uniform population initialization for gramar-guided genetic programming, Soft Comput. 24 (2020), 11265-11282