Polinomios ortogonales matriciales. Teoría y aplicaciones

La teoría de polinomios ortogonales matriarcales ha experimentado un desarrollo importante en las últimas décadas. El primer contacto de nuestro grupo de investigación con el tema surgio al dearrollar un método de Frobenius matriarcal para resolver ecuaciones diferenciales matriarcales de segundo orden sin aumentar la dimensión del problema. De esta forma, aparecieron soluciones de tipo polinomial matriarcal de ecuaciones diferenciales matriarcales que generalizaban las ecuaciones escalares clásicas de Hermite, Laguerre; Legendre.En la Tesis doctoral de R. Company [3] y en los trabajos siguientes [34],[35],[40],se introdujeron los polinomios matriarcales de Laguerre, Gegenbauer y Hermite, que verificaban ciertas propiedades de ortogonalidad de naturaleza no del todo transparente. Nos encontramos entonces, al disponer de ejemplos de clases concretas de polinomios ortogonales, sin estructurar la idea de ortogonalidad, a pesar de que ya se habían publicado, incluso en un contexto abstracto, pero próximo, resultados sobre ortogonalidad de polinomios en un álgebra no conmutativa [10],[11]. El objetivo de esta tesis es bidireccional; por una parte se trata de estructurar satisfactoriamente la idea de ortogonalidad para polinomios matriarcales, pero, con la intención dirigida a conseguir la utilidad en las aplicaciones que suministran las familias clásicas de polinomios ortogonales escalares. Estamos pensando, a corto plazo, en este trabajo, en utilizar la idea de ortoganalidad de polinomios matriarcales para aproximar integrales matriarcales y, también en desarrollar funciones matriarcales en serie de polinomios ortogonales matriarcales. Estas ambiciones han estado influidas por el enfoque de Chihara [5] y los trabajos de Stone [70] y Ghizzetti [29]. en la memoria se resuelven algunas de las dificultades que aparecen y, se suministran algunas respuestas, parcialmente publicadas en [36], [38], [39], [41], que no son ni mucho menos, el final de los muchos objetivos que en esta línea, pensamos se pueden conseguir. Entre las cuestiones a resolver objeto de este trabajo se encuentran: - Definición del concepto de ortogonalidad para polinomios matriarcales y funciones matriarcales. - Estructurar un espacio normado base donde yacen las funciones ortogonales matriarcales. - Estudio de la relación de la norma del espacio base y el concepto de ortogonalidad en ausencia de espacio Hilbert. - Solución del problema de la mejor aproximación matriarcal respecto a un funcional matriarcal definido positivo. - Series de Fourier matriarcales. - Obtención de análogos de Lema de Riemann-Lebesgue y de la igualdad (desigualdad) de Bessel-Parseval, en ausencia de estructura hilbertiana. - Introducción del concepto de totalidad para una familia de funciones ortogonales matriarcales en ausencia de estructura hilbertiana. - Posibilidad de desarrollo en serie de polinomios ortogonales matriarcales (solamente para el caso de Hermite) - Aplicación al desarrollo de la exponencial de una matriz.Defez Candel, E. (1996). Polinomios ortogonales matriciales. Teoría y aplicaciones [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/5641Palanci

A new type of Hermite matrix polynomial series

[EN] Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new generating functions for Hermite matrix polynomials and examines existence and convergence of their associated series expansion by using Mehler¿s formula for the general matrix case. Moreover, we derive interesting new relations for even- and odd-power summation in the generating-function expansion containing Hermite matrix polynomials. Some new results for the scalar case are also presented.The authors thank the Spanish Ministerio de Economia y Competitividad and the European Regional Development Fund (ERDF) for financial support under grant TIN2014-59294-P.Defez Candel, E.; Tung, MM. (2018). A new type of Hermite matrix polynomial series. Quaestiones Mathematicae. 41(2):205-212. https://doi.org/10.2989/16073606.2017.1376231S20521241

Boosting the computation of the matrix exponential

[EN] This paper presents new Taylor algorithms for the computation of the matrix exponential based on recent new matrix polynomial evaluation methods. Those methods are more efficient than the well known Paterson-Stockmeyer method. The cost of the proposed algorithms is reduced with respect to previous algorithms based on Taylor approximations. Tests have been performed to compare the MATLAB implementations of the new algorithms to a state-of-the-art Pade algorithm for the computation of the matrix exponential, providing higher accuracy and cost performances.This work has been supported by Spanish Ministerio de Economia y Competitividad and European Regional Development Fund (ERDF) grant TIN2014-59294-P.Sastre, J.; Ibáñez González, JJ.; Defez Candel, E. (2019). Boosting the computation of the matrix exponential. Applied Mathematics and Computation. 340:206-220. https://doi.org/10.1016/j.amc.2018.08.017S20622034

A Method to Solve Non-homogeneous Strongly Coupled Mixed Parabolic Boundary Value Systems with Non-homogeneous Boundary Conditions

In this paper, a method to construct the solution of non-homogeneous parabolic coupled systems with non-homogeneous boundary conditions of the type ut−Auxx = G(x, t), A1u(0, t)+B1ux(0, t) = P(t), A2u(l, t)+ B2ux(l, t) = Q(t), 0 0, u(x, 0) = f(x), where A is a positive stable matrix and A1, A2, B1, B2 are arbitrary matrices for which the block matrix A1 B1 A2 B2 is non-singular, is proposed. Two illustrative examples of the method are given.Soler Basauri, V.; Defez Candel, E.; Capilla Lladró, R. (2015). A Method to Solve Non-homogeneous Strongly Coupled Mixed Parabolic Boundary Value Systems with Non-homogeneous Boundary Conditions. International Journal of Mathematical Analysis. 9(40):1955-1970. doi:10.12988/ijma.2015.57176S1955197094

Solving engineering models using hyperbolic matrix functions

In this paper a method for computing hyperbolic matrix functions based on Hermite matrix polynomial expansions is outlined. Hermite series truncation together with Paterson-Stockmeyer method allow to compute the hyperbolic matrix cosine efficiently. A theoretical estimate for the optimal value of its parameters is obtained. An efficient and highly-accurate Hermite algorithm and a MATLAB implementation have been developed. The MATLAB implementation has been compared with the MATLAB function funm on matrices of different dimensions, obtaining lower execution time and higher accuracy in most cases. To do this we used an NVIDIA Tesla K20 GPGPU card, the CUDA environment and MATLAB. With this implementation we get much better performance for large scale problems. (C) 2015 Elsevier Inc. All rights reserved.This work has been supported by Spanish Ministerio de Educacion TIN2014-59294-P.Defez Candel, E.; Sastre, J.; Ibáñez González, JJ.; Peinado Pinilla, J. (2016). Solving engineering models using hyperbolic matrix functions. Applied Mathematical Modelling. 40(4):2837-2844. https://doi.org/10.1016/j.apm.2015.09.050S2837284440

On the Construction of Analytic-Numerical Approximations for a Class of Coupled Differential Models in Engineering

In this paper, a method to construct an analytic-numerical solution for homogeneous parabolic coupled systems with homogeneous boundary conditions of the type ut = Auxx, A1u(o,t) + B1ux(o,t) = 0, A2u(1,t) + B2ux(1,t) = 0, ot>0, u (x,0) = f(x), where A is a positive stable matrix and A1, B1, B1, B2, are arbitrary matrices for which the block matrix is non-singular, is proposed.Defez Candel, E.; Soler Basauri, V.; Capilla Lladró, R. (2015). On the Construction of Analytic-Numerical Approximations for a Class of Coupled Differential Models in Engineering. Open Journal of Modelling and Simulation. 3:1-18. doi:10.4236/ojmsi.2015.31001S118

Approximating a Special Class of Linear Fourth-Order Ordinary Differential Problems

[EN] Differential matrix models are an important component of many interesting applications in science and engineering. This work elaborates a procedure to approximate the solutions of special non linear fourth-order matrix differential problems by suitable matrix splinesThis work has been supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF) under grant TIN2014-59294-PDefez Candel, E.; Tung, MM.; Ibáñez González, JJ.; Sastre, J. (2016). Approximating a Special Class of Linear Fourth-Order Ordinary Differential Problems. Springer. 577-584. https://doi.org/10.1007/978-3-319-63082-3_89S577584Defez, E., Tung, M.M., Ibáñez, J., Sastre, J.: Approximating and computing nonlinear matrix differential models. Math. Comput. Model. 55(7), 2012–2022 (2012)Famelis, I., Tsitouras, C.: On modifications of Runge–Kutta–Nyström methods for solving y (4) = f(x, y). Appl. Math. Comput. 273, 726–734 (2016)Golub, G.H., Loan, C.F.V.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore, MD (1996)Hussain, K., Ismail, F., Senu, N.: Two embedded pairs of Runge-Kutta type methods for direct solution of special fourth-order ordinary differential equations. Math. Probl. Eng. 2015 (2015). doi:10.1155/2015/196595Loscalzo, F.R., Talbot, T.D.: Spline function approximations for solutions of ordinary differential equations. SIAM J. Numer. Anal. 4(3), 433–445 (1967)Olabode, B., et al.: Implicit hybrid block Numerov-type method for the direct solution of fourth-order ordinary differential equations. Am. J. Comput. Appl. Math. 5(5), 129–139 (2015)Papakostas, S.N., Tsitmidelis, S., Tsitouras, C.: Evolutionary generation of 7th order Runge - Kutta - Nyström type methods for solving y (4) = f(x, y). In: American Institute of Physics Conference Series, vol. 1702 (2015). doi: 10.1063/1.493898

Accurate and efficient matrix exponential computation

[EN] This work gives a new formula for the forward relative error of matrix exponential Taylor approximation and proposes new bounds for it depending on the matrix size and the Taylor approximation order, providing a new efficient scaling and squaring Taylor algorithm for the matrix exponential. A Matlab version of the new algorithm is provided and compared with Pad´e state-of-the-art algorithms obtaining higher accuracy in the majority of tests at similar or even lower cost.This work has been supported by the Programa de Apoyo a la Investigacion y el Desarrollo of the Universitat Politecnica de Valencia grant PAID-06-11-2020Sastre, J.; Ibáñez González, JJ.; Ruiz Martínez, PA.; Defez Candel, E. (2014). Accurate and efficient matrix exponential computation. International Journal of Computer Mathematics. 91(1):97-112. https://doi.org/10.1080/00207160.2013.791392S97112911Al-Mohy, A. H., & Higham, N. J. (2010). A New Scaling and Squaring Algorithm for the Matrix Exponential. SIAM Journal on Matrix Analysis and Applications, 31(3), 970-989. doi:10.1137/09074721xArioli, M., Codenotti, B., & Fassino, C. (1996). The Padé method for computing the matrix exponential. Linear Algebra and its Applications, 240, 111-130. doi:10.1016/0024-3795(94)00190-1S. Blackford and J. Dongarra,Installation guide for LAPACK, LAPACK Working Note 411, Department of Computer Science, University of Tenessee, 1999.Dieci, L., & Papini, A. (2000). Padé approximation for the exponential of a block triangular matrix. Linear Algebra and its Applications, 308(1-3), 183-202. doi:10.1016/s0024-3795(00)00042-2Dieci, L., & Papini, A. (2001). Numerical Algorithms, 28(1/4), 137-150. doi:10.1023/a:1014071202885Dolan, E. D., & Moré, J. J. (2002). Benchmarking optimization software with performance profiles. Mathematical Programming, 91(2), 201-213. doi:10.1007/s101070100263C. Fassino,Computation of matrix functions, Ph.D. thesis TD-7/93, Università di Pisa, Genova, 1993.Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms. doi:10.1137/1.9780898718027Higham, N. J. (2005). The Scaling and Squaring Method for the Matrix Exponential Revisited. SIAM Journal on Matrix Analysis and Applications, 26(4), 1179-1193. doi:10.1137/04061101xHigham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Higham, N. J., & Tisseur, F. (2000). A Block Algorithm for Matrix 1-Norm Estimation, with an Application to 1-Norm Pseudospectra. SIAM Journal on Matrix Analysis and Applications, 21(4), 1185-1201. doi:10.1137/s0895479899356080Moler, C., & Van Loan, C. (2003). Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later. SIAM Review, 45(1), 3-49. doi:10.1137/s00361445024180Paterson, M. S., & Stockmeyer, L. J. (1973). On the Number of Nonscalar Multiplications Necessary to Evaluate Polynomials. SIAM Journal on Computing, 2(1), 60-66. doi:10.1137/0202007Sastre, J., Ibáñez, J., Defez, E., & Ruiz, P. (2011). Accurate matrix exponential computation to solve coupled differential models in engineering. Mathematical and Computer Modelling, 54(7-8), 1835-1840. doi:10.1016/j.mcm.2010.12.04

On the Exact Solution of a Class of Homogeneous Strongly Coupled Mixed Parabolic Problems

[EN] In this paper an exact series solution for homogeneous parabolic coupled systems is constructed using a projection method. An illustrative example is given.This work has been supported by Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF) TIN2014-59294-P.Defez Candel, E.; Soler Basauri, V.; Romero Vivó, S.; Verdoy González, JA. (2019). On the Exact Solution of a Class of Homogeneous Strongly Coupled Mixed Parabolic Problems. Filomat. 33(3):897-915. https://doi.org/10.2298/FIL1903897DS89791533

Approximating and computing nonlinear matrix differential models

NOTICE: this is the author’s version of a work that was accepted for publication in Mathematical and Computer Modelling. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Mathematical and Computer Modelling Volume 55, Issues 7–8, April 2012, Pages 2012–2022 DOI: 10.1016/j.mcm.2011.11.060Differential matrix models are an essential ingredient of many important scientific and engineering applications. In this work, we propose a procedure to represent the solutions of first-order matrix differential equations Y(x) = f(x, Y(x)) with approximate matrix splines. For illustration of the method, we choose one scalar example, a simple vector model, and finally a Sylvester matrix differential equation as a test.This work has been supported by grant PAID-06-11-2020 from the Universitat Politecnica de Valencia, Spain.Defez Candel, E.; Tung ., MM.; Ibáñez González, JJ.; Sastre, J. (2012). Approximating and computing nonlinear matrix differential models. Mathematical and Computer Modelling. 55(7):2012-2022. https://doi.org/10.1016/j.mcm.2011.11.0602012202255