A Rodrigues-type formula for Gegenbauer matrix polynomials

This paper centers on the derivation of a Rodrigues-type formula for the Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is givenThis research has been supported by the Universitat Politecnica de Valencia under grant PAID-06-11-2020.Defez Candel, E. (2013). A Rodrigues-type formula for Gegenbauer matrix polynomials. Applied Mathematics Letters. 26:899-903. https://doi.org/10.1016/j.aml.2013.04.001S8999032

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

Serie de Chebyshev para un operador Schrödinger 1-D ergódico

Las autofunciones {Ψn(ǫ)} (n ∈ Z +) de algunos operadores Schrödinger unidimensionales, de interés tanto académico como tecnológico, tienen una representación en forma de ecuación en diferencias lineal de segundo orden conocida como ecuación de Harper. La dificultad del análisis del espectro de estos operadores en el caso ergódico y la producción científica que ha generado puede consultarse en [9] J. Puig, Cantor Spectrum for the Almost Mathieu Operator, Comm. Math. Phys. 244(2):297-309, 2004. Con las condiciones iniciales adecuadas, se puede asociar a este tipo de operador una familia {Ψn(ǫ)} de autofunciones en forma de polinomios mónicos ortonormales. En [1] J.C. Abderramán, Chebyshev expansion for the eigenfunctions of the almost Mathieu operator, 6th Int. Congress on Industrial and Applied Math. ICIAM07, Zurich, 2007. se usan las propiedades algebraicas de los polinomios de Chebyshev de primera clase en la familia ortonormal de autofunciones, para separar variables y obtener para cada Ψn(ǫ) una expansión en serie de {Tk(ω)}. Los coeficientes de la serie {a (n) k (ǫ, λ)} se obtienen de forma recurrente y la energía ǫ depende de aquellos. En este trabajo se obtienen las matrices de transferencia entre los vectores de coeficientes y se comenta brevemente las propiedades del espectro y de las soluciones según valores de θ

Efficient computation of the matrix cosine

Trigonometric matrix functions play a fundamental role in second order differential equation systems. This work presents an algorithm for computing the cosine matrix function based on Taylor series and the cosine double angle formula. It uses a forward absolute error analysis providing sharper bounds than existing methods. The proposed algorithm had lower cost than state-of-the-art algorithms based on Hermite matrix polynomial series and Padé approximants with higher accuracy in the majority of test matrices.This work has been supported by Universitat Politecnica de Valencia Grant PAID-06-011-2020.Sastre, J.; Ibáñez González, JJ.; Ruiz Martínez, PA.; Defez Candel, E. (2013). Efficient computation of the matrix cosine. Applied Mathematics and Computation. 219:7575-7585. https://doi.org/10.1016/j.amc.2013.01.043S7575758521

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

Improvement on the bound of Hermite matrix polynomials

In this paper, we introduce an improved bound on the 2-norm of Hermite matrix polynomials. As a consequence, this estimate enables us to present and prove a matrix version of the Riemann-Lebesgue lemma for Fourier transforms. Finally, our theoretical results are used to develop a novel procedure for the computation of matrix exponentials with a priori bounds. A numerical example for a test matrix is provided. © 2010 Elsevier Inc. All rights reserved.This work has been partially supported by the Universidad Politecnica de Valencia under project PAID-06-07/3283 and the Generalitat Valenciana under project GVPRE/2008/340.Defez Candel, E.; Tung, MM.; Sastre, J. (2011). Improvement on the bound of Hermite matrix polynomials. Linear algebra and its applications. 434(8):1910-1919. https://doi.org/10.1016/j.laa.2010.12.015S19101919434

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

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