Efficient Evaluation of Matrix Polynomials beyond the Paterson-Stockmeyer Method

[EN] Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson-Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson-Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson-Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson-Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given.This research was partially funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad grant TIN2017-89314-P, and by the Programa de Apoyo a la Investigacion y Desarrollo 2018 of the Universitat Politecnica de Valencia grant PAID-06-18-SP20180016.Sastre, J.; Ibáñez González, JJ. (2021). Efficient Evaluation of Matrix Polynomials beyond the Paterson-Stockmeyer Method. Mathematics. 9(14):1-23. https://doi.org/10.3390/math9141600S12391

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

The walls of Cogotas and La Mesa de Miranda: Some notes on the Defensive Architecture of the vettones

El estudio sobre los sistemas defensivos de los castros vettones y, más específicamente, el de sus murallas, ha seguido la tendencia a considerarlas como un hecho único tanto en su concepción como en su construcción. De la observación de los procesos de limpieza, consolidación y restauración de algunos de los lienzos de las murallas de Las Cogotas (Cardeñosa, Ávila) y de La Mesa de Miranda (Chamartín de la Sierra, Ávila) así como de los datos obtenidos en la excavación de la casa C de este último castro, se deduce que los procesos de construcción se desarrollan a lo largo del tiempo de vida de los castros, no como un acto único sino como una serie de actos vinculados al propio devenir histórico de los asentamientos. Las remodelaciones, añadidos y reparaciones de los muros debieron ser una constante en el día a día de sus habitantes, más aún en aquellos momentos en que la presencia de ejércitos numerosos y bien estructurados generó los graves conflictos que sacudieron esta región desde finales del siglo III a. C. hasta la completa pacificación e incorporación a la estructura político/administrativa romana. El conocimiento de los distintos acontecimientos históricos y su vinculación a los procesos constructivos, de remodelación o de reparación de las murallas nos permitirá llegar a un mejor conocimiento de la historia de estos pueblos.The study on the defensive systems of the castros (hills forts) vettones and, more specifically, of its walls, has followed the tendency of considering them like a unique fact in its conception as in its construction. From the observation of the cleaning processes, consolidation and restoration of some sectors of the walls of Cogotas and La Mesa de Miranda as well as of the data collected in the excavation of house C of this castro is deduced that the construction processes are developed throughout the whole habitation time of the castros, not like a unique act but like a series of acts connected to the own historical development of the establishments. The rebuilding, adding and repairing of the walls had to be a constant in the everyday life of their inhabitants, even more in those moments in which the presence of numerous and well structured armies generated the serious conflicts that shook to this region from end of century III a. C. until the complete pacification and incorporation to the political/administrative Roman structure. The knowledge of the different historical events and its relation to the constructive processes, of remodeling or repairing of the walls, will allow us to reach a better knowledge of the history of these towns

La necrópolis de Trasguija: aproximación al estudio de la estructura social de las Cogotas

La necrópolis de Trasguija es, por el momento, la mejor conocida de todas las que se localizan en la Meseta Norte. La publicación íntegra efectuada por D. Juan Cabré de la misma, nos permite, a través de los datos que aporta, una mayor profundización en el conocimiento de la estructura social latente en el ritual representado en esta necrópolis.The necropolis of Trasguija is, for the moment, the best known of all those located in the North Meseta. The full publication by D. Juan Cabré of the same, allows us, through the data provided, a deeper understanding of the social structure latent in the ritual represented in this necropolis.peerReviewe

Assessing the value of the information provision for enhancing the autonomy of mobility impaired users. Madrid pilot Site Study.

A City is the space where every person acquires the citizen condition, which demands access to multiple services and facilities, and develops social relations in a free and equal condition of options. A lack of accessibility limits independency and autonomy. Thus, the relationship between “sustainable development” and “accessibility for all” becomes clearer, and both goals reinforce each other. In this sense, information plays a key role in order to overcome existing barriers, specially for people who rarely use public transport, have impaired mobility, or make a particular journey for the first time. The impact and benefits is linked with public transport as a “facilitator” of mobility, and, in particular, for the aim of intermodality. The usefulness of information that should be provided (both the information itself and how is offered) to mobility impaired users (MI users) is discussed on this paper based on following of the ASK-IT project that has being carry out on Madrid. The work was done in close cooperation with representatives of all different types of MI user groups

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

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