76 research outputs found
A Rodrigues-type formula for Gegenbauer matrix polynomials
No full textThis 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
Introducción al cálculo integral
No full textEl objetivo del presente libro es introducir en el estudio del cálculo integral para su posterior aplicación, mediante el uso de métodos básicos del cálculo integral. La intención principal es disponer de herramientas que permitan abordar el cálculo de integralse que aparecen en diferentes aplicacionesSoler Basauri, V.; Defez Candel, E. (2013). Introducción al cálculo integral. Editorial Universitat Politècnica de València. https://riunet.upv.es/handle/10251/2810
Computing Matrix Trigonometric Functions with GPUs through Matlab
No full text[EN] This paper presents an implementation of one of the most up-to-day algorithms proposed to compute the matrix trigonometric functions sine and cosine. The method used is based on Taylor series approximations which intensively uses matrix multiplications. To accelerate matrix products, our application can use from one to four NVIDIA GPUs by using the NVIDIA cublas and cublasXt libraries. The application, implemented in C++, can be used from the Matlab command line thanks to the mex files provided. We experimentally assess our implementation in modern and very high-performance NVIDIA GPUs.This work has been supported by Spanish Ministerio de Economia y Competitividad and the European Regional Development Fund (ERDF) Grants TIN2014-59294-P and TEC2015-67387-C4-1-RAlonso-Jordá, P.; Peinado Pinilla, J.; Ibáñez González, JJ.; Sastre, J.; Defez Candel, E. (2019). Computing Matrix Trigonometric Functions with GPUs through Matlab. The Journal of Supercomputing. 75(3):1227-1240. https://doi.org/10.1007/s11227-018-2354-1S12271240753Serbin SM (1979) Rational approximations of trigonometric matrices with application to second-order systems of differential equations. Appl Math Comput 5(1):75–92Serbin Steven M, Blalock Sybil A (1980) An algorithm for computing the matrix cosine. SIAM J Sci Stat Comput 1(2):198–204Hargreaves GI, Higham NJ (2005) Efficient algorithms for the matrix cosine and sine. Numer Algorithms 40:383–400Al-Mohy Awad H, Higham Nicholas J (2009) A new scaling and squaring algorithm for the matrix exponential. SIAM J Matrix Anal Appl 31(3):970–989Defez E, Sastre J, Ibáñez Javier J, Ruiz Pedro A (2011) Computing matrix functions arising in engineering models with orthogonal matrix polynomials. Math Comput Model 57:1738–1743Sastre J, Ibáñez J, Ruiz P, Defez E (2013) Efficient computation of the matrix cosine. Appl Math Comput 219:7575–7585Al-Mohy Awad H, Higham Nicholas J, Relton Samuel D (2015) New algorithms for computing the matrix sine and cosine separately or simultaneously. SIAM J Sci Comput 37(1):A456–A487Alonso P, Ibáñez J, Sastre J, Peinado J, Defez E (2017) Efficient and accurate algorithms for computing matrix trigonometric functions. J Comput Appl Math 309(1):325–332CUBLAS library (2017) http://docs.nvidia.com/cuda/cublas/index.html . Accessed May 2017Alonso Jordá P, Boratto M, Peinado Pinilla J, Ibáñez González JJ, Sastre Martínez J (2014) On the evaluation of matrix polynomials using several GPGPUs. Universitat Politècnica de València, 2014. http://hdl.handle.net/10251/39615 . Accessed Sept 2017Boratto Murilo, Alonso Pedro, Giménez Domingo, Lastovetsky Alexey L (2017) Automatic tuning to performance modelling of matrix polynomials on multicore and multi-gpu systems. J Supercomput 73(1):227–239Alonso P, Peinado J, Ibáñez J, Sastre J, Defez E (2017) A fast implementation of matrix trigonometric functions sine and cosine. In: Proceedings of the 17th International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2017), pp 51–55, Costa Ballena, Rota, Cadiz (Spain), July 4th–8thSastre Jorge, Ibáñez Javier, Alonso Pedro, Peinado Jesús, Defez Emilio (2017) Two algorithms for computing the matrix cosine function. Appl Math Comput 312:66–77Paterson Michael S, Stockmeyer Larry J (1973) On the number of nonscalar multiplications necessary to evaluate polynomials. SIAM J Comput 2(1):60–66Higham Nicholas J (2008) Functions of matrices: theory and computation. SIAM, PhiladelphiaSastre J, Ibáñez Javier J, Defez E, Ruiz Pedro A (2011) Efficient orthogonal matrix polynomial based method for computing matrix exponential. Appl Math Comput 217:6451–6463Sastre J, Ibáñez Javier J, Defez E, Ruiz Pedro A (2015) Efficient scaling-squaring Taylor method for computing matrix exponential. SIAM J Sci Comput 37(1):A439–455Higham NJ, Tisseur F (2000) A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra. SIAM J Matrix Anal Appl 21:1185–1201Demmel JW (1987) A counterexample for two conjectures about stability. IEEE Trans Autom Control 32:340–343Wright Thomas G (2002) EigTool library. http://www.comlab.ox.ac.uk/pseudospectra/eigtool/ . Accessed May 201
A stable numerical method for solving variable coefficient advection–diffusion models
No full textAbstractIn a recent paper [E. Defez, R. Company, E. Ponsoda, L. Jódar, Aplicación del método CE-SE a la ecuación de advección-difusión con coeficientes variables, Congreso de Métodos Numéricos en Ingenierá (SEMNI), Granada, Spain, 2005] a modified space–time conservation element and solution element scheme for solving the advection–diffusion equation with time-dependent coefficients, is proposed. This equation appears in many physical and technological models like gas flow in industrial tubes, conduction of heat in solids or the evaluation of the heating through radiation of microwaves when the properties of the media change with time. This method improves the well-established methods, like finite differences or finite elements: the integral form of the problem exploits the physical properties of conservation of flow, unlike the differential form. Also, this explicit scheme evaluates the variable and its derivative simultaneously in each knot of the partitioned domain. The modification proposed in [E. Defez, R. Company, E. Ponsoda, L. Jódar, Aplicación del método CE-SE a la ecuación de advección-difusión con coeficientes variables, Congreso de Métodos Numéricos en Ingenierá (SEMNI), Granada, Spain, 2005] with regard the original method [S.C. Chang, The method of space–time conservation element and solution element. A new approach for solving the Navier–Stokes and Euler equations, J. Comput. Phys. 119 (1995) 295–324] consists of keeping the variable character of the coefficients in the solution element, without considering the linear approximation. In this paper the stability of the proposed method is studied and a CFL condition is obtained
Boosting the computation of the matrix exponential
No full text[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
Simulation of harmonic oscillators on the lattice
No full text[EN] This work deals with the simulation of a two¿dimensional ideal lattice having simple tetragonal geometry. The harmonic character of the oscillators give rise to a system of second¿order linear differential equations, which can be recast into matrix form. The explicit solutions which govern the dynamics of this system can be expressed in terms of matrix trigonometric functions. For the derivation we employ the Lagrangian formalism to determine the correct solutions, which extremize the underlying action of the system. In the numerical evaluation we develop diverse state¿of¿the¿art algorithms which efficiently tackle equations with matrix sine and cosine functions. For this purpose, we introduce two special series related to trigonometric functions. They provide approximate solutions of the system through a suitable combination. For the final computation an algorithm based on Taylor expansion with forward and backward error analysis for computing those series had to be devised. We also implement several MATLAB programs which simulate and visualize the two¿dimensional lattice and check its energy conservation.This work has been supported by the Spanish Ministerio de Economia y Competitividad, the European Regional Development Fund (ERDF) under grant TIN2017-89314-P, and the Programa de Apoyo a la Investigacion y Desarrollo 2018 (PAID-06-18) of the Universitat Politecnica de Valencia under grant SP20180016.Tung, MM.; Ibáñez González, JJ.; Defez Candel, E.; Sastre, J. (2020). Simulation of Harmonic Oscillators on the Lattice. Mathematical Methods in the Applied Sciences. 43(14):8237-8252. https://doi.org/10.1002/mma.6510S823782524314Dehghan, M., & Hajarian, M. (2009). Determination of a matrix function using the divided difference method of Newton and the interpolation technique of Hermite. Journal of Computational and Applied Mathematics, 231(1), 67-81. doi:10.1016/j.cam.2009.01.021Dehghan, M., & Hajarian, M. (2010). Computing matrix functions using mixed interpolation methods. Mathematical and Computer Modelling, 52(5-6), 826-836. doi:10.1016/j.mcm.2010.05.013Kazem, S., & Dehghan, M. (2017). Application of finite difference method of lines on the heat equation. Numerical Methods for Partial Differential Equations, 34(2), 626-660. doi:10.1002/num.22218Kazem, S., & Dehghan, M. (2018). Semi-analytical solution for time-fractional diffusion equation based on finite difference method of lines (MOL). Engineering with Computers, 35(1), 229-241. doi:10.1007/s00366-018-0595-5Paterson, 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). Efficient orthogonal matrix polynomial based method for computing matrix exponential. Applied Mathematics and Computation, 217(14), 6451-6463. doi:10.1016/j.amc.2011.01.004Higham, N. J. (2008). Functions of Matrices. doi:10.1137/1.9780898717778Sastre, 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.2010.12.049Serbin, S. M., & Blalock, S. A. (1980). An Algorithm for Computing the Matrix Cosine. SIAM Journal on Scientific and Statistical Computing, 1(2), 198-204. doi:10.1137/0901013Ruiz, P., Sastre, J., Ibáñez, J., & Defez, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics, 291, 370-379. doi:10.1016/j.cam.2015.04.001Higham, N. J. (1988). FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation. ACM Transactions on Mathematical Software, 14(4), 381-396. doi:10.1145/50063.21438
A Method to Solve Non-homogeneous Strongly Coupled Mixed Parabolic Boundary Value Systems with Non-homogeneous Boundary Conditions
No full textIn 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
Two algorithms for computing the matrix cosine function
No full text[EN] The computation of matrix trigonometric functions has received remarkable attention in
the last decades due to its usefulness in the solution of systems of second order linear
differential equations. Several state-of-the-art algorithms have been provided recently for
computing these matrix functions. In this work, we present two efficient algorithms based
on Taylor series with forward and backward error analysis for computing the matrix cosine.
A MATLAB implementation of the algorithms is compared to state-of-the-art algorithms,
with excellent performance in both accuracy and cost.This work has been supported by Spanish Ministerio de Economia y Competitividad and the European Regional Development Fund (ERDF) grant TIN2014-59294-P.Sastre, J.; Ibáñez González, JJ.; Alonso-Jordá, P.; Peinado Pinilla, J.; Defez Candel, E. (2017). Two algorithms for computing the matrix cosine function. Applied Mathematics and Computation. 312:66-77. https://doi.org/10.1016/j.amc.2017.05.019S667731
Computing matrix functions solving coupled differential models
No full textIn this paper a modification of the method proposed in [E. Defez, L. Jódar, Some applications
of Hermite matrix polynomials series expansions, Journal of Computational and Applied
Mathematics 99 (1998) 105–117] for computing matrix sine and cosine based on Hermite
matrix polynomial expansions is presented. An algorithm and illustrative examples
demonstrate the performance of the new proposed method.
© 2009 Elsevier Ltd. All rights reserved.Defez Candel, E.; Sastre, J.; Ibáñez González, JJ.; Ruíz Martínez, PA. (2009). Computing matrix functions solving coupled differential models. Mathematical and Computer Modelling. 50(5):831-839. doi:10.1016/j.mcm.2009.05.012S83183950
Efficient computation of the matrix cosine
No full textTrigonometric 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
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