Computing Matrix Trigonometric Functions with GPUs through Matlab

[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

Efficient and accurate algorithms for computing matrix trigonometric functions

[EN] Trigonometric matrix functions play a fundamental role in second order differential equations. This work presents an algorithm based on Taylor series for computing the matrix cosine. It uses a backward error analysis with improved bounds. Numerical experiments show that MATLAB implementations of this algorithm has higher accuracy than other MATLAB implementations of the state of the art in the majority of tests. Furthermore, we have implemented the designed algorithm in language C for general purpose processors, and in CUDA for one and two NVIDIA GPUs. We obtained a very good performance from these implementations thanks to the high computational power of these hardware accelerators and our effort driven to avoid as much communications as possible. All the implemented programs are accessible through the MATLAB environment. (C) 2016 Elsevier B.V. All rights reserved.This work has been supported by Spanish Ministerio de Economía y Competitividad and European Regional Development Fund (ERDF) grant TIN2014-59294-PAlonso-Jordá, P.; Ibáñez González, JJ.; Sastre Martinez, J.; Peinado Pinilla, J.; Defez Candel, E. (2017). Efficient and accurate algorithms for computing matrix trigonometric functions. Journal of Computational and Applied Mathematics. 309(1):325-332. https://doi.org/10.1016/j.cam.2016.05.015S325332309

An efficient and accurate algorithm for computing the matrix cosine based on New Hermite approximations

[EN] In this work we introduce new rational-polynomial Hermite matrix expansions which allow us to obtain a new accurate and efficient method for computing the matrix cosine. This method is compared with other state-of-the-art methods for computing the matrix cosine, including a method based on Pade approximants, showing a far superior efficiency, and higher accuracy. The algorithm implemented on the basis of this method can also be executed either in one or two NVIDIA GPUs, which demonstrates its great computational capacity. (C) 2018 Elsevier B.V. All rights reserved.This work has been partially supported by Spanish Ministerio de Economia y Competitividad and European Regional Development Fund (ERDF) grants TIN2014-59294-P, and T1N2017-89314-P.Defez Candel, E.; Ibáñez González, JJ.; Peinado Pinilla, J.; Sastre, J.; Alonso-Jordá, P. (2019). An efficient and accurate algorithm for computing the matrix cosine based on New Hermite approximations. Journal of Computational and Applied Mathematics. 348:1-13. https://doi.org/10.1016/j.cam.2018.08.047S11334

Fast Taylor polynomial evaluation for the computation of the matrix cosine

[EN] In this work we introduce a new method to compute the matrix cosine. It is based on recent new matrix polynomial evaluation methods for the Taylor approximation and a mixed forward and backward error analysis. The matrix polynomial evaluation methods allow to evaluate the Taylor polynomial approximation of the matrix cosine function more efficiently than using Paterson-Stockmeyer method. A sequential Matlab implementation of the new algorithm is provided, giving better efficiency and accuracy than state-of-the-art algorithms. Moreover, we provide an implementation in Matlab that can use NVIDIA CPUs easily and efficiently. (C) 2018 Elsevier B.V. All rights reserved.This work has been partially supported by Spanish Ministerio de Economía y Competitividad and European Regional Development Fund (ERDF) grants TIN2014-59294-P, and TIN2017-89314-P.Sastre, J.; Ibáñez González, JJ.; Alonso-Jordá, P.; Peinado Pinilla, J.; Defez Candel, E. (2019). Fast Taylor polynomial evaluation for the computation of the matrix cosine. Journal of Computational and Applied Mathematics. 354:641-650. https://doi.org/10.1016/j.cam.2018.12.041S64165035

Applications of GPU Computing to Control and Simulate Systems

[Abstract] This work deals with the new programming paradigm that exploits the benefits of modern Graphics Processing Units (GPUs), specifically their capacity to carry heavy calculations out for simulating systems or solving complex control strategies in real time

A new efficient and accurate spline algorithm for the matrix exponential computation

[EN] In this work an accurate and efficient method based on matrix splines for computing matrix exponential is given. An algorithm and a MATLAB implementation have been developed and compared with the state-of-the-art algorithms for computing the matrix exponential. We also developed a parallel implementation for large scale problems. This implementation allowed us to get a much better performance when working with this kind of problems.This work has been supported by Spanish Ministerio de Economia y Competitividad and the European Regional Development Fund (ERDF) grant TIN2014-59294-P.Defez Candel, E.; Ibáñez González, JJ.; Sastre, J.; Peinado Pinilla, J.; Alonso-Jordá, P. (2018). A new efficient and accurate spline algorithm for the matrix exponential computation. Journal of Computational and Applied Mathematics. 337(1):354-365. https://doi.org/10.1016/j.cam.2017.11.029S354365337