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

A New Algorithm for Computing the Actions of Trigonometric and Hyperbolic Matrix Functions

A new algorithm is derived for computing the actions $f(tA)B$ and $f(tA^{1/2})B$, where $f$ is cosine, sinc, sine, hyperbolic cosine, hyperbolic sinc, or hyperbolic sine function. $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with $n_0 \ll n$. $A^{1/2}$ denotes any matrix square root of $A$ and it is never required to be computed. The algorithm offers six independent output options given $t$, $A$, $B$, and a tolerance. For each option, actions of a pair of trigonometric or hyperbolic matrix functions are simultaneously computed. The algorithm scales the matrix $A$ down by a positive integer $s$, approximates $f(s^{-1}tA)B$ by a truncated Taylor series, and finally uses the recurrences of the Chebyshev polynomials of the first and second kind to recover $f(tA)B$. The selection of the scaling parameter and the degree of Taylor polynomial are based on a forward error analysis and a sequence of the form $\|A^k\|^{1/k}$ in such a way the overall computational cost of the algorithm is optimized. Shifting is used where applicable as a preprocessing step to reduce the scaling parameter. The algorithm works for any matrix $A$ and its computational cost is dominated by the formation of products of $A$ with $n\times n_0$ matrices that could take advantage of the implementation of level-3 BLAS. Our numerical experiments show that the new algorithm behaves in a forward stable fashion and in most problems outperforms the existing algorithms in terms of CPU time, computational cost, and accuracy.Comment: 4 figures, 16 page

Map online system using internet-based image catalogue

Digital maps carry along its geodata information such as coordinate that is important in one particular topographic and thematic map. These geodatas are meaningful especially in military field. Since the maps carry along this information, its makes the size of the images is too big. The bigger size, the bigger storage is required to allocate the image file. It also can cause longer loading time. These conditions make it did not suitable to be applied in image catalogue approach via internet environment. With compression techniques, the image size can be reduced and the quality of the image is still guaranteed without much changes. This report is paying attention to one of the image compression technique using wavelet technology. Wavelet technology is much batter than any other image compression technique nowadays. As a result, the compressed images applied to a system called Map Online that used Internet-based Image Catalogue approach. This system allowed user to buy map online. User also can download the maps that had been bought besides using the searching the map. Map searching is based on several meaningful keywords. As a result, this system is expected to be used by Jabatan Ukur dan Pemetaan Malaysia (JUPEM) in order to make the organization vision is implemented

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

Group Iterative Spectrum Thresholding for Super-Resolution Sparse Spectral Selection

Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency, thereby resulting in a coherent design. The popular convex compressed sensing methods break down in presence of high coherence and large noise. We propose a new regularization approach to handle model collinearity and obtain parsimonious frequency selection simultaneously. It takes advantage of the pairing structure of sine and cosine atoms in the frequency dictionary. A probabilistic spectrum screening is also developed for fast computation in high dimensions. A data-resampling version of high-dimensional Bayesian Information Criterion is used to determine the regularization parameters. Experiments show the efficacy and efficiency of the proposed algorithms in challenging situations with small sample size, high frequency resolution, and low signal-to-noise ratio

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

Cross-subject dual-domain fusion network with task-related and task-discriminant component analysis enhancing one-shot SSVEP classification

This study addresses the significant challenge of developing efficient decoding algorithms for classifying steady-state visual evoked potentials (SSVEPs) in scenarios characterized by extreme scarcity of calibration data, where only one calibration is available for each stimulus target. To tackle this problem, we introduce a novel cross-subject dual-domain fusion network (CSDuDoFN) incorporating task-related and task-discriminant component analysis (TRCA and TDCA) for one-shot SSVEP classification. The CSDuDoFN framework is designed to comprehensively transfer information from source subjects, while TRCA and TDCA are employed to exploit the single available calibration of the target subject. Specifically, we develop multi-reference least-squares transformation (MLST) to map data from both source subjects and the target subject into the domain of sine-cosine templates, thereby mitigating inter-individual variability and benefiting transfer learning. Subsequently, the transformed data in the sine-cosine templates domain and the original domain data are separately utilized to train a convolutional neural network (CNN) model, with the adequate fusion of their feature maps occurring at distinct network layers. To further capitalize on the calibration of the target subject, source aliasing matrix estimation (SAME) data augmentation is incorporated into the training process of the ensemble TRCA (eTRCA) and TDCA models. Ultimately, the outputs of the CSDuDoFN, eTRCA, and TDCA are combined for SSVEP classification. The effectiveness of our proposed approach is comprehensively evaluated on three publicly available SSVEP datasets, achieving the best performance on two datasets and competitive performance on one. This underscores the potential for integrating brain-computer interface (BCI) into daily life.Comment: 10 pages,6 figures, and 3 table