Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector

Ibáñez González, JJ.; Alonso Abalos, JM.; Alonso-Jordá, P.; Defez Candel, E.; Sastre, J. (2022). Two Taylor Algorithms for Computing the Action of the Matrix Exponential on a Vector. Algorithms. 15(2):1-48. https://doi.org/10.3390/a1502004814815

Advances in the Approximation of the Matrix Hyperbolic Tangent

[EN] In this paper, we introduce two approaches to compute the matrix hyperbolic tangent. While one of them is based on its own definition and uses the matrix exponential, the other one is focused on the expansion of its Taylor series. For this second approximation, we analyse two different alternatives to evaluate the corresponding matrix polynomials. This resulted in three stable and accurate codes, which we implemented in MATLAB and numerically and computationally compared by means of a battery of tests composed of distinct state-of-the-art matrices. Our results show that the Taylor series-based methods were more accurate, although somewhat more computationally expensive, compared with the approach based on the exponential matrix. To avoid this drawback, we propose the use of a set of formulas that allows us to evaluate polynomials in a more efficient way compared with that of the traditional Paterson¿Stockmeyer method, thus, substantially reducing the number of matrix products (practically equal in number to the approach based on the matrix exponential), without penalising the accuracy of the resultThis research was funded by the Spanish Ministerio de Ciencia e Innovacion under grant number TIN2017-89314-P.Ibáñez González, JJ.; Alonso Abalos, JM.; Sastre, J.; Defez Candel, E.; Alonso-Jordá, P. (2021). Advances in the Approximation of the Matrix Hyperbolic Tangent. Mathematics. 9(11):1-20. https://doi.org/10.3390/math911121912091

Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials

[EN] This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results.This research was supported by the Vicerrectorado de Investigacion de la Universitat Politecnica de Valencia (PAID-11-21).Alonso Abalos, JM.; Ibáñez González, JJ.; Defez Candel, E.; Alvarruiz Bermejo, F. (2023). Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials. Mathematics. 11(3):1-22. https://doi.org/10.3390/math1103052012211

On the inverse of the Caputo matrix exponential

[EN] Matrix exponentials are widely used to efficiently tackle systems of linear differential equations. To be able to solve systems of fractional differential equations, the Caputo matrix exponential of the index a > 0 was introduced. It generalizes and adapts the conventional matrix exponential to systems of fractional differential equations with constant coefficients. This paper analyzes the most significant properties of the Caputo matrix exponential, in particular those related to its inverse. Several numerical test examples are discussed throughout this exposition in order to outline our approach. Moreover, we demonstrate that the inverse of a Caputo matrix exponential in general is not another Caputo matrix exponential.This work has been partially supported by Spanish Ministerio de Economia y Competitividad and European Regional Development Fund (ERDF) grants TIN2017-89314-P and by the Programa de Apoyo a la Investigacion y Desarrollo 2018 of the Universitat Politecnica de Valencia (PAID-06-18) grant SP20180016.Defez Candel, E.; Tung, MM.; Chen-Charpentier, BM.; Alonso Abalos, JM. (2019). On the inverse of the Caputo matrix exponential. Mathematics. 7(12):1-11. https://doi.org/10.3390/math7121137S111712Moler, 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/s00361445024180Ortigueira, M. D., & Tenreiro Machado, J. A. (2015). What is a fractional derivative? Journal of Computational Physics, 293, 4-13. doi:10.1016/j.jcp.2014.07.019Caputo, M. (1967). Linear Models of Dissipation whose Q is almost Frequency Independent--II. Geophysical Journal International, 13(5), 529-539. doi:10.1111/j.1365-246x.1967.tb02303.xRodrigo, M. R. (2016). On fractional matrix exponentials and their explicit calculation. Journal of Differential Equations, 261(7), 4223-4243. doi:10.1016/j.jde.2016.06.023Garrappa, R., & Popolizio, M. (2018). Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus. Journal of Scientific Computing, 77(1), 129-153. doi:10.1007/s10915-018-0699-

An Improved Taylor Algorithm for Computing the Matrix Logarithm

[EN] The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Pade approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson-Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison.This research was funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad Grant TIN2017-89314-P.Ibáñez González, JJ.; Sastre, J.; Ruíz Martínez, PA.; Alonso Abalos, JM.; Defez Candel, E. (2021). An Improved Taylor Algorithm for Computing the Matrix Logarithm. Mathematics. 9(17):1-19. https://doi.org/10.3390/math9172018S11991

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

[EN] Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form Y-(3)(x)=f(x,Y(x)). Some numerical test problems are also included, whose solutions are computed by our method.This research was partially funded by the European Regional Development Fund (ERDF) and the Spanish Ministerio de Economia y Competitividad Grant TIN2017-89314-PDefez Candel, E.; Ibáñez González, JJ.; Alonso Abalos, JM.; Tung, MM.; Real-Herraiz, TP. (2021). On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem. Mathematics. 9(18):1-17. https://doi.org/10.3390/math9182262S11791

EducaFarma 11.0

Memoria ID2022-036. Ayudas de la Universidad de Salamanca para la innovación docente, curso 2022-2023

Educafarma 10.0

Memoria ID-030. Ayudas de la Universidad de Salamanca para la innovación docente, curso 2021-2022

EducaFarma 9.0

Memoria ID-020 Ayudas de la Universidad de Salamanca para la innovación docente, curso 2020-2021

Multiancestry analysis of the HLA locus in Alzheimer’s and Parkinson’s diseases uncovers a shared adaptive immune response mediated by HLA-DRB1*04 subtypes

Across multiancestry groups, we analyzed Human Leukocyte Antigen (HLA) associations in over 176,000 individuals with Parkinson’s disease (PD) and Alzheimer’s disease (AD) versus controls. We demonstrate that the two diseases share the same protective association at the HLA locus. HLA-specific fine-mapping showed that hierarchical protective effects of HLA-DRB1*04 subtypes best accounted for the association, strongest with HLA-DRB1*04:04 and HLA-DRB1*04:07, and intermediary with HLA-DRB1*04:01 and HLA-DRB1*04:03. The same signal was associated with decreased neurofibrillary tangles in postmortem brains and was associated with reduced tau levels in cerebrospinal fluid and to a lower extent with increased Aβ42. Protective HLA-DRB1*04 subtypes strongly bound the aggregation-prone tau PHF6 sequence, however only when acetylated at a lysine (K311), a common posttranslational modification central to tau aggregation. An HLA-DRB1*04-mediated adaptive immune response decreases PD and AD risks, potentially by acting against tau, offering the possibility of therapeutic avenues