Numerical approximations of second-order matrix differential equations using higher-degree splines

Many studies of mechanical systems in engineering are based on second-order matrix models. This work discusses the second-order generalization of previous research on matrix differential equations dealing with the construction of approximate solutions for non-stiff initial problems Y 00(x) = f(x, Y (x), Y 0 (x)) using higher-degree matrix splines without any dimensional increase. An estimation of the approximation error for some illustrative examples are presented by using Mathematica. Several MatLab functions have also been developed, comparing, under equal conditions, accuracy and execution times with built-in MatLab functions. Experimental results show the advantages of solving the above initial problem by using the implemented MatLab functions.The authors wish to thank for financial support by the Universidad Politecnica de Valencia [grant number PAID-06-11-2020].Defez Candel, E.; Tung ., MM.; Solis Lozano, FJ.; Ibáñez González, JJ. (2015). Numerical approximations of second-order matrix differential equations using higher-degree splines. Linear and Multilinear Algebra. 63(3):472-489. https://doi.org/10.1080/03081087.2013.873427S472489633Loscalzo, F. R., & Talbot, T. D. (1967). Spline Function Approximations for Solutions of Ordinary Differential Equations. SIAM Journal on Numerical Analysis, 4(3), 433-445. doi:10.1137/0704038Al-Said, E. A. (2001). The use of cubic splines in the numerical solution of a system of second-order boundary value problems. Computers & Mathematics with Applications, 42(6-7), 861-869. doi:10.1016/s0898-1221(01)00204-8Al-Said, E. A., & Noor, M. A. (2003). Cubic splines method for a system of third-order boundary value problems. Applied Mathematics and Computation, 142(2-3), 195-204. doi:10.1016/s0096-3003(02)00294-1Kadalbajoo, M. K., & Patidar, K. C. (2002). Numerical solution of singularly perturbed two-point boundary value problems by spline in tension. Applied Mathematics and Computation, 131(2-3), 299-320. doi:10.1016/s0096-3003(01)00146-1Micula, G., & Revnic, A. (2000). An implicit numerical spline method for systems for ODEs. Applied Mathematics and Computation, 111(1), 121-132. doi:10.1016/s0096-3003(98)10111-xDefez, E., Soler, L., Hervás, A., & Santamaría, C. (2005). Numerical solution ofmatrix differential models using cubic matrix splines. Computers & Mathematics with Applications, 50(5-6), 693-699. doi:10.1016/j.camwa.2005.04.012Defez, E., Hervás, A., Soler, L., & Tung, M. M. (2007). Numerical solutions of matrix differential models using cubic matrix splines II. Mathematical and Computer Modelling, 46(5-6), 657-669. doi:10.1016/j.mcm.2006.11.027Ascher, U., Pruess, S., & Russell, R. D. (1983). On Spline Basis Selection for Solving Differential Equations. SIAM Journal on Numerical Analysis, 20(1), 121-142. doi:10.1137/0720009Brunner, H. (2004). On the Divergence of Collocation Solutions in Smooth Piecewise Polynomial Spaces for Volterra Integral Equations. BIT Numerical Mathematics, 44(4), 631-650. doi:10.1007/s10543-004-3828-5Tung, M. M., Defez, E., & Sastre, J. (2008). Numerical solutions of second-order matrix models using cubic-matrix splines. Computers & Mathematics with Applications, 56(10), 2561-2571. doi:10.1016/j.camwa.2008.05.022Defez, E., Tung, M. M., Ibáñez, J. J., & Sastre, J. (2012). Approximating and computing nonlinear matrix differential models. Mathematical and Computer Modelling, 55(7-8), 2012-2022. doi:10.1016/j.mcm.2011.11.060Claeyssen, J. R., Canahualpa, G., & Jung, C. (1999). A direct approach to second-order matrix non-classical vibrating equations. Applied Numerical Mathematics, 30(1), 65-78. doi:10.1016/s0168-9274(98)00085-3Froese, C. (1963). NUMERICAL SOLUTION OF THE HARTREE–FOCK EQUATIONS. Canadian Journal of Physics, 41(11), 1895-1910. doi:10.1139/p63-189Marzulli, P. (1991). Global error estimates for the standard parallel shooting method. Journal of Computational and Applied Mathematics, 34(2), 233-241. doi:10.1016/0377-0427(91)90045-lShore, B. W. (1973). Comparison of matrix methods applied to the radial Schrödinger eigenvalue equation: The Morse potential. The Journal of Chemical Physics, 59(12), 6450-6463. doi:10.1063/1.1680025ZHANG, J. F. (2002). OPTIMAL CONTROL FOR MECHANICAL VIBRATION SYSTEMS BASED ON SECOND-ORDER MATRIX EQUATIONS. Mechanical Systems and Signal Processing, 16(1), 61-67. doi:10.1006/mssp.2001.1441Flett, T. M. (1980). Differential Analysis. doi:10.1017/cbo978051189719

The Relationship Between Heart Rate Variability and Electroencephalography Functional Connectivity Variability Is Associated With Cognitive Flexibility

[EN] The neurovisceral integration model proposes a neuronal network that is related to heart rate activity and cognitive performance. The aim of this study was to determine whether heart rate variability (HRV) and variability in electroencephalographic (EEG) functional connectivity in the resting state are related to cognitive flexibility. Thirty-eight right-handed students completed the CAMBIOS test, and their heart and EEG activity was recorded during 6 min in the resting state with their eyes open. We calculated correlations, partial correlations and multiple linear regressions among HRV indices, functional brain connectivity variability and CAMBIOS scores. Furthermore, the sample was divided into groups according to CAMBIOS performance, and one-way ANOVA was applied to evaluate group differences. Our results show direct and inverse correlations among cognitive flexibility, connectivity (positive and negative task networks) and heartbeat variability. Partial correlations and multiple linear regressions suggest that the relation between HRV and CAMBIOS performance is mediated by neuronal oscillations. ANOVA confirms that HRV and variability in functional brain connectivity is related to cognitive performance. In conclusion, the levels of brain signal variability might predict cognitive flexibility in a cognitive task, while HRV might predict cognitive flexibility only when it is mediated by neuronal oscillations.This work was supported by the Spanish Ministry of Economy and Competitiveness (PSI2014-57231-R). Research was funded by Grant Nos. PSI2014-57231-R and PSI2017-88388-C4-3-R from Spanish Ministry of Science. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. The authors have declared that no competing interests exist.Alba, G.; Vila, J.; Rey, B.; Montoya, P.; Muñoz, MÁ. (2019). The Relationship Between Heart Rate Variability and Electroencephalography Functional Connectivity Variability Is Associated With Cognitive Flexibility. Frontiers in Human Neuroscience. 13(64):1-12. https://doi.org/10.3389/fnhum.2019.00064S1121364Alba, G., Pereda, E., Mañas, S., Méndez, L. D., Duque, M. R., González, A., & González, J. J. (2016). The variability of EEG functional connectivity of young ADHD subjects in different resting states. Clinical Neurophysiology, 127(2), 1321-1330. doi:10.1016/j.clinph.2015.09.134Albinet, C. T., Abou-Dest, A., André, N., & Audiffren, M. (2016). Executive functions improvement following a 5-month aquaerobics program in older adults: Role of cardiac vagal control in inhibition performance. Biological Psychology, 115, 69-77. doi:10.1016/j.biopsycho.2016.01.010Albinet, C. T., Boucard, G., Bouquet, C. A., & Audiffren, M. (2010). Increased heart rate variability and executive performance after aerobic training in the elderly. 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Right frontal event related EEG coherence (ERCoh) differentiates good from bad performers of the Wisconsin Card Sorting Test (WCST). Neurophysiologie Clinique/Clinical Neurophysiology, 37(2), 63-75. doi:10.1016/j.neucli.2007.02.002Catarino, A., Churches, O., Baron-Cohen, S., Andrade, A., & Ring, H. (2011). Atypical EEG complexity in autism spectrum conditions: A multiscale entropy analysis. Clinical Neurophysiology, 122(12), 2375-2383. doi:10.1016/j.clinph.2011.05.004Chang, C., Liu, Z., Chen, M. C., Liu, X., & Duyn, J. H. (2013). EEG correlates of time-varying BOLD functional connectivity. NeuroImage, 72, 227-236. doi:10.1016/j.neuroimage.2013.01.049Chang, C., Metzger, C. D., Glover, G. H., Duyn, J. H., Heinze, H.-J., & Walter, M. (2013). Association between heart rate variability and fluctuations in resting-state functional connectivity. NeuroImage, 68, 93-104. doi:10.1016/j.neuroimage.2012.11.038Chen, Y., Wang, W., Zhao, X., Sha, M., Liu, Y., Zhang, X., … Ming, D. (2017). 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Efficient resolution of the Colebrook equation

A robust, fast and accurate method for solving the Colebrook-like equations is presented. The algorithm is efficient for the whole range of parameters involved in the Colebrook equation. The computations are not more demanding than simplified approximations, but they are much more accurate. The algorithm is also faster and more robust than the Colebrook solution expressed in term of the Lambert W-function. Matlab and FORTRAN codes are provided

Automatic linearity detection

Given a function, or more generally an operator, the question "Is it linear?" seems simple to answer. In many applications of scientific computing it might be worth determining the answer to this question in an automated way; some functionality, such as operator exponentiation, is only defined for linear operators, and in other problems, time saving is available if it is known that the problem being solved is linear. Linearity detection is closely connected to sparsity detection of Hessians, so for large-scale applications, memory savings can be made if linearity information is known. However, implementing such an automated detection is not as straightforward as one might expect. This paper describes how automatic linearity detection can be implemented in combination with automatic differentiation, both for standard scientific computing software, and within the Chebfun software system. The key ingredients for the method are the observation that linear operators have constant derivatives, and the propagation of two logical vectors, $\ell$ and $c$, as computations are carried out. The values of $\ell$ and $c$ are determined by whether output variables have constant derivatives and constant values with respect to each input variable. The propagation of their values through an evaluation trace of an operator yields the desired information about the linearity of that operator

The automatic solution of partial differential equations using a global spectral method

A spectral method for solving linear partial differential equations (PDEs) with variable coefficients and general boundary conditions defined on rectangular domains is described, based on separable representations of partial differential operators and the one-dimensional ultraspherical spectral method. If a partial differential operator is of splitting rank $2$, such as the operator associated with Poisson or Helmholtz, the corresponding PDE is solved via a generalized Sylvester matrix equation, and a bivariate polynomial approximation of the solution of degree $(n_x,n_y)$ is computed in $\mathcal{O}((n_x n_y)^{3/2})$ operations. Partial differential operators of splitting rank $\geq 3$ are solved via a linear system involving a block-banded matrix in $\mathcal{O}(\min(n_x^{3} n_y,n_x n_y^{3}))$ operations. Numerical examples demonstrate the applicability of our 2D spectral method to a broad class of PDEs, which includes elliptic and dispersive time-evolution equations. The resulting PDE solver is written in MATLAB and is publicly available as part of CHEBFUN. It can resolve solutions requiring over a million degrees of freedom in under $60$ seconds. An experimental implementation in the Julia language can currently perform the same solve in $10$ seconds.Comment: 22 page