Numerical Solutions of Matrix Differential Models using Cubic Matrix Splines II

This paper presents the non-linear generalization of a previous work on matrix differential models. It focusses on the construction of approximate solutions of first-order matrix differential equations Y'(x)=f(x,Y(x)) using matrix-cubic splines. An estimation of the approximation error, an algorithm for its implementation and illustrative examples for Sylvester and Riccati matrix differential equations are given.Comment: 14 pages; submitted to Math. Comp. Modellin

Indole-3-acetic acid regulates the central metabolic pathways in Escherichia coli.

The physiological changes induced by indoleacetic acid (IAA) treatment were investigated in the totally sequenced Escherichia coli K-12 MG1655. DNA macroarrays were used to measure the mRNA levels for all the 4290 E. coli protein-coding genes; 50 genes (1-1 %) exhibited significantly different expression profiles. In particular, genes involved in the tricarboxylic acid cycle, the glyoxylate shunt and amino acid biosynthesis (leucine, isoleucine, valine and proline) were up-regulated, whereas the fermentative adhE gene was down-regulated. To confirm the indications obtained from the macroarray analysis the activity of 34 enzymes involved in central metabolism was measured; this showed an activation of the tricarboxylic acid cycle and the glyoxylate shunt. The malic enzyme, involved in the production of pyruvate, and pyruvate dehydrogenase, required for the channelling of pyruvate into acetyl-CoA, were also induced in IAA-treated cells. Moreover, it was shown that the enhanced production of acetyl-CoA and the decrease of NADH/NAD+ ratio are connected with the molecular process of the IAA response. The results demonstrate that IAA treatment is a stimulus capable of inducing changes in gene expression, enzyme activity and metabolite level involved in central metabolic pathways in E. col

Accurate matrix exponential computation to solve coupled differential models in engineering

NOTICE: this is the author’s version of a work that was accepted for publication in Mathematical and Computer Modelling. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Mathematical and Computer Modelling [Volume 54, Issues 7–8, October 2011] DOI: 10.1016/j.mcm.2010.12.049The matrix exponential plays a fundamental role in linear systems arising in engineering, mechanics and control theory. This work presents a new scaling-squaring algorithm for matrix exponential computation. It uses forward and backward error analysis with improved bounds for normal and nonnormal matrices. Applied to the Taylor method, it has presented a lower or similar cost compared to the state-of-the-art Padé algorithms with better accuracy results in the majority of test matrices, avoiding Padé's denominator condition problems. © 2011 Elsevier Ltd.This work has been supported by Universidad Politecnica de Valencia grants PAID-05-09-4338, PAID-06-08-3307 and Spanish Ministerio de Educacion grant MTM2009-08587.Sastre, J.; Ibáñez González, JJ.; Defez Candel, E.; Ruiz Martínez, PA. (2011). Accurate matrix exponential computation to solve coupled differential models in engineering. Mathematical and Computer Modelling. 54(7-8):1835-1840. https://doi.org/10.1016/j.mcm.2010.12.049S18351840547-

Efficient orthogonal matrix polynomial based method for computing matrix exponential

The matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields. This paper presents an efficient method for computing matrix exponentials based on Hermite matrix polynomial expansions. Hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods. A backward-error analysis of the approximation in exact arithmetic is given. This analysis is used to provide a theoretical estimate for the optimal scaling of matrices. Two algorithms based on this method have been implemented as MATLAB functions. They have been compared with MATLAB functions funm and expm obtaining greater accuracy in the majority of tests. A careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals. Numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs, reaching in numerical tests relative higher average costs than expm of only 4.43% for the final Hermite selected order, and obtaining better accuracy results in the 77.36% of the test matrices. The MATLAB implementation of the best Hermite matrix polynomial based algorithm has been made available online. © 2011 Elsevier Inc. All rights reserved.This work has been supported by the Programa de Apoyo a la Investigacion y el Desarrollo of the Universidad Politecnica de Valencia PAID-05-09-4338, 2009.Sastre, J.; Ibáñez González, JJ.; Defez Candel, E.; Ruiz Martínez, PA. (2011). Efficient orthogonal matrix polynomial based method for computing matrix exponential. Applied Mathematics and Computation. 217(14):6451-6463. https://doi.org/10.1016/j.amc.2011.01.004S645164632171

Sensory navigation device for blind people

[EN] This paper presents a new Electronic Travel Aid (ETA) 'Acoustic Prototype' which is especially suited to facilitate the navigation of visually impaired users. The device consists of a set of 3-Dimensional Complementary Metal Oxide Semiconductor (3-D CMOS) image sensors based on the three-dimensional integration and Complementary Metal-Oxide Semiconductor (CMOS) processing techniques implemented into a pair of glasses, stereo headphones as well as a Field-Programmable Gate Array (FPGA) used as processing unit. The device is intended to be used as a complementary device to navigation through both open known and unknown environments. The FPGA and the 3D-CMOS image sensor electronics control object detection. Distance measurement is achieved by using chip-integrated technology based on the Multiple Short Time Integration method. The processed information of the object distance is presented to the user via acoustic sounds through stereophonic headphones. The user interprets the information as an acoustic image of the surrounding environment. The Acoustic Prototype transforms the surface of the objects of the real environment into acoustical sounds. The method used is similar to a bat's acoustic orientation. Having good hearing ability, with few weeks training the users are able to perceive not only the presence of an object but also the object form (that is, if the object is round, if it has corners, if it is a car or a box, if it is a cardboard object or if it is an iron or cement object, a tree, a person, a static or moving object). The information is continuously delivered to the user in a few nanoseconds until the device is shut down, helping the end user to perceive the information in real time.The first author would like to acknowledge that this research was funded through the FP6 European project CASBLiP number 027063 and Project number 2062 of the Programa de Apoyo a la Investigacion y Desarrollo 2011 from the Universitat Politecnica de Valencia.Dunai, L.; Peris Fajarnes, G.; Lluna Gil, E.; Defez Garcia, B. (2013). Sensory navigation device for blind people. Journal of Navigation. 66(3):346-362. doi:10.1017/S0373463312000574S34636266

High performance computing of the matrix exponential

This work presents a new algorithm for matrix exponential computation that significantly simplifies a Taylor scaling and squaring algorithm presented previously by the authors, preserving accuracy. A Matlab version of the new simplified algorithm has been compared with the original algorithm, providing similar results in terms of accuracy, but reducing processing time. It has also been compared with two state-of-the-art implementations based on Fade approximations, one commercial and the other implemented in Matlab, getting better accuracy and processing time results in the majority of cases. (C) 2015 Elsevier B.V. All rights reserved.Ruíz Martínez, PA.; Sastre Martinez, J.; Ibáñez González, JJ.; Defez Candel, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics. 291:370-379. doi:10.1016/j.cam.2015.04.001S37037929

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

Velocity vector (3D) measurement for spherical objects using an electro-optical device

The present paper describes a procedure to measure the velocity vector (3D) of a spherical object using an electro-optical device configured as a single large detection area optical barrier. The proposed procedure allows a measurement accuracy up to 0.1% in some cases and presents several advantages in relation to other measurement procedures like image processing, doppler-radar and some other electro-optical devices. The procedure is independent of the relative position of the measurement device in relation to the object trajectory. The fact of using a single optical barrier reduces the space required in the movement direction and increase the cases where the device can be used. A prototype has been built and tested.Lluna Gil, E.; Santiago-Praderas, V.; Defez Garcia, B.; Dunai, L.; Peris Fajarnes, G. (2011). Velocity vector (3D) measurement for spherical objects using an electro-optical device. Measurement. 44(9):1723-1729. doi:10.1016/j.measurement.2011.07.006S1723172944