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

Efficient mixed rational and polynomial approximation of matrix functions

This paper presents an efficient method for computing approximations for general matrix functions based on mixed rational and polynomial approximations. A method to obtain this kind of approximation from rational approximations is given, reaching the highest efficiency when transforming nondiagonal rational approximations with a higher numerator degree than the denominator degree. Then, the proposed mixed rational and polynomial approximation can be successfully applied for matrix functions which have any type of rational approximation, such as Pade, Chebyshev, etc., with maximum efficiency for higher numerator degrees than the denominator degrees. The efficiency of the mixed rational and polynomial approximation is compared with the best existing evaluating schemes for general polynomial and rational approximations, providing greater theoretical accuracy with the same cost in terms of matrix multiplications. It is well known that diagonal rational approximants are generally more accurate than the corresponding nondiagonal rational approximants which have the same computational cost. Using the proposed mixed approximation we show that the above statement is no longer true, and nondiagonal rational approximants are in fact generally more accurate than the corresponding diagonal rational approximants with the same cost. (C) 2012 Elsevier Inc. All rights reserved.This work has been supported by Universitat Politecnica de Valencia grant PAID-06-011-2020.Sastre, J. (2012). Efficient mixed rational and polynomial approximation of matrix functions. Applied Mathematics and Computation. 218(24):11938-11946. https://doi.org/10.1016/j.amc.2012.05.064S11938119462182

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-

Synthesis of ZnO Nanoparticle and Utilized as a Drug Carrier to Treat Leukemia

This study includes two parts, and the first was the preparation of the Zn(II)complex by reacting N-[4-(5-{(Z)-[(5-oxo-2-sulfanyl-4,5-dihydro-1H-imidazol-1-yl)imino]methyl}furan-2-yl)phenyl]acetamide with ZnCl2. The complex was characterized by using microscopic analysis such as UV-Vis spectrum, LC-MS, FTIR spectrophotometer, measurements of conductivity, magnetic susceptibility, and atomic absorption. The second part was the preparation of the ZnO nanoparticles by dissolving the Zn(II) complex in HNO3 and HCl and its use as a drug transporter to treat leukemia. FSEM, TEM, and XRD were examined for the characterization of ZnO nanoparticles that will be used in the synthesis of most medicines and drugs in the future

Taylor's theorem for matrix functions with applications to condition number estimation

We derive an explicit formula for the remainder term of a Taylor polynomial of a matrix function. This formula generalizes a known result for the remainder of the Taylor polynomial for an analytic function of a complex scalar. We investigate some consequences of this result, which culminate in new upper bounds for the level-1 and level-2 condition numbers of a matrix function in terms of the pseudospectrum of the matrix. Numerical experiments show that, although the bounds can be pessimistic, they can be computed much faster than the standard methods. This makes the upper bounds ideal for a quick estimation of the condition number whilst a more accurate (and expensive) method can be used if further accuracy is required. They are also easily applicable to more complicated matrix functions for which no specialized condition number estimators are currently available

Asthma control factors in the Gulf Cooperation Council (GCC) countries and the effectiveness of ICS/LABA fixed dose combinations: a dual rapid literature review.

BACKGROUND: Asthma control is influenced by multiple factors. These factors must be considered when appraising asthma interventions and their effectiveness in the Gulf Cooperation Council (GCC) countries (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and United Arab Emirates [UAE]). Based on published studies, the most prevalent asthma treatment in these countries are fixed dose combinations (FDC) of inhaled corticosteroid and long-acting beta-agonist (ICS/LABA). This study is a rapid review of the literature on: (a) factors associated with asthma control in the GCC countries and (b) generalisability of ICS/LABA FDC effectiveness studies. METHODS: To review local factors associated with asthma control and, generalisability of published ICS/LABA FDC studies, two rapid reviews were conducted. Review 1 targeted literature pertaining to asthma control factors in GCC countries. Eligible studies were appraised, and clustering methodology used to summarise factors. Review 2 assessed ICS/LABA FDC studies in conditions close to actual clinical practice (i.e. effectiveness studies). Eligibility was determined by reviewing study characteristics. Evaluation of studies focused on randomised controlled trials (RCTs). In both reviews, initial (January 2018) and updated (November 2019) searches were conducted in EMBASE and PubMed databases. Eligible studies were appraised using the Critical Appraisal Skills Program (CASP) checklists. RESULTS: We identified 51 publications reporting factors associated with asthma control. These publications reported studies conducted in Saudi Arabia (35), Qatar (5), Kuwait (5), UAE (3), Oman (1) and multiple countries (2). The most common factors associated with asthma control were: asthma-related education (13 articles), demographics (11articles), comorbidities (11 articles) and environmental exposures (11 articles). Review 2 identified 61 articles reporting ICS/LABA FDC effectiveness studies from countries outside of the GCC. Of these, six RCTs were critically appraised. The adequacy of RCTs in informing clinical practice varied when appraised against previously published criteria. CONCLUSIONS: Asthma-related education was the most recurring factor associated with asthma control in the GCC countries. Moreover, the generalisability of ICS/LABA FDC studies to this region is variable. Hence, asthma patients in the region, particularly those on ICS/LABA FDC, will continue to require physician review and oversight. While our findings provide evidence for local treatment guidelines, further research is required in GCC countries to establish the causal pathways through which asthma-related education influence asthma control for patients on ICS/LABA FDC therapy

Efficient computation of the matrix cosine

Trigonometric matrix functions play a fundamental role in second order differential equation systems. This work presents an algorithm for computing the cosine matrix function based on Taylor series and the cosine double angle formula. It uses a forward absolute error analysis providing sharper bounds than existing methods. The proposed algorithm had lower cost than state-of-the-art algorithms based on Hermite matrix polynomial series and Padé approximants with higher accuracy in the majority of test matrices.This work has been supported by Universitat Politecnica de Valencia Grant PAID-06-011-2020.Sastre, J.; Ibáñez González, JJ.; Ruiz Martínez, PA.; Defez Candel, E. (2013). Efficient computation of the matrix cosine. Applied Mathematics and Computation. 219:7575-7585. https://doi.org/10.1016/j.amc.2013.01.043S7575758521

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