Study of Mechanism Keto-Enol Tautomerism (isomeric reaction) Structure Cyclohexanone by Using Ab initio Molecular Orbital and Density Functional Theory (DFT) Method with NBO Analysis

AbstractInitial quantum mechanic studies and density function theory (DFT) in the level of HF/6-311+G**, B3LYP/3-21+G* and B3LYP/6-311+G** on Keto-Enol Tautomerism (isomeric reaction) cyclohexanone structure, that results shows that activation energy for this reaction equal 64.6143(kcal.mol-1) and transition state has the highest energy level equivalent whit -194354.27(kcal.mol-1) that due to breaking of C-H bond and composing of O-H bond. The results of NBO analysis of showed that the bond π are in resonance condition with lone-pair electrons oxygen of and therefore providing enol state and the transition state in these reactions usually is the structure between ketone state and enol state. Bond order and density of electrons aren’t the same in structures enol state, transition state and ketone state. Also tautomerism cyclohexanone structure is a kind of endothermic isomeric reaction

Energy-efficient digital processing via Approximate Computing

Smart Systems applications often include error resilient computations, due to the presence of noisy input data, the lack of a unique golden output, etc. Therefore, computation accuracy constraints can be relaxed to improve a system's efficiency. Recently, a design paradigm called Approximate Computing (AC) has been proposed, that formalizes the exploitation of the accuracy dimension as a way to optimize efficiency in digital computing systems. AC configures configures as one of the most promising ways to reduce energy consumption in Smart Systems. In this chapter, we present an overview of the different AC techniques proposed in literature. Then, we focus on Algorithmic Noise Tolerance (ANT), one of the most suitable AC approaches for Smart Systems applications. In particular, we investigate for the first time the automatic application of this technique to an existing design. We show how this automation can be achieved with a flow that leverages standard EDA tools, with minimal input from the designer. Moreover, for a typical DSP circuit, we are able to obtain almost 45% total power saving

Some new bi-accelerator two-point methods for solving nonlinear equations

In this work, we extract some new and efficient two-point methods with memory from their corresponding optimal methods without memory, to estimate simple roots of a given nonlinear equation. Applying two accelerator parameters in each iteration, we try to increase the convergence order from four to seven without any new functional evaluation. To this end, firstly we modify three optimal methods without memory in such a way that we could generate methods with memory as efficient as possible. Then, convergence analysis is put forward. Finally, the applicability of the developed methods on some numerical examples is examined and illustrated by means of dynamical tools, both in smooth and in nonsmooth functions.The authors thank to the anonymous referees for their suggestions to improve the final version of the paper. The second author would like to thank Hamedan Brach of Islamic Azad University for partial financial support in this research.Cordero Barbero, A.; Lotfi, T.; Torregrosa Sánchez, JR.; Assari, P.; Mahdiani, K. (2016). Some new bi-accelerator two-point methods for solving nonlinear equations. Computational and Applied Mathematics. 35(1):251-267. doi:10.1007/s40314-014-0192-1S251267351Babajee DKR (2012) Several improvements of the 2-point third order midpoint iterative method using weight functions. Appl Math Comput 218:7958–7966Chicharro FI, Cordero A, Torregrosa JR (2013) Drawing dynamical and parameters planes of iterative families and methods. Sci World J. Article ID 780153, 11 ppChun C, Lee MY (2013) A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl Math Comput 223:506–519Cordero A, Hueso JL, Martínez E, Torregrosa JR (2010) New modifications of Potra–Ptàk’s method with optimal fourth and eighth orders of convergence. J Comput Appl Math 234:2969–2976Cordero A, Lotfi T, Bakhtiari P, Torregrosa JR (2014) An efficient two-parametric family with memory for nonlinear equations. Numer Algor. doi: 10.1007/s11075-014-9846-8Geum YH, Kim YI (2011) A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Appl Math Lett 24:929–935Heydari M, Hosseini SH, Loghmani GB (2011) On two new families of iterative methods for solving nonlinear equations with optimal order. Appl Anal Discret Math 5:93–109Jay IO (2001) A note on Q-order of convergence. BIT Numer Math 41:422–429Khattri SK, Steihaug T (2013) Algorithm for forming derivative-free optimal methods. Numer Algor. doi: 10.1007/s11075-013-9715-xKou J, Wang X, Li Y (2010) Some eighth-order root-finding three-step methods. Commun Nonlinear Sci Numer Simul 15:536–544Kung HT, Traub JF (1974) Optimal order of one-point and multipoint iteration. J Assoc Comput Math 21:634–651Liu X, Wang X (2012) A convergence improvement factor and higher-order methods for solving nonlinear equations. Appl Math Comput 218:7871–7875Lotfi T, Tavakoli E (2014) On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator parameter. Sci World J. Article ID 769758, 9 pp. doi: 10.1155/2014/769758Lotfi T, Soleymani F, Shateyi S, Assari P, Khaksar Haghani F (2014a) New mono- and biaccelerator iterative methods with memory for nonlinear equations. Abstr Appl Anal. Article ID 705674, 8 pp. doi: 10.1155/2014/705674Lotfi T, Soleymani F, Noori Z, Kiliman A, Khaksar Haghani F (2014b) Efficient iterative methods with and without memory possessing high efficiency indices. Discret Dyn Nat Soc. Article ID 912796, 9 pp. doi: 10.1155/2014/912796Magreñan AA (2014) A new tool to study real dynamics: the convergence plane. arXiv:1310.3986 [math.NA]Ortega JM, Rheimbolt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New YorkOstrowski AM (1966) Solutions of equations and systems of equations. Academic Press, New York-LondonPetković MS, Ilić S, Džunić J (2010) Derivative free two-point methods with and without memory for solving nonlinear equations. Appl Math Comput 217(5):1887–1895Petković MS, Neta B, Petković LD, Džunić J (2014) Multipoint methods for solving nonlinear equations: a survey. Appl Math Comput 226(2):635–660Ren H, Wu Q, Bi W (2009) A class of two-step Steffensen type methods with fourth-order convergence. Appl Math Comput 209:206–210Soleymani F, Sharifi M, Mousavi S (2012) An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J Optim Theory Appl 153:225–236Soleimani F, Soleymani F, Shateyi S (2013) Some iterative methods free from derivatives and their basins of attraction for nonlinear equations. Discret Dyn Nat Soc. Article ID 301718, 10 ppThukral R (2011) Eighth-order iterative methods without derivatives for solving nonlinear equation. ISRN Appl Math. Article ID 693787, 12 ppTraub JF (1964) Iterative methods for the solution of equations. Prentice Hall, New YorkZheng Q, Li J, Huang F (2011) An optimal Steffensen-type family for solving nonlinear equations. Appl Math Comput 217:9592–959