Farahnaz Soleimani

Some Iterative Methods Free from Derivatives and Their Basins of Attraction for Nonlinear Equations

First, we make the Jains derivative-free method optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jains method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins.

On the extension of Householder's method for weighted Moore-Penrose inverse

This note discusses how to extend and use the well-known method of Householder (1964) [6] for finding weighted Moore-Penrose inverse. The discussions cover both theoretical and numerical aspects. An approach for accelerating the initial phase of convergence will be contributed. Some tests will also be employed to check the validity and superiority of the results over the Schulz method.

Some Matrix Iterations for Computing Generalized Inverses and Balancing Chemical Equations

An application of iterative methods for computing the Moore–Penrose inverse in balancing chemical equations is considered. With the aim to illustrate proposed algorithms, an improved high order hyper-power matrix iterative method for computing generalized inverses is introduced and applied. The improvements of the hyper-power iterative scheme are based on its proper factorization, as well as on the possibility to accelerate the iterations in the initial phase of the convergence. Although the effectiveness of our approach is confirmed on the basis of the theoretical point of view, some numerical comparisons in balancing chemical equations, as well as on randomly-generated matrices are furnished.

Oxidation of Aliphatic Alcohols by Tetramethylammonium Flurochromate, a Kinetic and Mechanistic Study

The oxidation of methanol, ethanol, and n-propanol to their corresponding aldehydes were studied by tetramethylammonium fluorochromate in acidic solution using spectrophotometric technique. The reaction was arranged to be under pseudo-first order conditions with respect to the oxidant. A MichaelisMenten type kinetic was observed with respect to the substrates, indicating the formation of an oxidant-alcohol complex in the pre-equilibrium. The reaction is catalyzed by hydrogen ions. The formation constants and the rates of decomposition of complexes were determined. Dependences of the reaction rates on temperature have been studied and the activation parameters were computed using the Arrhenius and Eyring plots. The reaction was studied in different solvents. The solvent effect was analyzed using KamletTafts and Swains multiparametric equations. Finally, a mechanism consistent with the observed results has been proposed.