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Multiplicative calculus (MUC) measures the rate of change of function in terms of ratios, which makes the exponential functions significantly linear in the framework of MUC. Therefore, a generally non-linear optimization problem containing exponential functions becomes a linear problem in MUC. Taking this as motivation, this paper lays mathematical foundation of well-known classical Gauss-Newton minimization (CGNM) algorithm in the framework of MUC. This paper formulates the mathematical derivation of proposed method named as multiplicative Gauss-Newton minimization (MGNM) method along with its convergence properties. The proposed method is generalized for n number of variables, and all its theoretical concepts are authenticated by simulation results. Two case studies have been conducted incorporating multiplicatively-linear and non-linear exponential functions. From simulation results, it has been observed that proposed MGNM method converges for 12972 points, out of 19600 points considered while optimizing multiplicatively-linear exponential function, whereas CGNM and multiplicative Newton minimization methods converge for only 2111 and 9922 points, respectively. Furthermore, for a given set of initial value, the proposed MGNM converges only after 2 iterations as compared to 5 iterations taken by other methods. A similar pattern is observed for multiplicatively-non-linear exponential function. Therefore, it can be said that proposed method converges faster and for large range of initial values as compared to conventional methods.

We define a Bessel type equation in multiplicative calculus by some algebraic structures. Eigenfunctions of constructed problem are obtained by power series solution technique. While making these solutions, multiplicative Bessel functions are used strongly. We get a generator for that problem and construct integration representations for multiplicative Bessel functions. Finally, some basic spectral properties of multiplicative Bessel problem are examined in detail.

The classical calculus is viewed as additive calculus based on addition in the real line. Another known multiplicative calculus corresponding to multiplication in the positive real axis has been precisely introduced. Abstract multiplicative integration through positive measures has been newly introduced. Results of multiplicative differentiation and integration have been obtained for completion, when some of them have been obtained through multiplicative modulus function. Results have been obtained also for abstract multiplicative measure integration.

Numerical interpolation methods are essential for the estimation of nonlinear functions and they have a wide range of applications in economics and accounting. In this regard, the idea of using interpolation methods based on multiplicative calculus for suitable accounting problems is self-evident. The purpose of this study, therefore, is to develop a way to better estimate the learning curve, which is an exponentially decreasing function, based on multiplicative Lagrange interpolation. The results of this study show that the proposed multiplicative method of learning curve provides more accurate estimates of labour costs when compared to the conventional methods. This is because the exponential functions are linear in multiplicative calculus. Furthermore, the results reveal that using the proposed method enables cost and managerial accountants to better calculate both cost of unused capacity and product cost in a cumulative production represented by a nonlinear function. The results of this study are also expected to help researchers, practitioners, economists, business managers, and cost and managerial accountants to understand how to construct a multiplicative based learning curve to improve such decisions as pricing, profit planning, capacity management, and budgeting.

In this paper, we develop an efficient technique in the framework of multiplicative calculus and suggest a new class of numerical methods for solving multiplicative nonlinear equations \textit{g}(\textit{x}% ) = 1. We also develop the convergence criteria of the proposed methods. We solve the population growth model and minimization problem, which demonstrate the implementation and efficiency of the new techniques. We also show that these techniques perform much better as compared to the similar ordinary methods for solving ordinary nonlinear equations \textit{f}(\textit{x}) = 0.

In this paper, the multiplicative least square method is introduced and is applied to integrals for the finite product representation of the positive functions. Hence, many nonlinear functions can be represented by well-behaved exponential functions. Product representation produces an accurate representation of signals, especially where exponentials occur. Some real applications of nonlinear exponential signals will be selected to demonstrate the applicability and efficiency of proposed representation.

This paper focuses on optimum representation for both linear and non-linear type signals which have a wide range of applications in the analysis and processing of real-world signals, that is, noise, filtering, audio, image etc. Accurate representation of signals, usually is not an easy process. The optimum representation is achieved by introducing exponential bases within multiplicative calculus which enables direct processing to reveal the unknown fitting parameters. Simulation tests confirm that the newly introduced models produce accurate results while using substantially less computation and provide support for applying the new model in the field of parametric linear, non-linear signal representation for processing.

The problem of cylindrical wave incidence on a conducting half plane has been considered. A modal solution for Green's function of the problem is transformed into contour integral representations in a complex plane. Some contour deformations and changes of variables are then made for the integrals. Finally, the resultant integrals are transformed back into a series, which converges rapidly to the exact solution when the observation angles are close to the reflection/shadow boundaries (RSBs) of the conducting half plane. The multiplicative calculus is employed in deriving an expression that can be used for obtaining approximate solutions when the observation angles are away from the RSBs of the conducting half plane. The derived expressions are seen to be very simple for implementing in any computational environment.

In this paper, we propose q-analog of some basic concepts of multiplicative calculus and we called it as q-multiplicative calculus. We successfully introduced q-multiplicative calculus and some basic theorems about derivatives, integrals and infinite products are proved within this calculus.

The diffraction coefficient of the geometrical theory of diffraction is expressed in terms of contour integrals that are of the same form. A series of approximations are made for the evaluation of those integrals by employing a multiplicative calculus. Finally, two different expressions are derived for the evaluation of the diffraction coefficient: one is for a case where the field observation angle is away from the reflection/shadow boundaries (RSB) of a conducting surface, and the other is for a case where the field observation angle is close to the RSB. The derived expressions are seen to be relatively simple when compared with those existing in the literature and they can be used in place of them. The given approximation techniques for the contour integrals can be extended to other high-frequency techniques in electromagnetics.

This paper is devoted to investigate some characteristic features of complex numbers and functions in terms of non-Newtonian calculus. Following Grossman and Katz, (Non-Newtonian Calculus, Lee Press, Piegon Cove, Massachusetts, 1972), we construct the fieldℂ*of*-complex numbers and the concept of*-metric. Also, we give the definitions and the basic important properties of*-boundedness and*-continuity. Later, we define the spaceC*(Ω)of*-continuous functions and state that it forms a vector space with respect to the non-Newtonian addition and scalar multiplication and we prove thatC*(Ω)is a Banach space. Finally, Multiplicative calculus (MC), which is one of the most popular non-Newtonian calculus and created by the famous exp function, is applied to complex numbers and functions to investigate some advance inner product properties and give inclusion relationship betweenC*(Ω)and the set ofC*′(Ω)*-differentiable functions.

Theory and applications of multiplicative and Volterra calculi have been evolving rapidly over the recent years. As numerical minimization methods have a wide range of applications in science and engineering, the idea of the design of minimization methods based on multiplicative and Volterra calculi is self-evident. In this paper, the well-known Newton minimization method for one and two variables is developed in the frameworks of multiplicative and Volterra calculi. The efficiency of these proposed minimization methods is exposed by examples, and the results are compared with the original minimization method. One of the striking results of the proposed method is that the rate of convergence and the range of initial values are considerably larger compared to the original method. Due to copyright restrictions, the access to the publisher version (published version) of this article is only available via subscription. You may click URI and have access to the Publisher Version of this article through the publisher web site or online databases, if your Library or institution has subscription to the related journal or publication.

We propose a hybrid dynamical system as a continuous analog to the block-iterative multiplicative algebraic reconstruction technique (BI-MART), which is a well-known iterative image reconstruction algorithm for computed tomography. The hybrid system is described by a switched nonlinear system with a piecewise smooth vector field or differential equation and, for consistent inverse problems, the convergence of non-negatively constrained solutions to a globally stable equilibrium is guaranteed by the Lyapunov theorem. Namely, we can prove theoretically that a weighted Kullback-Leibler divergence measure can be a common Lyapunov function for the switched system. We show that discretizing the differential equation by using the first-order approximation (Euler's method) based on the geometric multiplicative calculus leads to the same iterative formula of the BI-MART with the scaling parameter as a time-step of numerical discretization. The present paper is the first to reveal that a kind of iterative image reconstruction algorithm is constructed by the discretization of a continuous-time dynamical system for solving tomographic inverse problems. Iterative algorithms with not only the Euler method but also the Runge-Kutta methods of lower-orders applied for discretizing the continuous-time system can be used for image reconstruction. A numerical example showing the characteristics of the discretized iterative methods is presented.

This work is aimed to show that various problems from different fields can be modeled more efficiently using multiplicative calculus, in place of Newtonian calculus. Since multiplicative calculus is still in its infancy, some effort is put to explain its basic principles such as exponential arithmetic, multiplicative calculus, and multiplicative differential equations. Examples from finance, actuarial science, economics, and social sciences are presented with solutions using multiplicative calculus concepts. Based on the encouraging results obtained it is recommended that further research into this field be vested to exploit the applicability of multiplicative calculus in different fields as well as the development of multiplicative calculus concepts. .Due to copyright restrictions, the access to the publisher version (published version) of this article is only available via subscription. You may click URI and have access to the Publisher Version of this article through the publisher web site or online databases, if your Library or institution has subscription to the related journal or publication

The multiplicative version of Adams Bashforth---Moulton algorithms for the numerical solution of multiplicative differential equations is proposed. Truncation error estimation for these numerical algorithms is discussed. A specific problem is solved by methods defined in multiplicative sense. The stability properties of these methods are analyzed by using the standart test equation.

Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches. © 2009 Brown University.

This paper presents analysis of Lyapunov type stability for multiplicative dynamical systems. It has been formally defined and numerical simulations were performed to explore nonlinear dynamics. Chaotic behavior manifested for exemplary multiplicative dynamical systems has been confirmed by calculated Lyapunov exponent values.

Numerical multiplicative algorithm of Runge–Kutta type for solving multiplicative differential equations is presented. The multiplicative Rossler system has been briefly examined to test the method proposed.

Multiplicative calculus is based on a multiplicative rate of change, as the usual calculus is based on an additive rate of change. Some student investigations into the multiplicative calculus are described, including an original student idea about a multiplicative Euler's Method.

A new type of calculus called “multiplicative calculus” is developed and some basic theorems about derivatives, integrals, and infinite products are proved within this calculus. Multiplicative calculus is based on a multiplicative mechanism in the same sense that the usual calculus is based on an additive mechanism. One consequence is that, just as the usual derivative of a linear function is constant, the multiplicative derivative of an exponential function is constant (and just as the usual derivative of a constant function is 0, the multiplicative derivative of a constant function is 1). Similarly, just as two functions that have a constant difference have the same usual derivative, two functions that have a constant ratio have the same multiplicative derivative. Finally, just as many functions have infinite series representations based on the usual derivative, the same functions have infinite product representations based on the multiplicative derivative. These in turn are derived from expone...

Chapter 11 discusses transcendental functions. These functions can be obtained as a sum of functional series and improper integrals, depending on a parameter. Logarithmic and exponential, trigonometric and hyperbolic functions are defined in these ways and their properties are deduced. Power series are discussed and analytic functions are defined. An interesting feature of this chapter is an introduction to multiplicative calculus, which presents a calculus alternative to calculus of Newton and Leibnitz. By use of methods of multiplicative calculus it is proved that an infinitely many times differentiable function may not be analytic. Infinite products are handled. Improper integrals, depending on a parameter, are discussed and used to define Euler’s Gamma and Beta functions.

Multiplicative calculus provides a natural framework in problems involving positive images and positivity preserving operators. In increasingly important, complex imaging frameworks, such as diffusion tensor imaging, it complements standard calculus in a nontrivial way. The purpose of this article is to illustrate the basics of multiplicative calculus and its application to the regularization of positive definite matrix fields.

The important point to be noted on the non-Newtonian calculus is a self- contained system independent of any other system of calculus. Therefore, the reader may be surprised to learn that there is a uniform relationship between the corresponding operators of this calculus and the classical calculus. In the present paper, some fundamental theorems and notions of the classical calculus are in- Terpreted from the view point of multiplicative calculus and the analogies between them are given. We propose a concrete approach based on some topological prop- erties with respect to the multiplicative calculus. Finally, we give ?-completeness results on some sets of specific sequences.