T. Malange - Academia.edu (original) (raw)
Uploads
Papers by T. Malange
The interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity
Journal of Computational and Theoretical Nanoscience, 2016
Methods like the maximum likelihood and Newton's iterative techniques are usually employe... more Methods like the maximum likelihood and Newton's iterative techniques are usually employed to estimate the parameters of the classical Rydberg interatomic potential function. However, such approaches require initial guess values (IGV) to compute the optimal solutions of the unknowns. In this work, a multiple objective function approach that is used to compute the parameter estimates via the least-squares method is presented. The new approach does not necessitate the uniqueness and existence of the initial guess values to compute the required estimates. The classical Rydberg interactomic potential is firstly considered as a general solution of a second order linear differential equation with constant coefficients. Secondly, the characteristic equation of the general solution of the potential function is formulated and it's constraints derived. Two objective functions are constructed from the assumed ordinary differential equation. The first objective function is constrained and selected parameters are estimated using the least-squares method and the second objective function is directly minimised by formulating normal equations that can be solved for exact solutions of the unknowns. The new method and the computer algebraic system (CAS) are tested using experimental datasets of copper, silver ions and silver-copper alloy. Estimates from both approaches are compared and used to construct potential energy curves of the atoms considered. Three potential surfaces, that is, for experimental data, new method and the CAS for each atom are plotted on the same axis for better comparison. It is observed that estimates from the new method are greater in absolute terms than those from the CAS. However, all the estimates have the same direction. The potential curves from the experimental data, the new method and the CAS all have the same shape and close to each other, especially at the minimum of the potential well and at that point they are all indistinguishable. Estimates from the new method can be trusted as they produce a potential curve that is close to that of the experimental data and the CAS. As the present iterative methods usually converge to the required solutions given " good " IGVs, it is proposed that the new method be used symbiotically with the present methods to systematically compute the necessary IGVs. Hence the new method can be used in both theoretical and practical applications to estimate parameters of the Rydberg potential function since it does not require provision of IGVs to the unknown parameters.
Methods like the maximum likelihood and Newton's iterative techniques are usually employed to est... more Methods like the maximum likelihood and Newton's iterative techniques are usually employed to estimate the parameters of the classical Rydberg interatomic potential function. However, such approaches require initial guess values (IGV) to compute the optimal solutions of the unknowns. In this work, a multiple objective function approach that is used to compute the parameter estimates via the least-squares method is presented. The new approach does not necessitate the uniqueness and existence of the initial guess values to compute the required estimates. The classical Rydberg interactomic potential is firstly considered as a general solution of a second order linear differential equation with constant coefficients. Secondly, the characteristic equation of the general solution of the potential function is formulated and it's constraints derived. Two objective functions are constructed from the assumed ordinary differential equation. The first objective function is constrained and selected parameters are estimated using the least-squares method and the second objective function is directly minimised by formulating normal equations that can be solved for exact solutions of the unknowns. The new method and the computer algebraic system (CAS) are tested using experimental datasets of copper, silver ions and silver-copper alloy. Estimates from both approaches are compared and used to construct potential energy curves of the atoms considered. Three potential surfaces, that is, for experimental data, new method and the CAS for each atom are plotted on the same axis for better comparison. It is observed that estimates from the new method are greater in absolute terms than those from the CAS. However, all the estimates have the same direction. The potential curves from the experimental data, the new method and the CAS all have the same shape and close to each other, especially at the minimum of the potential well and at that point they are all indistinguishable. Estimates from the new method can be trusted as they produce a potential curve that is close to that of the experimental data and the CAS. As the present iterative methods usually converge to the required solutions given " good " IGVs, it is proposed that the new method be used symbiotically with the present methods to systematically compute the necessary IGVs. Hence the new method can be used in both theoretical and practical applications to estimate parameters of the Rydberg potential function since it does not require provision of IGVs to the unknown parameters.
The interdisciplinary journal of Discontinuity, Nonlinearity, and Complexity
Journal of Computational and Theoretical Nanoscience, 2016
Methods like the maximum likelihood and Newton's iterative techniques are usually employe... more Methods like the maximum likelihood and Newton's iterative techniques are usually employed to estimate the parameters of the classical Rydberg interatomic potential function. However, such approaches require initial guess values (IGV) to compute the optimal solutions of the unknowns. In this work, a multiple objective function approach that is used to compute the parameter estimates via the least-squares method is presented. The new approach does not necessitate the uniqueness and existence of the initial guess values to compute the required estimates. The classical Rydberg interactomic potential is firstly considered as a general solution of a second order linear differential equation with constant coefficients. Secondly, the characteristic equation of the general solution of the potential function is formulated and it's constraints derived. Two objective functions are constructed from the assumed ordinary differential equation. The first objective function is constrained and selected parameters are estimated using the least-squares method and the second objective function is directly minimised by formulating normal equations that can be solved for exact solutions of the unknowns. The new method and the computer algebraic system (CAS) are tested using experimental datasets of copper, silver ions and silver-copper alloy. Estimates from both approaches are compared and used to construct potential energy curves of the atoms considered. Three potential surfaces, that is, for experimental data, new method and the CAS for each atom are plotted on the same axis for better comparison. It is observed that estimates from the new method are greater in absolute terms than those from the CAS. However, all the estimates have the same direction. The potential curves from the experimental data, the new method and the CAS all have the same shape and close to each other, especially at the minimum of the potential well and at that point they are all indistinguishable. Estimates from the new method can be trusted as they produce a potential curve that is close to that of the experimental data and the CAS. As the present iterative methods usually converge to the required solutions given " good " IGVs, it is proposed that the new method be used symbiotically with the present methods to systematically compute the necessary IGVs. Hence the new method can be used in both theoretical and practical applications to estimate parameters of the Rydberg potential function since it does not require provision of IGVs to the unknown parameters.
Methods like the maximum likelihood and Newton's iterative techniques are usually employed to est... more Methods like the maximum likelihood and Newton's iterative techniques are usually employed to estimate the parameters of the classical Rydberg interatomic potential function. However, such approaches require initial guess values (IGV) to compute the optimal solutions of the unknowns. In this work, a multiple objective function approach that is used to compute the parameter estimates via the least-squares method is presented. The new approach does not necessitate the uniqueness and existence of the initial guess values to compute the required estimates. The classical Rydberg interactomic potential is firstly considered as a general solution of a second order linear differential equation with constant coefficients. Secondly, the characteristic equation of the general solution of the potential function is formulated and it's constraints derived. Two objective functions are constructed from the assumed ordinary differential equation. The first objective function is constrained and selected parameters are estimated using the least-squares method and the second objective function is directly minimised by formulating normal equations that can be solved for exact solutions of the unknowns. The new method and the computer algebraic system (CAS) are tested using experimental datasets of copper, silver ions and silver-copper alloy. Estimates from both approaches are compared and used to construct potential energy curves of the atoms considered. Three potential surfaces, that is, for experimental data, new method and the CAS for each atom are plotted on the same axis for better comparison. It is observed that estimates from the new method are greater in absolute terms than those from the CAS. However, all the estimates have the same direction. The potential curves from the experimental data, the new method and the CAS all have the same shape and close to each other, especially at the minimum of the potential well and at that point they are all indistinguishable. Estimates from the new method can be trusted as they produce a potential curve that is close to that of the experimental data and the CAS. As the present iterative methods usually converge to the required solutions given " good " IGVs, it is proposed that the new method be used symbiotically with the present methods to systematically compute the necessary IGVs. Hence the new method can be used in both theoretical and practical applications to estimate parameters of the Rydberg potential function since it does not require provision of IGVs to the unknown parameters.