N. A. Baykara

Separate node ascending derivatives expansion (SNADE) for univariate functions: Univariate numerical integration

This work’s content has been built on the material created in the first two companion papers on the very newly proposed method named “Separate Node Ascending Derivatives Expansion (SNADE) for Univariate Functions”. Those four companion papers are to appear in this proceedings. This work focuses on the univariate numerical integration of a given univariate function by using SNADE. Almost trivial polynomial integration has been empowered by integrating the SNADE remainder integral. To this end the integral representation of the SNADE remainder has been reformulated in unit hypercubic format. The factor arising in the integrand to multiply the target function’s relevant derivative has been separated to appropriate pieces to get weighted integrals under strictly positive multivariate weight functions. Paper emphasizes totally onconceptuality and formulation. Hence we do not present any illustrative implementations here.

Separate node ascending derivatives expansion (SNADE) for univariate functions: Node optimization via partial fluctuation suppression

This work’s content finds its root in the material given in the first three companion papers on the very newly proposed method called “Separate Node Ascending Derivatives Expansion (SNADE) for Univariate Functions”. Those three and the present companion papers are to appear in this proceedings. This work focuses on the determination of the nodal values which make the Euclidean distance between the target function and the SNADE truncation polynomial under consideration. The minimization procedure uses certain elements of the mathematical fluctuation theory. We obtain nonlinear equations after the minimization of the Euclidean distance mentioned above. The solutions of these equations can be numerically obtained unless the target function has a very specific structure. This is a so-called “baby age” theory and needs very specific care for robustness and sophistication. Here, the purpose is just formalism and conceptuality. Practicality has been left to future works.

A basis set comparison in a variational scheme for the Yukawa potential

The variational method is used to obtain solutions to Schrödingers equation for a particle in the radially screened Yukawa potential. A basis set is presented. While the Laguerre basis set shows considerable improvement over the hydrogenic one, problems are still encountered as the screening parameter approaches its threshold value. Variational calculations are also presented using an Eckart-type basis set which looks promising near the critical screening region.

Separate node ascending derivatives expansion (SNADE) for univariate functions: Polynomial recursions, remainder bounds and the convergence

A novel method which has been developed in Demiralp’s group and has been named “Separate Node Ascending Derivatives Expansion (SNADE) for Univariate Functions” is the content of the first one of four companion papers to appear in this proceedings. This work focuses on the construction of basic recursions amongst the SNADE polynomials and also construction of a bound to SNADE remainder term. These goals are achieved. The convergence of SNADE could have also been proven by using the derived remainder bound. The recursions are constructed in both implicit and explicit form.

Fluctuation suppression to optimize initial data to increase the quality of truncation approximants in probabilistic evolution approach for ODEs: Basic philosophy

This work focuses on initial fluctuation optimization on truncation approximants of Probabilistic Evolution for Ordinary Differential Equations (ODEs). Probabilistic Evolution Approach (PEA) is a new method which is proposed by Metin Demiralp and his group (Group of Science and Methods of Computing). This approximation enables us to convert nonlinear differential equations to linear ones.

Multivariate numerical integration via fluctuationlessness theorem: Case study

In this work we come up with the statement of the Fluctuationlessness theorem recently conjectured and proven by M. Demiralp and its application to numerical integration of univariate functions by restructuring the Taylor expansion with explicit remainder term. The Fluctuationlessness theorem is stated. Following this step an orthonormal basis set is formed and the necessary formulae for calculating the coefficients of the three term recursion formula are constructed. Then for multivariate numerical integration, instead of dealing with a single formula for multiple remainder terms, a new approach that is already mentioned for bivariate functions is taken into consideration. At every step of a multivariate integration one variable is considered and the others are held constant. In such a way, this gives us the possibility to get rid of the complexity of calculations. The trivariate case is taken into account and its generalization is step by step explained. At the final stage implementations are done for some trivariate functions and the results are tabulated together with the implementation times.In this work we come up with the statement of the Fluctuationlessness theorem recently conjectured and proven by M. Demiralp and its application to numerical integration of univariate functions by restructuring the Taylor expansion with explicit remainder term. The Fluctuationlessness theorem is stated. Following this step an orthonormal basis set is formed and the necessary formulae for calculating the coefficients of the three term recursion formula are constructed. Then for multivariate numerical integration, instead of dealing with a single formula for multiple remainder terms, a new approach that is already mentioned for bivariate functions is taken into consideration. At every step of a multivariate integration one variable is considered and the others are held constant. In such a way, this gives us the possibility to get rid of the complexity of calculations. The trivariate case is taken into account and its generalization is step by step explained. At the final stage implementations are done for s...

Numerical integration based on nested Taylor decomposition of univariate functions under fluctuationlessness approximation

The application of the Fluctuationlessness theorem to the remainder term of Taylor decomposition on which both sides are integrated has been already worked on. In this work the novelty brought to the previous work is to apply the Fluctuationlessness theorem to the remainder part which, itself also is decomposed in Taylor sense.

Nested Taylor decomposition of univariate functions under fluctuationlessness approximation

Taylor decomposition of an analytic function and the use of the remainder part of this decomposition expressed in integral form on which Fluctuationlessness theorem is applied was already known in the litterature, but application of Fluctuationlessness approximation twice on the remainder part adds up an amelioration to the approximation. Organisation of the decomposition in such a way that this is made possible is explained in detail in this work.

Univariate approximate integration via nested Taylor multivariate function decomposition

This work is based on the idea of nesting one or more Taylor decompositions in the remainder term of a Taylor decomposition of a function. This provides us with a better approximation quality to the original function. In addition to this basic idea each side of the Taylor decomposition is integrated and the limits of integrations are arranged in such a way to obtain a universal [0;1] interval without losing from the generality. Thus a univariate approximate integration technique is formed at the cost of getting multivariance in the remainder term. Moreover the remainder term expressed as an integral permits us to apply Fluctuationlessness theorem to it and obtain better results.

Nested Taylor decomposition in multivariate function decomposition

Fluctuationlessness approximation applied to the remainder term of a Taylor decomposition expressed in integral form is already used in many articles. Some forms of multi-point Taylor expansion also are considered in some articles. This work is somehow a combination these where the Taylor decomposition of a function is taken where the remainder is expressed in integral form. Then the integrand is decomposed to Taylor again, not necessarily around the same point as the first decomposition and a second remainder is obtained. After taking into consideration the necessary change of variables and converting the integration limits to the universal [0;1] interval a multiple integration system formed by a multivariate function is formed. Then it is intended to apply the Fluctuationlessness approximation to each of these integrals one by one and get better results as compared with the single node Taylor decomposition on which the Fluctuationlessness is applied.