The fluctuationlessness approach to the numerical integration of functions with a single variable by integrating Taylor expansion with explicit remainder term

In this paper we give the definition of the Fluctuationlessness concept and using this concept we make approximations to univariate functions by using Taylor expansion with the explicit remainder term. Then integrating this approximate expression we obtain a new quadrature-like numerical integration method. The results of numerical experiments are compared with the results obtained from the corresponding Taylor series expansion without the remainder term and errors are analyzed

Multi Nodalset Fluctuation Free Approximation in Taylor Remainder's Evaluation

The matrix representation of a univariate function is equal to the image of the independent variable matrix representation under that function at the no fluctuation limit. In recent studies this fact is extended in such a way that the matrix representation of a univariate function can be expressed as a linear combination of the same function with two different matrix arguments. This idea makes us think for more than two matrices whose images under the target function are combined to get better approximation. This paper focuses on the application of this approximation method on the integral representation of the remainder term of the Taylor series expansion. In this work the basic conceptual background is given. Some illustrative implementations will be given at the relevant conference presentation

Analitik olmayan ve/veya yüksek salınımlı fonksiyonların ağırlık fonksiyonu altındaki Taylor açılımlarının integral formundaki kalan terimine sendelenimsizlik teoremi uygulanmak suretiyle yaklaştırımlarının yapılması ve buna bağlı olarak nümerik integrasyon yöntemlerinin geliştirilmesi

ÖZETGerek yapısal olarak gerekse doğa bilimlerine temel teşkil etmesi bakımından yaklaşık değer hesaplamaları, fonksiyon yaklaştırımları ve kuadratürler matematikle uğraşanların kaçınılmaz ilgi alanlarından olagelmistir. Bu tezde bir yaklaştırım yöntemi olarak önemli bir yer tutan ve bir çok yeni yöntemin oluşturulmasına esas olan Taylor açılımı esas alınmış ve Taylor açılımının integral formunda yazılmış kalan terimi üzerinde çalışılarak yeni ve daha verimli yaklaştırımlar geliştirilmeye çalışılmıştır. Tezin birinci bölümünde konuya kısa bir giriş yapılmış, yaklaştırımların sınıflandırılmasına değinilmiş, bunlardan gerek interpolasyon gerekse eğri uydurma yoluyla yapılan yaklaştırımların temel felsefesine değinilmiştir. Tezin ikinci bölümünde yapılan çalısmalara esas teşkil eden Sendelenimsizlik Yaklaştırımı, Sendelenimsizlik Teoreminin Taylor açılımına uygulanması, Taylor açılımının sendelenimsizlik uygulamasına imkan verecek biçimde düzenlenmesi ve son olarak da Çok Değişkenliliği Yükseltilmis Çarpımsal Gösterilim (ÇYÇG) konuları ile ilgili yapılmış çalışmalara değinilerek bu çerçevede gerekli bilgiler verilmistir. Tezin üçüncü bölümünde yeni çalışmalar ve bulgular ele alınarak varılan sonuçlar üzerinde tartısılmıstır. Öncelikle operatör çarpımları ve ayrıştırımlarının sonlu ve sonsuz aralıktaki matris gösterilimlerinde sendelenim incelemeleri ve etkileri araştırılmıştır. Sonrasında oldukça önemli bir sadeleştirme sağlayan üç terimli özyineleme yoluyla matris gösterilimi üzerinde durulmuştur. Sonrasında trigonometrik taban fonksiyonları kullanılarak Taylor açılımının kalan terimine sendelenimsizlikuygulanmak suretiyle çok salınımlı fonksiyonlara yaklaştırım ele alınmıstır. Bunu takiben ÇYÇG’nin Cauchy integral formunda ifade edilmiş Taylor açılımına uygulanması üzerinde çalışılmıs ve yine sendelenimsizlik yaklaştırımının dairesel konturlar yoluyla kontur integrasyonu tarafından desteklenmesi üzerinde çalışılmıstır. Daha sonrasında ise önce kontur integrasyonu ve ÇYÜDÇG yoluyla ve devamında aynı işin Taylor serisi kalan terimi üzerinde nasıl sonuç vereceği sorgulanmıstır. Son olarak ise Başlangıç Değer problemlerinde temel yöntemlerden biri olan Euler metodu ve yüksek mertebeli Taylor metodları ele alınarak sendelenimsizlik yoluyla bunlarda iyileştirme sağlanmıştır. Tezin dördüncü ve son bölümünde ise varılan sonuçlar kısaca ele alınarak özetlenmistir.--------------------ABSTRACTApproximate value calculations, function approximations and quadratures, either because of structural reasons or because they constitute a basis for natural sciences, have always been a center of attention for mathematicians. In this thesis Taylorexpansion which has an important place in the approximation theory and which makes an important starting point for the construction of many new methods, is taken into consideration and some more productive methods are searched by workingon the remainder term expressed in integral form. In the first part of the thesis a short introduction is given, approximations areclassified and the basic philosophies of interpolation and curve fitting are mentioned. In the second part of the thesis Fluctuation Theorem, its application to Taylor expansion, the reorganisation of Taylor expansion to make room for this applicationand finally theorems and applications concerning EMPR are given in detail. In the third part new research and findings are given and discussed in details. It starts with fluctuation studies on the matrix representation of operator multiplicationand decomposition. Then the approximation to highly oscillatory functions by applying trigonometric basis functions to the remainder term of Taylor expansion. This is followed by the application of EMPR to the remainder term of Taylor expansionexpressed as a contour integration and the application of Fluctuationlessness Theorem on contour integration with circular contours. Then using TKEMPR on the remainder term of Taylor expansion expressed in contour integral form is takeninto consideration. Finally Euler and higher order Taylor methods for initial value problems are worked on and beter results are reached through fluctuationlessness approximation applied on them. In the fourth section some corollaries are liste

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

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

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

Fluctuation Studies on the Finite Interval Matrix Representations of Operator Products and Their Decompositions

In this work an experimental study on finite dimensional matrix approximations to products of various operators under a basis set orthonormalized on a finite interval is conducted. The elements of the matrices corresponding to the matrix representation of various operators, are calculated based on various term recursive relations. It is shown that higher the values of n, lower the relative marginal change will be obtained from the norm of the difference of the matrix representation of a product of two operators and the product of the matrix representations of the same operators

Recovery of Missing Data via Wavelets Followed by High-Dimensional Modeling

In this article missing multi-dimensional data imputation is taken into consideration for unevenly spaced data. The only prerequisite information is intended to be the knowledge that would allow us to guess a matrix called a frame. As an example in image processing an inverse discrete cosine transform matrix would be a suitable frame. The main purpose here is to guess such a sparse frame that can represent complete data vector f. By a sparse representation we mean the majority of components being close to zero. In the present article the data imputation using the expected sparse representation is intended to be done in a wavelet or lifting scheme basis. Finally, the generalization to multivariate case will be discussed

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

Sendelenimsizlik teoreminin yaklaştırım ve sayısal integrasyona uygulanması

ÖZETSendelenimsizlik Teoreminin Yaklaştırım ve Sayısal İntegrasyona UygulanmasıM. Demiralp tarafından ortaya konarak ispatlanmış ve işlevi tek değişkenli bir fonksiyonun argümanını fonksiyonun kendisi ile çarpmak olan cebirsel bir operatörün matris gösteriliminin, aynı uzay ve aynı taban takımı altında argümanın matris gösteriliminin o fonksiyonun altındaki görüntüsüne eşit olduğunu ifade eden Sendelenimsizlik teoreminin, hata terimi integral biçiminde verilen Taylor açılımının hata terimine uygulanması söz konusudur. Elde edilen sonuçlar hata terimsiz Taylor açılımından elde edilenlerle karşılaştırılmıştır. Tek ve çok değişkenli integrasyon için ise fonksiyonun Taylor açılımı integre edilerek sendelenimsizlik uygulanmış ve böylelikle kuadratür benzeri yeni bir sayısal integrasyon metodu oluşturulmuştur. Sayısal sonuçlar hata terimsiz Taylor açılımının integrasyonundan elde edilenlerle karşılaştırılmış ve hataların analizi yapılmıştır. ABSTRACTApplication of the Fluctuationlessness Theorem to the Approx-imation of Functions and IntegrationBased on the Fluctuationlessness theorem which was conjectured and proven by M. Demiralp, which states that the matrix representation of an algebraic operator which multiplies its argument by a scalar univariate function, is identical to the image of the independent variable’s matrix representation over the same space via the same basis set, under that univariate function, when the fluctuation terms are ignored. Approximations to the functions of a single variable as well as to multivariable functions are made by using the Taylor expansion with the remainder term expressed in integral form. Results are compared with those obtained from the corresponding Taylor series expansion without the error term. As for the single and multivariable integration, the approximate expression obtained from the Taylor expansion is integrated and a new quadrature-like numerical integration method is obtained. The results of numerical experiments are compared with the results obtained from the corresponding Taylor series expansion without the remainder term, and errors are analyze