scholarly journals An extension of Lagrange interpolation to approximate derivative, integral and bivariate function

2021 ◽  
Author(s):  
Loc Nguyen
Keyword(s):  
Arbitrary Function ◽  
Lagrange Interpolation ◽  
Interpolation Polynomial ◽  
Lagrange Polynomial ◽  
Approximate Derivative ◽  
Bivariate Function ◽  
Point Data ◽  
Set Of Points

Lagrange interpolation is the effective method to approximate an arbitrary function by a polynomial. But there is a need to estimate derivative and integral given a set of points. Although it is possible to make Lagrange interpolation first, which produces Lagrange polynomial; after that we take derivative or integral on such polynomial. However this approach does not result out the best estimation of derivative and integral. This research proposes a different approach that makes approximation of derivative and integral based on point data first, which in turn applies Lagrange interpolation into the approximation. Moreover, the research also proposes an extension of Lagrange interpolation to bivariate function, in which interpolation polynomial is converted as two-variable polynomial.

scholarly journals Fuzzy Volterra Integro-Differential Equations Using General Linear Method

Symmetry ◽  
2019 ◽  
Vol 11 (3) ◽  
pp. 381 ◽  
Author(s):  
Zanariah Abdul Majid ◽  
Faranak Rabiei ◽  
Fatin Abd Hamid ◽  
Fudziah Ismail
Keyword(s):  
Differential Equations ◽  
Lagrange Interpolation ◽  
Linear Method ◽  
Interpolation Polynomial ◽  
General Linear ◽  
Kutta Method ◽  
Lagrange Interpolation Polynomial ◽  
Runge Kutta Method ◽  
General Linear Method ◽  
Generalized Hukuhara Differentiability

In this paper, a fuzzy general linear method of order three for solving fuzzy Volterra integro-differential equations of second kind is proposed. The general linear method is operated using the both internal stages of Runge-Kutta method and multivalues of a multisteps method. The derivation of general linear method is based on the theory of B-series and rooted trees. Here, the fuzzy general linear method using the approach of generalized Hukuhara differentiability and combination of composite Simpson’s rules together with Lagrange interpolation polynomial is constructed for numerical solution of fuzzy volterra integro-differential equations. To illustrate the performance of the method, the numerical results are compared with some existing numerical methods.

scholarly journals Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

2016 ◽  
Vol 8 (4) ◽  
pp. 118 ◽  
Author(s):  
Maha Youssef ◽  
Hany A. El-Sharkawy ◽  
Gerd Baumann
Keyword(s):  
Uniform Convergence ◽  
Error Estimation ◽  
Upper Bound ◽  
Lagrange Interpolation ◽  
Lebesgue Constant ◽  
Explicit Construction ◽  
Interpolation Operator ◽  
Lebesgue Constants ◽  
Set Of Points

This paper gives an explicit construction of multivariate Lagrange interpolation at Sinc points. A nested operator formula for Lagrange interpolation over an $m$-dimensional region is introduced. For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior. For the uniform convergence the growth of the associated norms of the interpolation operator, i.e., the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature $O((log n)^m)$. We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.

scholarly journals On Secret Sharing with Newton’s Polynomial for Multi-Factor Authentication

Cryptography ◽  
2020 ◽  
Vol 4 (4) ◽  
pp. 34
Author(s):  
Sergey Bezzateev ◽  
Vadim Davydov ◽  
Aleksandr Ometov
Keyword(s):  
Access Control ◽  
Secret Sharing ◽  
Lagrange Interpolation ◽  
Interpolation Formula ◽  
Building Blocks ◽  
Interpolation Polynomial ◽  
Lagrange Interpolation Polynomial ◽  
Crucial Component

Security and access control aspects are becoming more and more essential to consider during the design of various systems and the tremendous growth of digitization. One of the related key building blocks in this regard is, essentially, the authentication process. Conventional schemes based on one or two authenticating factors can no longer provide the required levels of flexibility and pro-activity of the access procedures, thus, the concept of threshold-based multi-factor authentication (MFA) was introduced, in which some of the factors may be missing, but the access can still be granted. In turn, secret sharing is a crucial component of the MFA systems, with Shamir’s schema being the most widely known one historically and based on Lagrange interpolation polynomial. Interestingly, the older Newtonian approach to the same problem is almost left without attention. At the same time, it means that the coefficients of the existing secret polynomial do not need to be re-calculated while adding a new factor. Therefore, this paper investigates this known property of Newton’s interpolation formula, illustrating that, in specific MFA cases, the whole system may become more flexible and scalable, which is essential for future authentication systems.

scholarly journals Matrix Transformations and Disk of Convergence in Interpolation Processes

2008 ◽  
Vol 2008 ◽  
pp. 1-11
Author(s):  
Chikkanna R. Selvaraj ◽  
Suguna Selvaraj
Keyword(s):  
Lagrange Interpolation ◽  
Hermite Interpolation ◽  
Matrix Transformation ◽  
Matrix Transformations ◽  
Taylor Polynomial ◽  
Lagrange Polynomial ◽  
Birkhoff Interpolation ◽  
Partial Sum ◽  
The Difference ◽  
Definition Of

Let denote the set of functions analytic in but not on . Walsh proved that the difference of the Lagrange polynomial interpolant of and the partial sum of the Taylor polynomial of converges to zero on a larger set than the domain of definition of . In 1980, Cavaretta et al. have studied the extension of Lagrange interpolation, Hermite interpolation, and Hermite-Birkhoff interpolation processes in a similar manner. In this paper, we apply a certain matrix transformation on the sequences of operators given in the above-mentioned interpolation processes to prove the convergence in larger disks.

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