China
Africa
Explore chapters and articles related to this topic
Interpolation and Approximation Theory
Published in A. C. Faul, A Concise Introduction to Numerical Analysis, 2018
The problem with polynomial interpolation is that with increasing degree the polynomial ‘wiggles’ from data point to data point. Low-order polynomials do not display this behaviour. Let’s interpolate data by fitting two cubic polynomials, p1(x) and p2(x), to different parts of the data meeting at the point x*. Each cubic polynomial has four coefficients and thus we have 8 degrees of freedom and hence can fit 8 data points. However, the two polynomial pieces are unlikely to meet at x*. We need to ensure some continuity. If we let p1(x*) = p2(x*), then the curve is at least continuous, but we are losing one degree of freedom. The fit p′1(x∗)=p′2(x∗)p″1(x∗)=p″2(x∗)
Interpolation, Differentiation, and Integration
Published in Julio Sanchez, Maria P. Canton, Software Solutions for Engineers and Scientists, 2018
Julio Sanchez, Maria P. Canton
In theory, if there are n data points in a data set, it is possible to find a polynomial expression of degree n – 1 that defines a curve that passes through every data point. In this sense linear interpolation, as discussed in Section 12.1.1, is a form of polynomial interpolation when the polynomial is of the first degree. The general problem can be expressed as follows: given a set of n points, (x1, y1), (xy, y2) … (xn ,yn), find a unique polynomial of degree n – 1 that defines the curve.
Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
Definition 127. Polynomial interpolation is a method of estimating values between known data points. When graphical data contains a gap, but data is available on either side of the gap or at a few specific points within the gap, an estimate of values within the gap can be made by interpolation.
An Equivalent Fiber Frame Model for Nonlinear Analysis of Masonry Structures
Published in International Journal of Architectural Heritage, 2021
Behrooz Yousefi, Masoud Soltani
In a classical displacement-based methodology, interpolations of transverse and axial frame element displacements are applied by a cubic and linear Hermitian polynomials, respectively, which are only approximations of the actual displacement fields in the presence of non-uniform beam cross-section with nonlinear material behavior (Vanin et al. 2017). Polynomial interpolation introduces field inconsistencies in this formulation and dense meshes are often practically insufficient to obtain accurate responses, or the convergence is carried out at a low speed (Alemdar and White 2005), or numerical instabilities issues may occur (Gerasimidis et al. 2015). Several approaches have been proposed in the past to provide a mathematical formulation to estimate mentioned vulnerabilities. Dvorkin, Onte, and Oliver (1988) and Crivelli and Felippa (1993) developed an incremental form of total Lagrangian formulation for beam elements that included the effect of shear influence and the effect of large rotation. Also, Kabeyasawa et al. (1983) introduced a macro model called Three Vertical Line Element Model (TVLEM) for numerical modeling of reinforced concrete shear walls and subsequent development of TVLEM was Multiple-Vertical-Line Element Model (MVLEM) which was presented by Vulcano, Bertero, and Colotti (1988). Further progress has been done for incorporating shear-flexure interaction effects of EFM in Mergos and Beyer (2013).
Deterministic and fractional analysis of a newly developed dengue epidemic model
Published in Waves in Random and Complex Media, 2023
Rahat Zarin, Mohabat Khan, Amir Khan, Abdullahi Yusuf
Newton polynomial interpolation is a method of constructing a polynomial that passes through a set of given data points. It is named after Sir Isaac Newton, who developed the method in the 17th century. The basic idea behind this method is to create a polynomial that is as close as possible to the given data points. The polynomial is constructed by using the difference between the function and the polynomial at the given data points. The Newton polynomial is a unique polynomial that passes through all the given data points and can be computed efficiently using the divided difference method.
Parameter estimation for models of chemical reaction networks from experimental data of reaction rates
Published in International Journal of Control, 2023
Manvel Gasparyan, Arnout Van Messem, Shodhan Rao
An important tool in our parameter estimation method is the parametric Bézier curve that is used to approximate the concentrations of the species participating in the considered CRN. An alternative to the Bézier curve may be the B-spline, which is a generalisation of the Bézier curve. A B-spline is a spline function (a function defined piecewise by polynomials) defined by its order and a pre-specified set of given knots. As a matter of fact, a Bézier curve is a B-spline with no interior knots. For a detailed description of B-splines we refer to Prautzsch et al. (2002). However, even though B-splines are more powerful and flexible curves than Bézier curves, the theory behind such curves is more complicated. Moreover, the application of B-splines is computationally more expensive. Another powerful technique in approximating continuous curves is a polynomial interpolation, which is the interpolation of a given dataset by the polynomial of the lowest possible degree that passes through the points of the dataset. For any given dataset, there is a unique interpolating polynomial, known as the Lagrange interpolating polynomial (Waring, 1779). Even though approximating the species' concentrations with Lagrange interpolating polynomials sounds logical, it has a serious shortcoming compared to Bézier curves. In our case, the considered datasets correspond to time-series data of species' concentrations, which are always non-negative. It is therefore assured that the Bézier curves are also non-negative. This is due to the fact that any Bézier curve is contained in the convex hull of the corresponding Bézier polygon, which is the polygon formed by connecting the control points with lines, starting at the first control point and finishing with the last control point. In general, it is not assured that the Lagrange interpolating polynomial corresponding to the available dataset will also be non-negative. As such, it is preferred to use Bézier curves for approximating the species' concentrations. In the supplementary material, we include the definition of Lagrange interpolating polynomials and also provide an example of a dataset consisting of non-negative data points, for which the corresponding Lagrange interpolating polynomial is negative over some time interval. The associated Matlab function is included in our Matlab library that is provided as supplementary material.