China
Africa
Explore chapters and articles related to this topic
Data Sources and Models
Published in Yuri S. Popkov, Alexey Yu. Popkov, Yuri A. Dubnov, Alexander Yu. Mazurov, Entropy Randomization in Machine Learning, 2023
Yuri S. Popkov, Alexey Yu. Popkov, Yuri A. Dubnov, Alexander Yu. Mazurov
The higher-order divided differences are calculated recursively. The divided difference of order m has the general formula Δi−1,i,i+1,…,i+m=Δi,i+1,…,i+m−Δi−1,i,i+1,…,i+m−1ti+m−ti.
Finite difference methods
Published in Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee, Numerical Methods in Mechanics of Materials, 2017
Ken P. Chong, Arthur P. Boresi, Sunil Saigal, James D. Lee
In the theory of interpolating polynomials, divided differences play an important role. In this section, we present a few useful properties of divided differences. Consider arbitrary pivotal points a0, a1, a2, …, an of the independent variable x and the corresponding function values f(a0), f(a1), f(a2), …, f(an). The zero-order divided difference is defined as f[a0] = f(a0). First-, second-, and higher-order divided differences are defined by
Algebraic Interpolation
Published in Victor S. Ryaben’kii, Semyon V. Tsynkov, A Theoretical Introduction to Numerical Analysis, 2006
Victor S. Ryaben’kii, Semyon V. Tsynkov
Having defined the Newton divided differences1 according to (2.2), we can now represent the interpolating polynomial Pn(x, f, x0, x1,…,xn) in the following Newton form:Pn(x,f,x0,x1,…,xn)=f(x0)+(x−x0)f(x0,x1)+…+(x−x0)(x−x1)…(x−xn−1)f(x0,x1,…,xn).
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.
Automatic differentiation of a finite-volume-based transient heat conduction code for sensitivity analysis
Published in Numerical Heat Transfer, Part B: Fundamentals, 2018
Solution sensitivities can be computed using a large variety of methods, perhaps the most obvious being by finite differences. Using finite differences, each input is perturbed by a small amount, the solution is recalculated, and the derivative is estimated using a divided difference. This method, however, suffers from truncation error if the difference in inputs is too large or cancellation error if the difference between inputs is too small [6,7]. Although finite differences are known to be useful and practical in many situations, it is of interest to find more accurate and efficient methods. Derivatives could also be calculated by manually coding the derivative of each function within the code and propagating the derivatives through the code by chain rule. Although this method could provide machine-accurate derivatives and can be written to be very efficient, it requires significant programming effort to produce and maintain. Further, manually coding derivatives does not generalize to computing sensitivities with respect to arbitrary input variables and is prone to coding error. The continuous sensitivity equation (CSE) approach involves formally deriving solution sensitivities by implicit differentiation of the respective governing equations and the solution of the additional equations [3,4]. Although this approach is quite elegant and generalizes to arbitrary sensitivity parameters, the solution of the sensitivity equations does need to be implemented manually in the code and would require additional maintenance effort as the primary code is updated. Symbolic differentiation is another option that is available, but is not practical for large-scale computations where the complexity of the symbolic expressions would be too great [7]. The response surface methodology is another way to explore the sensitivity of a solution to its inputs by conducting a series of numerical simulations [8,9].
Complexity analysis and numerical implementation of large-update interior-point methods for SDLCP based on a new parametric barrier kernel function
Published in Optimization, 2018
Mohamed Achache, Nesrine Tabchouche
For any function , let us denote by the divided difference of :