Numerical differentiation for two-dimensional scattered data on arbitrary domain base on Hermite extension with an implicit iteration process

<abstract><p>In this paper, we develop a method for numerical differentiation of two-dimensional scattered input data on arbitrary domain. A Hermite extension approach is used to realize the approximation and a modified implicit iteration method is proposed to stabilize the approximation process. For functions with various smooth conditions, the numerical solution process of the method is uniform. The error estimates are obtained and numerical results show that the new method is effective. The advantage of the method is that it can solve the problem in any domain.</p></abstract>

IMPLEMENTATION OF THE TWO-FREQUENCY METHOD FOR MEASURING THE DERIVATIVE OF CAPACITANCE-VOLTAGE CHARACTERISTIC

The article presents an example of the implementation of the two-frequency method for measuring the derivative of the capacitance-voltage characteristic. The authors describe the diagram of the measuring system and the software package for the measurement process control. The article also gives the results of measurements of the capacitance-voltage characteristic and its derivative of the silicon-based metal-insulator-semiconductor structure. The qualitative differences between the derivatives of the capacitive characteristics of the metal-insulator-semiconductor structure were obtained as a result of numerical differentiation and its direct measurement by the two-frequency method. The authors offer an explanation for this difference.

Two Simultaneous Leak Diagnosis in Pipelines Based on Input–Output Numerical Differentiation

This paper addresses the two simultaneous leak diagnosis problem in pipelines based on a state vector reconstruction as a strategy to improve water shortages in large cities by only considering the availability of the flow rate and pressure head measurements at both ends of the pipeline. The proposed algorithm considers the parameters of both leaks as new state variables with constant dynamics, which results in an extended state representation. By applying a suitable persistent input, an invertible mapping in x can be obtained as a function of the input and output, including their time derivatives of the third-order. The state vector can then be reconstructed by means of an algebraic-like observer through the computation of time derivatives using a Numerical Differentiation with Annihilatorsconsidering its inherent noise rejection properties. Experimental results showed that leak parameters were reconstructed with accuracy using a test bed plant built at Cinvestav Guadalajara.

Precise Numerical Differentiation of Thermodynamic Functions with Multicomplex Variables

The multicomplex finite-step method for numerical differentiation is an extension of the popular Squire–Trapp method, which uses complex arithmetics to compute first-order derivatives with almost machine precision. In contrast to this, the multicomplex method can be applied to higher-order derivatives. Furthermore, it can be applied to functions of more than one variable and obtain mixed derivatives. It is possible to compute various derivatives at the same time. This work demonstrates the numerical differentiation with multicomplex variables for some thermodynamic problems. The method can be easily implemented into existing computer programs, applied to equations of state of arbitrary complexity, and achieves almost machine precision for the derivatives. Alternative methods based on complex integration are discussed, too.

Vanishing of the atomic form factor derivatives in non-spherical structural refinement – a key approximation scrutinized in the case of Hirshfeld atom refinement

When calculating derivatives of structure factors, there is one particular term (the derivatives of the atomic form factors) that will always be zero in the case of tabulated spherical atomic form factors. What happens if the form factors are non-spherical? The assumption that this particular term is very close to zero is generally made in non-spherical refinements (for example, implementations of Hirshfeld atom refinement or transferable aspherical atom models), unless the form factors are refinable parameters (for example multipole modelling). To evaluate this general approximation for one specific method, a numerical differentiation was implemented within the NoSpherA2 framework to calculate the derivatives of the structure factors in a Hirshfeld atom refinement directly as accurately as possible, thus bypassing the approximation altogether. Comparing wR 2 factors and atomic parameters, along with their uncertainties from the approximate and numerically differentiating refinements, it turns out that the impact of this approximation on the final crystallographic model is indeed negligible.

Numerical differentiation on scattered data through multivariate polynomial interpolation

We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor’s formula monomial basis. Error bounds for the approximation of partial derivatives of any order compatible with the function regularity are provided, as well as sensitivity estimates to functional perturbations, in terms of the inverse Vandermonde coefficients that are active in the differentiation process. Several numerical tests are presented showing the accuracy of the approximation.

Modified iterative method with red-black ordering for image composition using poisson equation

Image composition involves the process of embedding a selected region of the source image to the target image to produce a new desirable image seamlessly. This paper presents an image composition procedure based on numerical differentiation using the laplacian operator to obtain the solution of the poisson equation. The proposed method employs the red-black strategy to speed up the computation by using two acceleration parameters. The method is known as modified two-parameter over-relaxation (MTOR) and is an extension of the existing relaxation methods. The MTOR was extensively studied in solving various linear equations, but its usefulness in image processing was never explored. Several examples were tested to examine the effectiveness of the proposed method in solving the poisson equation for image composition. The results showed that the proposed MTOR performed faster than the existing methods.