First-order Derivative Approximation Research Articles

Convergence of derivative of Szász type operators involving Charlier polynomials

<p style='text-indent:20px;'>The paper deals with the approximation of first order derivative of a function by the first order derivative of Szász-type operators based on Charlier polynomials introduced by Varma and Taşdelen [<xref ref-type="bibr" rid="b20">20</xref>]. The uniform convergence theorem, Voronovskaja type asymptotic theorem and an estimate of error in terms of the second order modulus of continuity of the derivative of the function are investigated. Further, it is shown that linear combinations of the derivative of the above operators converge to the derivative of function at a faster rate. Finally, an estimate of error in the approximation is obtained in terms of the <inline-formula><tex-math id="M1">\begin{document}$ (2k+2)th $\end{document}</tex-math></inline-formula> order modulus of continuity using Steklov mean.</p>

4倍精度演算による複素数階微分の2階微分に関する基礎的検討

Methods for improving the accuracy of second order derivative calculations using the complex first-order derivative approximation (CSDA method) is proposed. In this paper, the following two approaches are investigated: (1) perturbation direction on the complex number plane, (2) use of double and quadruple precision floating point arithmetics. The above approaches were applied to several numerical examples to discuss the accuracy and computational costs. As the results, the perturbation of CSDA on θ = 45° had the best accuracy and stability throughout all the examples. Moreover, it was also found that when combined with perturbation on θ = 45° and quadruple precision floating point arithmetics, the second-order derivatives could be computed with more than accuracy of the double precision arithmetic in the examples.

Discrete-time zeroing neural network of [formula omitted] pattern for online solving time-varying nonlinear optimization problem: Application to manipulator motion generation

In this paper, a new Taylor-type differentiation formula is investigated for the first-order derivative approximation to generate higher numerical accuracy in the implementation of zeroing neural network model discretization. Then, two novel discrete-time zeroing neural network models are first developed, analyzed and verified for online solving time-varying nonlinear optimization problem. Moreover, the 0-stability, consistency and convergence reveal that the developed discrete-time zeroing neural network models converge to the theoretical solution to the time-varying nonlinear optimization problem with the residual error O(τ4), where τ signifies the sampling gap. In addition, according to the smooth solution accuracy, the proposed discrete-time zeroing neural network models have better performance than traditional computing models by adopting the Taylor-type computational differentiation formula with O(τ3) pattern. Finally, two illustrative numerical examples are further provided to demonstrate the efficiency and superiority of the proposed discrete-time zeroing neural network models for time-varying nonlinear optimization problems solving compared with the classical neural network models, furthermore, the proposed models are also applied on two-link mechanical arm with motion generation.

Five-step discrete-time noise-tolerant zeroing neural network model for time-varying matrix inversion with application to manipulator motion generation

In this paper, a novel Taylor-type difference rule with O(τ4) pattern error is provided for the first-order derivative approximation. Then, a high accuracy noise-tolerant five-step discrete-time zeroing neural network (ZNN) (termed as FDNTZNN model) is proposed to solve the time-varying matrix inversion problem in real-time. In addition, to obtain the derivative value of time-varying variables in real-world applications, the backward-difference rule is exploited to develop the FD-NTZNN model when the derivative information is unknown (FD-NTZNN-U). Theoretical analysis demonstrates that the proposed FD-NTZNN models have the properties of 0−stability, consistency and convergence. For comparative analysis, the classical Euler-type discrete-time ZNN model (EDZNN), five-step Taylor-type discrete-time ZNN model (FDZNN) and Euler-type discrete-time noise-tolerant ZNN (NTZNN) model (ED-NTZNN) are reconsidered. Ultimately, two illustrative numerical simulations and an application example to motion generation of manipulator are simulated to substantiate the feasibility and effectiveness of the proposed FD-NTZNN model and FD-NTZNN-U model for online time-varying matrix inversion in the presence of different types of noise.

Discrete-Time Zeroing Dynamics With Quadruplicate Error Pattern for Time-Varying Linear Inequality

Linear inequality (LI) plays an important role in many fields of science and engineering. Recently, a typical neural dynamics called zeroing dynamics (ZD) has been reported for online solution of time-varying LI (TVLI). On the basis of the previous work, the discrete-time form of the ZD with superior computational property is studied in this paper. Specifically, a Taylor-type difference rule is first presented for the first-order derivative approximation. By utilizing such a difference rule to discrete the previous ZD model, the new discrete-time ZD (DTZD) algorithm is thus established and proposed for TVLI solving. Such an algorithm performs better computational performance than the existing DTZD algorithm. Theoretical results show that the proposed DTZD algorithm has a quadruplicate error pattern on solving the TVLI. Comparative numerical results with two illustrative examples further substantiate the efficacy and superiority of the proposed DTZD algorithm over the existing DTZD algorithm.

Linear Interpolation on a Tetrahedron

The standard method for the linear interpolation on a tetrahedron of a function with continuous second-order partial derivatives bounded by a given constant is considered. Estimates of the approximation of first-order derivatives that are more exact than the known estimates are derived.

New Discrete-Time Models of Zeroing Neural Network Solving Systems of Time-Variant Linear and Nonlinear Inequalities

In this paper, a new one-step-ahead numerical differentiation rule termed 5-instant discretization formula is proposed for the first-order derivative approximation with higher computational precision. Then, by exploiting the proposed formula to discretize the continuous-time zeroing neural network [or termed, continuous-time Zhang neural network (ZNN)] models, two new discrete-time zeroing neural network [or termed, discrete-time ZNN (DTZNN)] models are proposed, analyzed and investigated for solving systems of discrete time-variant inequalities, including the system of discrete time-variant linear inequalities and the system of discrete time-variant nonlinear inequalities. For comparative purposes, the recently developed Taylor-type DTZNN models and the widely used Euler-type DTZNN models are also presented. Theoretical analyses show that the proposed DTZNN models are convergent, and their steady-state residual errors have an O(g <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) pattern with g denoting the sampling gap. Comparative numerical experimental results further substantiate the efficacy and superiority of the proposed DTZNN models for solving the systems of discrete time-variant inequalities.

General Six-Step Discrete-Time Zhang Neural Network for Time-Varying Tensor Absolute Value Equations

This article presents a general six-step discrete-time Zhang neural network (ZNN) for time-varying tensor absolute value equations. Firstly, based on the Taylor expansion theory, we derive a general Zhang et al. discretization (ZeaD) formula, i.e., a general Taylor-type 1-step-ahead numerical differentiation rule for the first-order derivative approximation, which contains two free parameters. Based on the bilinear transform and the Routh–Hurwitz stability criterion, the effective domain of the two free parameters is analyzed, which can ensure the convergence of the general ZeaD formula. Secondly, based on the general ZeaD formula, we design a general six-step discrete-time ZNN (DTZNN) for time-varying tensor absolute value equations (TVTAVEs), whose steady-state residual error changes in a higher order manner than those presented in the literature. Meanwhile, the feasible region of its step size, which determines its convergence, is also studied. Finally, experiment results corroborate that the general six-step DTZNN model is quite efficient for TVTAVE solving.

Accelerated complex-step finite difference for expedient deformable simulation

In deformable simulation, an important computing task is to calculate the gradient and derivative of the strain energy function in order to infer the corresponding internal force and tangent stiffness matrix. The standard numerical routine is the finite difference method, which evaluates the target function multiple times under a small real-valued perturbation. Unfortunately, the subtractive cancellation prevents us from setting this perturbation sufficiently small, and the regular finite difference is doomed for computing problems requiring a high-accuracy derivative evaluation. In this paper, we graft a new finite difference scheme, namely the complex-step finite difference (CSFD), with physics-based animation. CSFD is based on the complex Taylor series expansion, which avoids subtractions in first-order derivative approximation. As a result, one can use a very small perturbation to calculate the numerical derivative that is as accurate as its analytic counterpart. We accelerate the original CSFD method so that it is also as efficient as the analytic derivative. This is achieved by discarding high-order error terms, decoupling real and imaginary calculations, replacing costly functions based on the theory of equivalent infinitesimal, and isolating the propagation of the perturbation in composite/nesting functions. CSFD can be further augmented with multicomplex Taylor expansion and Cauchy-Riemann formula to handle higher-order derivatives and tensor-valued functions. We demonstrate the accuracy, convenience, and efficiency of this new numerical routine in the context of deformable simulation - one can easily deploy a robust simulator for general hyperelastic materials, including user-crafted ones to cater specific needs in different applications. Higher-order derivatives of the energy can be readily computed to construct modal derivative bases for reduced real-time simulation. Inverse simulation problems can also be conveniently solved using gradient/Hessian-based optimization procedures.

Novel numerical method of the fractional cable equation

In this article, mainly based on the second order compact approximation of first order derivative, the novel numerical method with second order temporal accuracy and fourth order spatial accuracy is proposed to solve the fractional cable equation. The numerical analysis involving convergence and stability of the novel numerical method subject to strict and detailed discussion. In addition, the numerical experiment strongly support the theoretical analysis results.

Solving future equation systems using integral-type error function and using twice ZNN formula with disturbances suppressed

In this paper, for solving future equation systems, two novel discrete-time advanced zeroing neural network models are proposed, analyzed and investigated. First of all, by using integral-type error function and twice zeroing neural network (or termed, Zhang neural network) formula, as the preliminaries and bases of future problems solving, two continuous-time advanced zeroing neural network models are presented for solving continuous time-variant equation systems. Secondly, a one-step-ahead numerical differentiation rule termed 5-instant discretization formula is presented for the first-order derivative approximation with higher computational precision. By exploiting the presented 5-instant discretization formula to discretize the continuous-time advanced zeroing neural network models, two novel discrete-time advanced zeroing neural network models are proposed. Theoretical analyses on the convergence and precision of the discrete-time advanced zeroing neural network models are proposed. In addition, in the presence of disturbance, the proposed discrete-time advanced zeroing neural network models still possess excellent performance. Comparative numerical experimental results further substantiate the efficacy and superiority of the proposed discrete-time advanced zeroing neural network models for solving the future equation systems.

New Discrete-Time ZNN Models for Least-Squares Solution of Dynamic Linear Equation System With Time-Varying Rank-Deficient Coefficient.

In this brief, a new one-step-ahead numerical differentiation rule called six-instant -cube finite difference (6I CFD) formula is proposed for the first-order derivative approximation with higher precision than existing finite difference formulas (i.e., Euler and Taylor types). Subsequently, by exploiting the proposed 6I CFD formula to discretize the continuous-time Zhang neural network model, two new-type discrete-time ZNN (DTZNN) models, namely, new-type DTZNNK and DTZNNU models, are designed and generalized to compute the least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient in real time, which is quite different from the existing ZNN-related studies on solving continuous-time and discrete-time (dynamic or static) linear equation systems in the context of full-rank coefficients. Specifically, the corresponding dynamic normal equation system, of which the solution exactly corresponds to the least-squares solution of dynamic linear equation system, is elegantly introduced to solve such a rank-deficient least-squares problem efficiently and accurately. Theoretical analyses show that the maximal steady-state residual errors of the two new-type DTZNN models have an pattern, where denotes the sampling gap. Comparative numerical experimental results further substantiate the superior computational performance of the new-type DTZNN models to solve the rank-deficient least-squares problem of dynamic linear equation systems.

Numerical simulation with the second order compact approximation of first order derivative for the modified fractional diffusion equation

In this paper, based on the second order compact approximation formula of first order derivative, the numerical simulation method with second order temporal accuracy and fourth order spatial accuracy is developed to simulate the modified fractional diffusion equation; the convergence, stability and solvability of the numerical simulation method are analyzed by Fourier analysis and algebraic theory, respectively; the theoretical results extremely consistent with the numerical experiment.

Novel Discrete-Time Zhang Neural Network for Time-Varying Matrix Inversion

In the previous work, Zhang et al. developed a special type of recurrent neural networks called Zhang neural network (ZNN) with continuous-time and discrete-time forms for time-varying matrix inversion. In this paper, a novel discrete-time ZNN (DTZNN) model for time-varying matrix inversion is proposed and investigated. Specifically, a new numerical difference rule based on Taylor series expansion is established in this paper for first-order derivative approximation. Then, by exploiting this Taylor-type difference rule, the novel DTZNN model, which is a five-step iteration algorithm, is thus proposed for time-varying matrix inversion. Theoretical results are also presented for the proposed DTZNN model to show its excellent computational property. Comparative numerical results with three illustrative examples further substantiate the efficacy and superiority of the proposed DTZNN model for time-varying matrix inversion compared with previous DTZNN models.

Numerical algorithm for solving the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative

In this paper, based on the second-order compact approximation of first-order derivative, the numerical algorithm with second-order temporal accuracy and fourth-order spatial accuracy is developed to solve the Stokes’ first problem for a heated generalized second grade fluid with fractional derivative; the solvability, convergence, and stability of the numerical algorithm are analyzed in detail by algebraic theory and Fourier analysis, respectively; the numerical experiment support our theoretical analysis results.

An improved KGF‐SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows

SummaryThe kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF‐SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF‐SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF‐SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF‐SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two‐dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF‐SPH and improved KGF‐SPH. The numerical results show that the improved KGF‐SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF‐SPH. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Taylor-type 1-step-ahead numerical differentiation rule for first-order derivative approximation and ZNN discretization

In order to achieve higher computational precision in approximating the first-order derivative and discretize more effectively the continuous-time Zhang neural network (ZNN), a Taylor-type numerical differentiation rule is proposed and investigated in this paper. This rule not only greatly remedies some intrinsic weaknesses of the backward and central numerical differentiation rules, but also overcomes the limitation of the Lagrange-type numerical differentiation rules in ZNN discretization. In addition, a formula is proposed to obtain the optimal step-length of the Taylor-type numerical differentiation rule. Moreover, based on the proposed numerical differentiation rule, the stability, convergence and residual error of the Taylor-type discrete-time ZNN (DTZNN) are analyzed. Numerical experimental results further substantiate the efficacy and advantages of the proposed Taylor-type numerical differentiation rule for first-order derivative approximation and ZNN discretization.

Presentation, error analysis and numerical experiments on a group of 1-step-ahead numerical differentiation formulas

In order to achieve higher computational precision in approximating the first-order derivative of the target point, the 1-step-ahead numerical differentiation formulas are presented. These formulas greatly remedy some intrinsic weaknesses of the backward numerical differentiation formulas, and overcome the limitation of the central numerical differentiation formulas. In addition, a group of formulas are proposed to obtain the optimal step length. Moreover, the error analysis of the 1-step-ahead numerical differentiation formulas and the backward numerical differentiation formulas is further investigated. Numerical studies show that the proposed optimal step-length formulas are effective, and the performance of the 1-step-ahead numerical differentiation formulas is much better than that of the backward numerical differentiation formulas in the first-order derivative approximation.