Trigonometric Term Research Articles

A new trigonometric kernel function for support vector machine

In the last few years, various types of machine learning algorithms, such as Support Vector Machine (SVM), Support Vector Regression (SVR), and Non-negative Matrix Factorization (NMF) have been introduced. The kernel approach is an effective method for increasing the classification accuracy of machine learning algorithms. This paper introduces a family of one-parameter kernel functions for improving the accuracy of SVM classification. The proposed kernel function consists of a trigonometric term and differs from all existing kernel functions. We show this function is a positive definite kernel function. Finally, we evaluate the SVM method based on the new trigonometric kernel, the Gaussian kernel, the polynomial kernel, and a convex combination of the new kernel function and the Gaussian kernel function on various types of datasets. Empirical results show that the SVM based on the new trigonometric kernel function and the mixed kernel function achieve the best classification accuracy. Moreover, some numerical results of performing the SVR based on the new trigonometric kernel function and the mixed kernel function are presented.

Complexity analysis of primal-dual interior-point methods for linear optimization based on a new efficient Bi-parameterized kernel function with a trigonometric barrier term

In this paper we are generalizing the efficient kernel function with trigonometric barrier term given by (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Using an elegant and simple analysis and under some easy to check conditions, we explore the best complexity result for the large update primal-dual interior point methods for linear optimization. This complexity estimate improves results obtained in (X. Li and M. Zhang, Oper. Res. Lett. 43 (2015) 471–475; M.R. Peyghami and S.F. Hafshejani, Numer. Algo. 67 (2014) 33–48; M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545). Our comparative numerical experiments on some test problems consolidate and confirm our theoretical results according to which the new kernel function has promising applications compared to the kernel function given by (M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672). Moreover, the comparative numerical study that we have established favors our new kernel function better than other best trigonometric kernel functions (M. Bouafia, D. Benterki and A. Yassine, J. Optim. Theory Appl. 170 (2016) 528–545; M. Bouafia and A. Yassine, Optim. Eng. 21 (2020) 651–672).

Three dimensional stabilization controller based on improved quaternion transformation for fixed-wing UAVs

To realize the three-dimensional stabilization control of fixed-wing unmanned aerial vehicles (UAVs), we analyze the nonholonomic characteristics, constraint non-integrability, and controllability of UAVs. To simplify the trigonometric function term in the dynamics of the UAV and avoid the singularity problem of the Euler angle in describing attitude, we use the quaternion theory to transform the dynamics of the UAV to avoid the complex trigonometric function derivation, which makes the dynamic matrix more concise. Based on this, a continuous periodic time-varying controller (CPTVC) is designed, and the effectiveness of the controller is proved using the homogeneous method. Finally, the results of the hardware in a loop simulation indicated that the exponential stability provided by the feedback controller can realize the three-dimensional stabilization of any initial position.

Shear-buckling of omega-stringer-stiffened panels

The local shear-buckling behaviour of omega-stringer-stiffened composite panels is analysed introducing a novel approximate analytical computational model. This type of stability analysis is critical for thin-walled structures like the proposed panel, which is applied, among other things, mainly in aerospace structures and marine vessels. In this context, computationally efficient methods are especially necessary for preliminary design phases that often involve optimization and extensive parameter studies. In the present work, a simple analytical model for the critical shear-buckling load is developed and evaluated taking bending-twisting coupling and different load directions into account. The modelling approach is based on energy methods and employs the principle of the stationary value of the total elastic potential. Hereby, the buckling shape functions are based on a combination of a polynomial term and a trigonometric Timoshenko-type term, where the latter is commonly used for closed-form analytical shear-buckling analysis. The approximate computational model is evaluated for four different laminate configurations and a wide range of different panel dimensions in comparison to finite element analyses. The parameter studies yield promising results with errors very acceptable in the context of preliminary design.

STUDYING THE STABILITY BY USING LOCAL LINEARIZATION METHOD

Abstract:
 In this paper we study the stability of one of a non linear autoregressive model with trigonometric term by using local linearization method proposed by Tuhro Ozaki .We find the singular point ,the stability of the singular point and the limit cycle. We conclude that the proposed model under certain conditions have a non-zero singular point which is a asymptotically salable ( when 0 ) and have an orbitaly stable limit cycle . Also we give some examples in order to explain the method.
 Key Words : Non-linear Autoregressive model; Limit cycle; singular point; Stability.

An efficient twice parameterized trigonometric kernel function for linear optimization

Recently, Bouafia et al. (J Optim Theory Appl 170:528–545, 2016) investigated a new efficient kernel function that differs from self-regular kernel functions. The kernel function has a trigonometric barrier term. This paper introduces a new efficient twice parametric kernel function that combines the parametric classic function with the parametric kernel function trigonometric barrier term given by Bouafia et al. (J Optim Theory Appl 170:528–545, 2016) to develop primal–dual interior-point algorithms for solving linear programming problems. In addition, we obtain the best complexity bound for large and small-update primal–dual interior point methods. This complexity estimate improves results obtained in Li and Zhang (Oper Res Lett 43(5):471–475, 2015), Peyghami and Hafshejani (Numer Algorithms 67:33–48, 2014) and matches the best bound obtained in Bai et al. (J Glob Optim 54:353–366) and Peng et al. (J Comput Technol 6:61–80, 2001). Finally, our numerical experiments on some test problems confirm that the new kernel function has promising applications compared to the kernel function given by Fathi-Hafshejani (Optimization 67(10):1605–1630, 2018).

Self-excited and hidden attractors in a novel chaotic system with complicated multistability

In this paper, a new 3D chaotic system with trigonometric function term as a nonlinear controller is proposed. Depending on the nonlinear controller and the value of the parameters, the proposed system exhibits self-excited attractor with an unstable equilibrium, and hidden attractor with no equilibrium or a stable equilibrium. In addition, the unusual and striking dynamic behavior of the coexistence of self-excited chaotic with periodic attractors, two self-excited chaotic attractors with periodic attractor, three periodic attractors, hidden chaotic with point attractors, two hidden chaotic attractors, and four hidden chaotic attractors are explored by selecting appropriate initial values. Consequently, the proposed system has high sensitivity to its initial values and parameters, hence it can be applied in chaos-based cryptographic applications. Thus, the non-periodicity of coexisting attractors of the system is investigated through Lyapunov exponents and Sample Entropy. To demonstrate the performance of the system in real applications, we construct a pseudo-random number generator (PRNG) based on the hidden attractor case. The randomness test results show that the generated PRNG pass all the statistical tests.

Primal–dual interior-point method for linear optimization based on a kernel function with trigonometric growth term

In this paper, we propose a large-update primal–dual interior-point algorithm for linear optimization problems based on a new kernel function with a trigonometric growth term. By simple analysis, we prove that in the large neighbourhood of the central path, the worst case iteration complexity of the new algorithm is bounded above by , which matches the currently best known iteration bound for large-update methods. Moreover, we show that, most of the so far proposed kernel functions can be rewritten as a kernel function with trigonometric growth term. Finally, numerical experiments on some test problems confirm that the new kernel function is well promising in practice in comparison with some existing kernel functions in the literature.

Complexity of Interior Point Methods for a Class of Linear Complementarity Problems Using a Kernel Function with Trigonometric Growth Term

In this paper, we propose a large-update primal-dual interior point method for solving a class of linear complementarity problems based on a new kernel function. The main aspects distinguishing our proposed kernel function from the others are as follows: Firstly, it incorporates a specific trigonometric function in its growth term, and secondly, the corresponding barrier term takes finite values at the boundary of the feasible region. We show that, by resorting to relatively simple techniques, the primal-dual interior point methods designed for a specific class of linear complementarity problems enjoy the so-called best-known iteration complexity for the large-update methods.

Complexity analysis of infeasible interior-point method for semidefinite optimization based on a new trigonometric kernel function

In this paper, a full Nesterov–Todd step infeasible interior-point method for solving semidefinite optimization problems based on a new kernel function is analyzed. In each iteration, the algorithm involves a feasibility step and several centrality steps. The centrality step is focused on Nesterov–Todd search directions, while we used a kernel function with trigonometric barrier term to induce the feasibility step. The complexity result coincides with the best-known iteration bound for infeasible interior-point methods.

Bessel phase functions: calculation and application

The Bessel phase functions are used to represent the Bessel functions as a positive modulus and an oscillating trigonometric term. This decomposition can be used to aid root-finding of certain combinations of Bessel functions. In this article, we give some new properties of the modulus and phase functions and some asymptotic expansions derived from differential equation theory. We find a bound on the error of the first term of this asymptotic expansion and give a simple numerical method for refining this approximation via standard routines for the Bessel functions. We then show an application of the phase functions to the root finding problem for linear and cross-product combinations of Bessel functions. This method improves upon previous methods and allows the roots in ascending order of these functions to be calculated independently. We give some proofs of correctness and global convergence.

A primal-dual interior-point algorithm for symmetric optimization based on a new kernel function with trigonometric barrier term yielding the best known iteration bounds

Kernel functions play an important role in defining new search directions for primal-dual interior-point algorithms for solving symmetric optimization problems. In this paper we present a new kernel function for which interior point method yields iteration bounds $${\mathcal {O}}(\sqrt{r}\log r\log \frac{r}{\epsilon })$$ and $${\mathcal {O}}(\sqrt{r}\log \frac{r}{\epsilon })$$ for large-and small-update methods, respectively, which matches currently the best known bounds for such methods.

An infeasible interior-point algorithm for monotone linear complementarity problem based on a specific kernel function

In this paper, a full-Newton step infeasible kernel-based interior-point algorithm for solving monotone linear complementarity problems is proposed. In each iteration, the algorithm computes the new feasibility search directions by using a specific kernel function with the trigonometric barrier term and obtains the centering search directions using the classical kernel function. The algorithm takes only full-Newton steps and therefore no line-searches are needed for generating the new iterations. The convergence of the algorithm is shown and it is proved that the iteration bound of the algorithm coincides with the currently best iteration bound for monotone linear complementarity problem.

A large-update primal–dual interior-point algorithm for second-order cone optimization based on a new proximity function

In this paper, we propose a large-update primal–dual interior-point algorithm for second-order cone optimization (SOCO) based on a class of kernel functions consisting of a trigonometric barrier term. The algorithm starts from a strictly feasible point and generates a sequence of points converging to an optimal solution of the problem. Using a simple analysis, we show that the algorithm has worst case iteration complexity for large-update primal–dual interior point methods which coincides with the so far best-known iteration bound for SOCO.

An Efficient Primal–Dual Interior Point Method for Linear Programming Problems Based on a New Kernel Function with a Trigonometric Barrier Term

In this paper, we present a primal---dual interior point method for linear optimization problems based on a new efficient kernel function with a trigonometric barrier term. We derive the complexity...

A full-Newton step infeasible interior-point method for linear optimization based on a trigonometric kernel function

An infeasible interior-point method (IIPM) for solving linear optimization problems based on a kernel function with trigonometric barrier term is analysed. In each iteration, the algorithm involves a feasibility step and several centring steps. The centring step is based on classical Newton’s direction, while we used a kernel function with trigonometric barrier term in the algorithm to induce the feasibility step. The complexity result coincides with the best-known iteration bound for IIPMs. To our knowledge, this is the first full-Newton step IIPM based on a kernel function with trigonometric barrier term.

Complexity analysis of primal-dual interior-point methods for semidefinite optimization based on a parametric kernel function with a trigonometric barrier term

In this paper, we present a class of large- and small-update primal-dual interior-point methods forsemidefinite optimization based on a parametric kernel function with a trigonometric barrier term.For both versions of the kernel-based interior-point methods, the worst case iteration bounds are established, namely,$O(n^{\frac{2}{3}}\log\frac{n}{\varepsilon})$ and $O(\sqrt{n}\log\frac{n}{\varepsilon})$, respectively.These results match the ones obtained in the linear optimization case.

A polynomial-time algorithm for linear optimization based on a new kernel function with trigonometric barrier term

In this paper, we propose a large-update interior-point algorithm for linear optimization based on a new kernel function. New search directions and proximity measure are defined based on this kernel function. We show that if a strictly feasible starting point is available, then the new algorithm has O(3/4log n/?) iteration complexity.

Flexure of cross-ply laminated plates using equivalent single layer trigonometric shear deformation theory

An equivalent single layer trigonometric shear deformation theory taking into account transverse shear deformation effect as well as transverse normal strain effect is presented for static flexure of cross-ply laminated composite and sandwich plates. The inplane displacement field uses sinusoidal function in terms of thickness coordinate to include the transverse shear deformation effect. The cosine function in thickness coordinate is used in transverse displacement to include the effect of transverse normal strain. The kinematics of the present theory is much richer than those of the other higher order shear deformation theories, because if the trigonometric term (involving thickness coordinate z) is expanded in power series, the kinematics of higher order theories (which are usually obtained by power series in thickness coordinate z) are implicitly taken into account to good deal of extent. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. The closed-form solutions of simply supported cross-ply laminated composite and sandwich plates have been obtained. The results of present theory are compared with those of the classical plate theory (CPT), first order shear deformation theory (FSDT), higher order shear deformation theory (HSDT) of Reddy and exact three dimensional elasticity theory wherever applicable. The results predicted by present theory are in good agreement with those of higher order shear deformation theory and the elasticity theory.

An interior-point method for $$P_*(\kappa )$$ P ∗ ( κ ) -linear complementarity problem based on a trigonometric kernel function

Recently, El Ghami (Optim Theory Decis Mak Oper Res Appl 31:331–349, 2013) proposed a primal dual interior point method for $$P_*(\kappa )$$ -Linear Complementarity Problem (LCP) based on a trigonometric barrier term and obtained the worst case iteration complexity as $$O\left( (1+2\kappa )n^{\frac{3}{4}}\log \frac{n}{\epsilon }\right) $$ for large-update methods. In this paper, we present a large update primal–dual interior point algorithm for $$P_{*}(\kappa )$$ -LCP based on a new trigonometric kernel function. By a simple analysis, we show that our algorithm based on the new kernel function enjoys the worst case $$O\left( (1+2\kappa )\sqrt{n}\log n\log \frac{n}{\epsilon }\right) $$ iteration bound for solving $$P_*(\kappa )$$ -LCP. This result improves the worst case iteration bound obtained by El Ghami for $$P_*(\kappa )$$ -LCP based on trigonometric kernel functions significantly.