A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

A complete MAPLE procedure is designed to effectively implement an algorithm for approximating trigonometric functions. The algorithm gives a piecewise polynomial approximation on an arbitrary interval, presenting a special partition that we can get its parts, subintervals with ending points of finite rational numbers, together with corresponding approximate polynomials. The procedure takes a sequence of pairs of interval–polynomial as its output that we can easily exploit in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both rational and irrational arguments as well as drawing the graph of the piecewise approximate functions are presented. Moreover, from the approximate integration on [ a , b ] with integrands of the form x m sin x , another MAPLE procedure is proposed to find the desired polynomial estimates in norm for the best L 2 -approximation of the sine function in the vector space P ℓ of polynomials of degree at most ℓ, a subspace of L 2 ( a , b ) .

A Computational Method with MAPLE for a Piecewise Polynomial Approximation to the Trigonometric Functions

A complete MAPLE procedure is designed to implement effectively an algorithm for approximating the trigonometric functions. The algorithm gives a piecewise polynomial approximation on an arbitrary interval, presenting a special partition that we can get its parts, subintervals with ending points of finite rational numbers, together with corresponding approximate polynomials. The procedure takes a sequence of pairs of interval-polynomial as its output that we can easily explore in some useful ways. Examples on calculating approximate values of the sine function with arbitrary accuracy for both of rational and irrational arguments as well as drawing the graph of the piecewise approximate functions will be presented. Moreover, from the approximate integration of integrands of the form $x^m\sin x$ on $[a,b]$, another MAPLE procedure is proposed to find the desired polynomial estimates in norm for the best $L^2$-approximation of the sine function in the vector space $\mathcal{P}_{\ell}$ of polynomials of degree at most $\ell$, a subspace of $L^2(a,b)$.

New algorithm for the design of robust PI controller for plants with parametric uncertainty

This paper proposes a new algorithm for the design of robust PI controller for plants with parametric uncertainty using new necessary and sufficient stability conditions. Most of the control systems operate under large uncertainty causing degradation of system performance and destabilization. In order to compensate these shortcomings, a robust PI controller is designed based on new necessary and sufficient conditions for stability of a plant with parametric uncertainty, a class of interval polynomial. New necessary and sufficient conditions for the determination of robust stability of interval polynomials have been developed using the results of Routhe’s theorem and Karitonov theorem. A set of inequalities are derived based on these developed new necessary and sufficient conditions to obtain robust controller parameters. The proposed method is simple and involves less computational complexity compared with the available methods in the literature. The efficacy of the proposed methodology is demonstrated with a numerical example for successful implementation.

Properties of the Set of Hadamardized Hurwitz Polynomials

We say that a Hurwitz polynomialptis a Hadamardized polynomial if there are two Hurwitz polynomialsftandgtsuch thatf∗g=p, wheref∗gis the Hadamard product offandg. In this paper, we prove that the set of all Hadamardized Hurwitz polynomials is an open, unbounded, nonconvex, and arc-connected set. Furthermore, we give a result so that a fourth-degree Hurwitz interval polynomial is a Hadamardized polynomial family and we discuss an approach of differential topology in the study of the set of Hadamardized Hurwitz polynomials.