International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014
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mapping from the...
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Polynomial evaluations in

Techniques for the evaluation of complex polynomials with one and two variables are introduced.Polynomials arise in may areas such as control systems, image and signal processing, coding theory,electrical networks, etc., and their evaluations are time consuming. This paper introduces new evaluationalgorithms that are straightforward with fewer arithmetic operations and a fast matrix exponentiation technique.
Published on: Mar 4, 2016
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Transcripts - Polynomial evaluations in

  • 1. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 DOI:10.5121/ijitca.2014.4302 7 POLYNOMIAL EVALUATIONS IN ENGINEERING Khier Benmahammed1 , Saeed badran2 and Bassam Kourdi2 1 Department of electronics, LSI, University Ferhat Abbas Setif, 19000 Setif, Algeria. 2 Electrical engineering department at Al Baha University, Al Baha, Saudia Arabia. Senior Member, IEEE ABSTRACT Techniques for the evaluation of complex polynomials with one and two variables are introduced.Polynomials arise in may areas such as control systems, image and signal processing, coding theory,electrical networks, etc., and their evaluations are time consuming. This paper introduces new evaluationalgorithms that are straightforward with fewer arithmetic operations and a fast matrix exponentiation technique. Keywords: Complex polynomial, mapping, polynomial arithmetic, one or two variables, control systems, fast matrix exponentiation. 1. INTRODUCTION Polynomials play an important role in almost all areas of engineering. Polynomials have wielded an enormous influence on the development of mathematics, since ancient times. Now-days, polynomial models are ubiquitous and widely used across the sciences. They arise in robotics, coding theory, control systems, electrical networks, image and signal processing, mathematical biology, computer vision, game theory, statistics, and numerous other areas. In the last few decades, a tremendous improvements in microelectronics technology have led to an advancement in microprocessors which in their turn have increased the availability of low cost personal computers. The personal computers have had multiplicative effects on a number of areas such as control systems, signal processing, etc.. However, at the beginning, many of the personal computers had compilers without the capability of complex arithmetic. Recent developments in computer software for computation have revolutionized the aforementioned fields. Formerly inaccessible problems are now tractable, providing fertile ground for simulations. This is a paper about the use of polynomials in engineering. We can do justice to only very small part of the subject and we confine most of our attention to the polynomial evaluation on a point of the complex plane. However, as the paper requires some basic notions of complex analysis, we have concentrated in the first two sections on building some theoretical foundations. Among which a
  • 2. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 8 mapping from the complex variable domain to the skew symmetric matrix domain is studied and used in combination with the Horner technique to develop polynomial evaluation algorithms. These algorithms do not involve any complex arithmetic and require fewer floating point arithmetic operations than the conventional techniques. Furthermore, the algorithms can be used in the stability analysis of linear continuous time invariant systems. Following similar arguments as before, one can develop algorithms for the two variable polynomials. If the system is given in its state space representation, then using the Leverrier Faddeva algorithm, one can compute the transfer function. In this system framework, one often needs to calculate a matrix to a certain power. The latter operation is time consuming. Therefore, we have also provided a fast matrix exponentiation algorithm. The paper is structured as follows: the third section introduces the key concepts of complex variables needed in the forthcoming sections. The fourth section is devoted to the development of the real polynomial evaluation. The fifth section is focused on the generalization of the previously developed algorithm to include the complex polynomials. The sections six and seven are devoted to the arithmetics of a polynomial with its complex conjugate. The arithmetic operations, including the addition, subtraction, multiplication and division, and the differentiation of a polynomial evaluated at a given point on a complex plane, are often used in variety of areas, namely the adaptive robust optimal control and signal processing.The section eight shows how the algorithms introduced in the previous sections can be in control systems.The algorithms are used in the computation of the system frequency response. The latter is needed in order to draw the Bode plots or the Nyquist plot in the system stability analysis. In this section, a fast matrix exponentiation technique for a very large exponents is also introduced. Finally, section nine, devoted to the generalization of the previous techniques, presents an algorithm for the evaluation of 2 D polynomials.
  • 3. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 9 2. NOTIONS OF COMPLEX VARIABLES
  • 4. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 10
  • 5. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 11 3. POLYNOMIALS WITH REAL COEFFICIENTS
  • 6. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 12 4. POLYNOMIALS WITH COMPLEX COEFFICIENTS
  • 7. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 13
  • 8. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 14 5. ARITHMETIC OF COMPLEX POLYNOMIALS 6. DERIVATIVE OF A POLYNOMIAL
  • 9. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 15 7. APPLICATIONS AND MATRIX EXPONENTIATION
  • 10. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 16
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  • 13. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 19 Notice that the use of equations (76), (77) and (78) requires the computation of a matrix to a certain power. The matrix exponentiation is very important subject in control, image and signal processing. Thusif the exponent of the matrix exponentiation is large enough, the calculation becomes tedious. Therefore,a technique of conversion from a binary number to an integer is used to develop a fast algorithm for the computation of a matrix exponentiation. The method of conversion is given as follows: 8. TWO DIMENSIONAL POLYNOMIALS The algorithms given in [4] and repeated in the previous sections will be extended to the two dimensional case. Although, a vast quantity of results has been found for the one variable polynomial, relatively fewer of these results can be extended to the two variables case. The most serious problem in the generalization of the one variable techniques to the two variable counterpart, is the fact that there is no fundamental theorem of algebra for polynomials in two independent variables. Consider the linear continuous time invariant two dimensional (2D) system described by the transfer function
  • 14. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 20
  • 15. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 21
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  • 18. International Journal of Information Technology, Control and Automation (IJITCA) Vol.4, No.3, July 2014 24 9. CONCLUSION In this paper, we have introduced algorithms for the evaluation of complex polynomials with one and two variables which can used for the stability analysis of linear continuous time invariant systems. We have also presented a fast matrix exponentiation technique. These algorithms require fewer arithmetic operations than the existing techniques. REFERENCES [1] J. W. Brown and R. V. Churchill, Complex variables and its applications. McGraw Hill 7th edition, New York 2003. [2] G. Strang, Linear algebra and its applications. 4th edition, Thomson Brooks/Cole, Belmont California 2006. [3] C. T. Chen, Linear system theory and design. Holt Reinhart and Winston, New York 1984. [4] K. Benmahammed, Complex polynomials and control systems, Proceedings of the first IEEE Conference on Control Applications,Vol.2, pp. 677 - 681, 1992. DOI: 10.1109/CCA.1992.269766.

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