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The Exact Root Algorithm for Computing the Real Roots of an Nth Degree Polynomial
Abstract
Problem statement: The need to find an efficient and reliable algorithm for computing the exact real roots of the steady-state polynomial encountered in the investigation of temperature profiles in biological tissues during Microwave heating and other similar cases as found in the literature gave rise to this study. Approach: The algorithm (simply called ERA-Exact Root Algorithm) adopted polynomial deflation technique and uses Newton-Raphson iterative procedure though with a modified termination rule. A general formula was specified for finding the initial approximation so as to overcome the limitation of local convergence which is inherent in Newton’s method. Results: A new algorithm for finding the real roots of an nth degree polynomial at a practically low computational cost was obtained. Conclusion/Recommendations: ERA is simple, flexible, easy to use and has clear benefits and preferences to a number of existing methods.
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We introduce a new three-stage process for calculating the zeros of a polynomial with real coefficients. The algorithm finds either a linear or quadratic factor, working completely in real arithmetic. In the third stage the algorithm uses one of two variable-shift iterations corresponding to the linear or quadratic case. The iteration for a linear factor is a real arithmetic version of the third stage of the algorithm for complex polynomials which we studied in an earlier paper. A new variable-shift iteration is introduced in this paper which is suitable for quadratic factors. If the complex algorithm and the new real algorithm are applied to the same real polynomial, then the real algorithm is about four times as fast.
We prove that the mathematical algorithm always converges and show that the rate of convergence of the third stage is faster than second order.
The problem and algorithm may be recast into matrix form. The third stage is a quadratic form of shifted inverse powering and a quadratic form of generalized Rayleigh iteration.
The results of extensive testing are summarized. For an ALGOL W program run on an.IBM 360/67 we found that for polynomials ranging in degree from 20 to 50, the time required to calculate all zeros averaged milliseconds.
An ALGOL 60 implementation of the algorithm and a program which calculates a posteriors bounds on the zeros may be found in Jenkins’ 1969 Stanford dissertation [2].
We show that Bairstow's method is just one member of a family of similar algorithms for determining a quadratic factor of a polynomial. We suggest a way of choosing an appropriate member of this family for a particular problem. Numerical results indicate that our strategy compares favorably with the classical Bairstow algorithm.
In classical linear algebra, the eigenvalues of a matrix are sometimes defined as the roots of the characteristic polynomial. An algorithm to compute the roots of a polynomial by computing the eigenvalues of the corresponding companion matrix turns the tables on the usual definition. We derive a firstorder error analysis of this algorithm that sheds light on both the underlying geometry of the problem as well as the numerical error of the algorithm. Our error analysis expands on work by Van Dooren and Dewilde in that it states that the algorithm is backward normwise stable in a sense that must be defined carefully. Regarding the stronger concept of a small componentwise backward error, our analysis predicts a small such error on a test suite of eight random polynomials suggested by Toh and Trefethen. However, we construct examples for which a small componentwise relative backward error is neither predicted nor obtained in practice. We extend our results to polynomial matrices, where the result is essentially the same, but the geometry becomes more complicated.
Appropriate use of computer technology in the learning process has been lacking and many web-based and stand alone tutorial systems in today's market-place do not provide material that teaches students in an enjoyable and effective way. This paper proposes a Web-based Intelligent Tutorial System that will present material in an intuitive, interactive, and innovative manner while focusing on ways to integrate qualitative and quantitative methods throughout the learning experience. In this system, the lessons are guided by the actual skill set of the student taking the course. The system evaluates the student's state of knowledge through examination and, depending on the specific percentage and level of questions they answer correctly, it will guide them through a lesson plan
The problem of finding the roots of a polynomial is equivalent to finding the eigenvalues of an upper Hessenberg matrix, which can be done with the QR algorithm. It is shown that the QR algorithm has considerable advantages over other standard algorithms to find the roots of a polynomial.
We show that Bairstow's method is just one member of a family of similar algorithms for determining a quadratic factor of a polynomial. We suggest a way of choosing an appropriate member of this family for a particular problem. Numerical results indicate that our strategy compares favorably with the classical Bairstow al-gorithm.
A categorized bibliography on roots of polynomials is presented, covering the period from the earliest times until late 1990. The actual bibliography is on a computer diskette accompanying this issue, together with programs designed to print parts of the main work, such as entries belonging to a certain category.
The classical problem of solving an nth degree polynomial equation has substantially influenced the development of mathematics throughout the centuries and still has several important applications to the theory and practice of present-day computing. We briefly recall the history of the algorithmic approach to this problem and then review some successful solution algorithms. We end by outlining some algorithms of 1995 that solve this problem at a surprisingly low computational cost.
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DOI: 10.1016/0377-0427(93)90064-I