A Three-Stage Algorithm for Real Polynomials Using Quadratic Iteration

Abstract

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 2n milliseconds. An ALGOL 60 implementation of the algorithm and a program which calculates a posteriori bounds on the zeros may be found in Jenkins' 1969 Stanford dissertation [2]. TABLE OF CONTENTS 1. Introduction 545 2. Fixed-shift K polynomials 546 3. A variable-shift iteration for a quadratic factor 550 4. Convergence of the variable-shift iteration for a quadratic factor 550 5. Rate of convergence 554 6. The three-stage algorithm 555 7. Proof of global convergence of the three-stage algorithm 557 8. Variable-shift iteration for a quadratic factor is quadratic generalized Rayleigh iteration 559 9. Computer implementation of the algorithm 561 10. Numerical results 564 Acknowledgment 566 References 566 1, Introduction. We introduce a new three-stage algorithm for calculating the zeros of a real polynomial P, P(z)= ai Zn-i, ao=l, a. 4:0, i=o P(z)-I (zpi) mi i=1 *

A New and Efficient Program for Finding all Polynomial Roots

“…+ h k q k : (8) The new estimate x (k+1) is determined such that it is the one root ofP (x) closer to x (k) . An example is given in Fig.…”

“…(2){(7). This is followed by the iteration equation (8). We have observed that values q k computed according to Eq.…”

Divided differences, shift transformations and Larkin’s root finding method

“…Therefore, the zeros are determined in increasing order, which usually keeps the deflation stable (see Wilkinson [8], Jenkins and Traub [9]). An iteration was terminated if monotonicity was violated.…”

A Generalization of the Jenkins-Traub Method

“…The one modification that might be considered is that of applying the Jenkins-Traub method and its dual alternately, thus locating, in turn, a relatively small and then a relatively large zero. This would help towards alleviating the problem of deflation instability referred to by Jenkins and Traub [4]. Of course, this modification may also be considered as the Jenkins-Traub method applied to P, say, to produce a zero pt, and then the same method applied to the inverse polynomial ofPj(z) = P(z)/(z -Pj).…”