Higher Order Bairstow Method for Solving Polynomial Equation

Abstract

1. I n t r o d u c t i o n : The BAIRSTOW Process The basic idea for finding the roots of real polynomials by finding a quadratic factor makes use of the following identity.@ + a, f l -1 + a, @-2 + . . . + an (9 + p 5 + q) ( f l -2 + b, ~n -3 + + 4 f l -4 + . . . + bn-2) + R@, 4 ) z + S(p7 q ) . B~~RSTOW method [2] consists in solving the systemStarting with the initial approximation (p, q), the correctioy A p and d q are found so that Expanding in T A Y~R series and truncating after the first order terms, we get R (P + LIP, P + Aq) = 0 9 (P + AP, q + Aq) = 0 * (2) R(P, q) -S(P, q ) -I JTp All + Sq Aq = 0 9 11, AP + 1 1 9 4 = 0 3(3)where R,, S, and R,, S, designate partial order derivates with respect to p and q. This is regarded as a linear system of equations for A p and Aq and when this system has been solved the procedure is repeated with the corrected values @, , q,) wherepIFp+ A p and q, = q + Aq. Similarly the subsequent approxlmatiom (pi, 4.1) are found out. Coefficients b,, 4,. . . , bn-2; R and S are computed from ( 1 ) by comparison.where n represents the degree of the polynomial with b, , = 1 and b-, = 0 .We get and bn = anp bn-1q bn-2 = Sp bn-1 Thus beginning with arbitrary values p and q, (3), (4) and ( 5 ) can be used to define an iterative process for getting a quadratic factor. This is usually known m LIN's method [3] which was extended and studied by FRIEDMAN [4] and LUKE and UPFORD [B]. The convergence properties have not been fully established, but the method is often slowly convergent. The theorem of KANTOKOVICII [l] gives conditions on R and S and on the starting values p , q which ensure convergence. The verification of these conditions is not computationally feasible, but the method is usually quadratically convergent. bk = akp bk-1 -4 bk-2 (k = 1,2, . . . , n) , (4) bn-1 = an-1 -p bn-2 -q bn-3 = R R = bn-1, S = bn + p bn-1.( 5 ) Higher Order B A~R S T O WProcess We introduce the following notations bk = CO, k where n is the degree of the polynomial. (4) and (5) are transformed to (k = 1,2,. . . , n), CO,k = akp CO,k-1 -qC0,k-2 c0,o = 1, Go, -1 = 0 R = C o , n -l , S = C o , n + p C o , n -~. (6)(k = 1,2, . . . , n) I n general when the process considers the effect of all the terms (say upto m-th order) in the expansion of (2) by means of TAYLOR series, we get the following recurrence relation ( k = 1 , 2 ,..., m -T ; T = l , 2 ,...)The partial derivatives upto m-th order are computed from the following recurrence relations R,(9) AP = pi -P = A,@, q) + A~( P , q) + . . * + 4) A q = 41 -4 = B~( P , 4) + B~P , q) + . . . + Btnb 4) where At and Bt (i = 1,2, . . . , m) are increment functions of the first, 2-nd and m-th order respectively. The equations (2), (7), (8) and (9) yield to the following system of equations which will determine the increment functions At and Bi. RpAt+Rg&+Ri=O, S p A( + Sq & + 8 1 = 0 9 where and R , = R ,

Preface

“…A two-stage implicit fifth-order method, which is both accurate and robust, has been developed. He [14] has also obtained higher order Bairstow's method for polynomial equation.…”