Some Improved Inclusion Methods for Polynomial Roots with Weierstrass' Corrections

1. I N T R O D U C T I O N Let P be a monic complex polynomial of degree n >_ 3 with simple zeros (1 , . . , (,, and let Zl , . . . , z , be distinct approximations of these zeros. Then an arbitrary zero can be expressed by (j zj _ ,,WJ , ( j 1,. . . , n) (1) 1 w k=l,k~j (see [3,4]), where = . P(z ) (2) k= l ,k#j is the so-called Weierstrass' correction [5]. Suppose that we have found disjoint disks ZI , . . . , Z~ in the complex plane such that (j E Zj for any j E {1 , . . . , n}. Starting from the fixed-point relation (1) and using circular complex arithmetic Petkovi~ stated in [1] the third order method for the simultaneous inclusion of all zeros of P, Zj = zj , Wj , (i = 1, . . . , n), (3) 1 w k=l ,kCj where Zj denotes the new circular approximation for (j. Considering the fixed-point relation (1), we observe that the exact zero ~j on the right-hand side can be substituted by the Weierstrass' approximation zj Wj. In this way, we obtain the fourth order method in ordinary complex arithmetic, ~j = zj _ , Wj , (j = 1, . . . , n) (4) 1 w. k= 1,k:~j zk-zj+Wj Typeset by . ~ T ~