Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence

Table 1

Achieved roots of Wilkinson polynomial by considering initial values of Schmeisser’s method.

Roots of 𝑢(𝑥)	Initial values of 𝑏[𝑖]	Eigenvalues of 𝑈	Schmeisser error	Fiedler error
1	0.99999999965	0.99999999988800000663	0.00𝐸−15	1.199𝐸−9
2	1.99999999999	1.99999999680000058	5.33𝐸−15	3.199𝐸−10
3	2.99999999998	2.9999999999359999597	2.66𝐸−15	6.400𝐸−10
4	3.99999999999	3.999999999679999014	1.421𝐸−14	3.20𝐸−10
5	5.00000000001	4.999999999999999988	2.66𝐸−15	1.200𝐸−17
6	6.00000000001	5.999999999999999626	0.00𝐸−15	3.74𝐸−16
7	7.00000000007	6.999999999999999984	1.68𝐸−14	1.600𝐸−17
8	8.00000000004	7.999999999999999551	1.15𝐸−14	4.490𝐸−16
9	9.00000000005	8.999999999999999997	3.55𝐸−15	3.000𝐸−18
10	10.00000000001	9.999999999999998875	1.78𝐸−15	1.125𝐸−15
11	11.00000000001	10.999999999999999904	5.33𝐸−15	9.600𝐸−16
12	12.00000000001	12.00000000000000001	7.11𝐸−15	1.000𝐸−17
13	13.00000000002	12.99999999999999974	5.33𝐸−15	2.600𝐸−16
14	14.00000000002	14.00000000000000019	1.78𝐸−15	1.900𝐸−16
15	15.00000000002	14.99999999999999982	7.11𝐸−15	1.800𝐸−16
16	16.00000000002	15.99999999999999977	1.776𝐸−14	2.300𝐸−16
17	17.00000000002	16.99999999999999949	3.55𝐸−15	5.100𝐸−16
18	18.00000000002	17.99999999999999949	7.11𝐸−15	5.100𝐸−16
19	19.00000000002	18.99999999999999977	1.421𝐸−14	2.230𝐸−15
20	20.00000000002	19.99999999999999916	2.487𝐸−14	8.400𝐸−16