Abstract: We propose the numerical method defined by
xn+1 = xn − λ[f(xn − μh(xn))/]f'(xn) , n ∈ N,
and determine the control parameter λ and μ to converge cubically. In addition, we derive the asymptotic error constant. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-precision computability.
Abstract: A given polynomial, possibly with multiple roots, is
factored into several lower-degree distinct-root polynomials with
natural-order-integer powers. All the roots, including multiplicities,
of the original polynomial may be obtained by solving these lowerdegree
distinct-root polynomials, instead of the original high-degree
multiple-root polynomial directly.
The approach requires polynomial Greatest Common Divisor
(GCD) computation. The very simple and effective process, “Monic
polynomial subtractions" converted trickily from “Longhand
polynomial divisions" of Euclidean algorithm is employed. It
requires only simple elementary arithmetic operations without any
advanced mathematics.
Amazingly, the derived routine gives the expected results for the
test polynomials of very high degree, such as p( x) =(x+1)1000.