Modified Newton's method with third-order convergence and multiple roots (Q1398708)

Some modified Newton methods, with order of convergence three in the case of simple roots, obtained by using interpolatory quadrature formula and not requiring the second or higher derivatives of the function, are studied in the case of multiple roots. A typical result (Theorem 3) reads as follows: If \(\xi \) is a root of \(f(x)\) with multiplicity \(p>1\) (\(f(\xi)=0,\;f^{\prime }(\xi)=0,\dots ,f^{(p)}(\xi)\neq 0\)) then the modified Newton method obtained by using a quadrature formula of order at least \(p-1\) and using the corrected Newton's method has order of convergence two.