Iterative methods for multiple roots with memory using self-accelerating technique

Recently, self-accelerating technique has been widely used to improve the convergence order of iterative methods for simple roots. However, it has not been applied in those for multiple ones, yet. In this paper, for the first time, this technique is used to construct higher-order multiple root finding iterations with memory. It is shown that the R-order of the considering iterations are improved from 2 to 1+2≈2.4142, 4 to 5+2≈4.2361 and (5+17)/2≈4.5616, respectively. Comparison of the methods with and without self-accelerating parameters is given on numerical tests, which supports our theoretical analysis very well. •Self-accelerating technique is first used to construct iterations for multiple roots.•Using Newton’s interpolation formula, new approximation functions are derived.•The R-order of the considering iterations are improved.•Numerical tests show our self-accelerating iterative methods perform excellently.•The technique can enlarge the convergence region and make the iteration stable.