Extension of Murakami's high-order non-linear solver to multiple roots

Abstract

Several one-parameter families of fourth-order methods for finding multiple zeros of non-linear functions are developed. The methods are based on Murakami's fifth-order method (for simple roots) and they require one evaluation of the function and three evaluations of the derivative. The informational efficiency of the methods is the same as the previously developed methods of lower order. For a double root, the method is more efficient than all previously known schemes. All these methods require the knowledge of multiplicity.

Keywords:

• multiple roots
• non-linear
• high-order
• efficiency

2000 AMS Subject Classification::

• 65D99
• 41A25

Notes

‘Maple is a system for mathematical computation – symbolic, numerical and graphical’ 14. For example, the command ‘rfp:=series(1/fp1,e,6);’ expands the reciprocal of the function fp1 into a power series in the variable e keeping terms up to O(e ⁶). Maple can convert that expansion into a polynomial by using the command ‘rfp:=convert(rfp,polynom):’