Efficient higher order derivative-free multipoint methods with and without memory for systems of nonlinear equations

ABSTRACT

We present a family of multipoint methods without memory with sixth-order convergence for solving systems of nonlinear equations. The methods use first-order divided difference operators and are derivative free. Extending the work further, we explore a family of methods with memory with increasing order of convergence from the basic family without memory. The increase in convergence order is attained by varying a free parameter step to step using information available from the previous step. It is proved that the convergence order of the family with memory is increased from 6 to $3+\sqrt{1}0\approx 6.162$, and in some special cases to $3+2\sqrt{3}\approx 6.464$ and $3+\sqrt{1}3\approx 6.606$. Computational efficiency of the methods is discussed and compared with existing methods. Numerical examples, including those arise from integral equations and boundary value problems, are considered to verify the theoretical results. A comparison with the existing methods shows that the new methods are more efficient than existing ones and hence use the minimum computing time in multiprecision arithmetic.

KEYWORDS:

• Systems of nonlinear equations
• derivative-free methods
• Traub–Steffensen method
• order of convergence
• computational efficiency

2010 AMS Subject Classifications:

• 65H10
• 65Y20
• 41A58

Disclosure statement

No potential conflict of interest was reported by the authors.