A New Newton Method with Memory for Solving Nonlinear Equations

• Main Authors: Xiaofeng Wang, Yuxi Tao
• Format: Article
• Language: English
• Published: MDPI AG 2020-01-01
• Series: Mathematics
• Subjects: simple roots; newton method; nonlinear equation; self-accelerating parameter; computational efficiency
• Online Access: https://www.mdpi.com/2227-7390/8/1/108

A new Newton method with memory is proposed by using a variable self-accelerating parameter. Firstly, a modified Newton method without memory with invariant parameter is constructed for solving nonlinear equations. Substituting the invariant parameter of Newton method without memory by a variable self-accelerating parameter, we obtain a novel Newton method with memory. The convergence order of the new Newton method with memory is <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </semantics> </math> </inline-formula>. The acceleration of the convergence rate is attained without any additional function evaluations. The main innovation is that the self-accelerating parameter is constructed by a simple way. Numerical experiments show the presented method has faster convergence speed than existing methods.