An adaptive Newton-method based on a dynamical systems approach

• Adaptive Newton method for general nonlinear operator equations.
• Dynamical systems approach leads to better understanding of chaos and convergence.
• Computationally inexpensive and easy implemental prediction strategy.
• Extensive tests from algebraic and differential equations.
• Graphical visualization of chaos reduction.

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.