The Solution of Nonlinear Systems of Equations by A-Stable Integration Techniques

It is well known that a damped or underrelaxed Newton’s method will sometimes solve a system of nonlinear equations when the full Newton’s method cannot. This happens, for example, when only a poor initial approximation to the solution is known. By considering Newton’s method as Euler’s method applied to the corresponding differential equation, we ask if there is a better way to integrate that differential equation, i.e., we seek a more rapidly converging and more stable integration technique than damped Newton's method. It is shown here by an extension of the Dahlquist A-stability theory that the A-stable methods allow us to achieve our goals.