Variable Order Adams-Bashforth Predictors with an Error-Stepsize Control for Continuation Methods

Variable order Adams–Bashforth predictors with an error-stepsize control are developed and used for a class of predictor-corrector continuation methods which trace solution paths of an underdetermined nonlinear system $F(w) = 0$, where $F:\mathbb{R}^{n + 1} \to \mathbb{R}^n $. The predictor method developed here employs consecutive order Adams–Bashforth predictors to determine the stepsize by estimating the local truncation error in the lower order formula, a comparison of three consecutive order methods to choose the order that yields the largest stepsize, and local extrapolation to improve accuracy. Unlike many differential equation algorithms, the stepsize is changed at each step to give more uniformity in the predictor error and the global error is controlled at each step through the use of a Newton-like correction back to the path after each prediction. The philosophy of following the path closely with only one or two corrections per step is shown to yield an efficient and robust algorithm which is fairly insensitive to parameter settings. The key to the success and robustness of these multistep predictors is the approximation of arclength along the path at an accuracy commensurate with that of the tangents to the path, which is achieved by a local parameterization suggested by the corrector equations. To demonstrate the efficiency and robustness of the resulting continuation algorithm, several numerical comparisons are made with state-of-the-art software packages HOMPACK and PITCON.