## Certified predictor-corrector tracking for Newton homotopies.(English)Zbl 1329.65110

Summary: We develop certified tracking procedures for Newton homotopies, which are homotopies for which only the constant terms are changed. For these homotopies, our certified procedures include using a constant predictor with Newton corrections, an Euler predictor with no corrections, and an Euler predictor with Newton corrections. In each case, the predictor is guaranteed to produce a point in the quadratic convergence basin of Newton’s method. We analyze the complexity of a tracking procedure using a constant predictor with Newton corrections, with the number of steps bounded above by a constant multiple of the length of the path in the $$\gamma$$-metric. Examples are included to compare the behavior of these certified tracking methods.

### MSC:

 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations 65H10 Numerical computation of solutions to systems of equations 65H04 Numerical computation of roots of polynomial equations

### Software:

HOM4PS; PHCpack; Bertini; NumericalAlgebraicGeometry; NAG4M2
Full Text:

### References:

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