Certified predictor–corrector tracking for Newton homotopies. (May 2016)

• Record Type: Journal Article
• Authors: Hauenstein, Jonathan D.; Liddell, Alan C.
• Is Part Of: Journal of symbolic computation. Volume 74(2016)
• Page Start: 239
• Page End: 254
• Publication Date: 2016-05
• Subjects: Certified tracking -- Alpha theory -- Numerical algebraic geometry -- Homotopy continuation -- Newton's method; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2015.07.001
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: 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 γ -metric. Examples are included to compare the behavior of these certified tracking methods.