Certified predictor-corrector tracking for Newton homotopies

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• 作者：Jonathan D. Hauenstein ^{hauenstein@nd.edu" class="auth_mail" title="E-mail the corresponding author} ; Alan C. Liddell Jr. ^{aliddel1@nd.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Certified tracking ; Alpha theory ; Numerical algebraic geometry ; Homotopy continuation ; Newton's method
• 刊名：Journal of Symbolic Computation
• 出版年：2016
• 出版时间：May-June 2016
• 年：2016
• 卷：74
• 期：Complete
• 页码：239-254
• 全文大小：688 K

文摘

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.