Pivot element

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The pivot element (from the French pivot , pivot point; in military jargon, pivot designates the wingman of a formation during a turning maneuver) is that element of a set of numbers that is selected first by an algorithm (e.g. Gaussian elimination method , quicksort or basic exchange method ) is used to perform certain calculations.

In order for matrix algorithms such as the Gaussian elimination method to work, it is often necessary that non-zero elements exist. Depending on the algorithm, a search is then made not only for a non-vanishing element, but also for the largest (in terms of amount) element in the respective row or column. The selection of the element made in this way is then called pivotization . The row in which the pivot element is located is called the pivot line , the column of the pivot element is called the pivot column . Before pivoting, equilibration may have to be carried out in order to improve the condition number .

When sorting using Quicksort, the pivot element refers to the element that is selected as the division limit. Quicksort sorts ( recursively ) all elements to the left and right of the pivot element. The median element, which creates two sublists of equal size, is optimal .

In the case of the basic exchange method (also known as pivot method) in linear optimization, the pivot element of an iteration is the non-disappearing entry that is selected to exchange an independent variable for a basic variable.

Individual evidence

  1. G. Pictet: Essai sur la tactique de l'infanterie: Ouvrage méthodique où l'on trouve en détail et par ordre les principes, les règles & les maximes qui sont propres à cette partie de l'art de la guerre, avec des applications continuelles de la theory à la pratique. Volume 2, 1761, p. 130.

literature

  • Hans-Joachim Kowalsky, Gerhard O. Michler: Lineare Algebra. de Gruyter, Berlin / New York 2003, ISBN 3-11-017963-6 , p. 88.