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Encyclopedia > Pivot element

The pivot or pivot element is the element of a matrix, which is selected first by an algorithm (e.g. Gaussian elimination, Quicksort, Simplex algorithm), to do certain calculations with the matrix. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, computing, linguistics, and related disciplines, an algorithm is a procedure (a finite set of well-defined instructions) for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... In mathematics, Gaussian elimination (not to be confused with Gaussâ€“Jordan elimination), named after Carl Friedrich Gauss, is an algorithm in linear algebra for determining the solutions of a system of linear equations, for determining the rank of a matrix, and for calculating the inverse of an invertible square matrix. ... Quicksort in action on a list of random numbers. ... In mathematical optimization theory, the simplex algorithm, created by the North American mathematician George Dantzig in 1947, is a popular technique for numerical solution of the linear programming problem. ...

The above mentioned matrix algorithms require an entry distinct from zero in pivot position to work properly or at all respectively. Depending on the algorithm either one (random) element distinct from zero or the element with the greatest absolute value in a row or column is chosen. This is called pivotization. The row containing the pivot element is called pivot row, the pivot element's column is called pivot column.

Pivot element in Quicksort means the element that is selected as boundary for partitioning. Quicksort sorts all elements „left“ and „right“ of the pivot element recursively. Quicksort in action on a list of random numbers. ... A visual form of recursion known as the Droste effect. ...

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