Pivottallet

Pivottallet is a term used in numerical linear algebra to denote the pivot value or pivot element in Gaussian elimination and related factorization methods. In a matrix A, during LU decomposition or Gauss elimination, the pivot is the entry that is used to eliminate other entries in its column, by dividing the row by the pivot and subtracting multiples of the pivot row from others. The choice of pivot is critical for numerical stability, hence pivoting strategies are employed: partial pivoting, which swaps rows to bring the largest (in absolute value) element in the current column to the pivot position; complete pivoting, which may also swap columns; and rook pivoting, among others.

Correctly performing pivoting avoids division by very small numbers and reduces round-off error, improving the accuracy of the solution to a linear system. The pivot sequence determines the LU factors L and U, possibly with permutation matrices P (and sometimes Q in complete pivoting) such that PAQ = LU.

The term pivottallet is commonly used in Danish texts as a direct translation of “pivot element” or “pivot value,” with the English equivalent being pivot element. The concept is foundational in numerical linear algebra and is implemented in many computational linear algebra libraries, where robust pivoting routines help ensure stable and accurate solutions for large and ill-conditioned systems.