Nulpivot

Nulpivot is a term used in numerical linear algebra to describe a pivot element that is treated as zero for the purposes of pivot selection, or more generally a pivoting strategy that explicitly handles zero or near-zero pivots during matrix factorization. The concept is procedural rather than a fixed matrix property and is encountered in algorithm descriptions, software implementations, and discussions of numerical stability.

Definition and usage

In Gaussian elimination and LU factorization, a pivot is the element used to eliminate entries below it. A nulpivot occurs when the intended pivot a[k,k] is exactly zero or falls below a chosen tolerance ε. When this happens, a nulpivot strategy selects a nonzero pivot from the remaining submatrix, typically by swapping rows or columns, or by applying a regularization step to continue the factorization. The term “nulpivot” emphasizes that the standard pivot is effectively treated as zero for the purposes of the current step.

Variants and methods

Nulpivot strategies include threshold-based pivoting (treating pivots with |a[k,k]| < ε as null and seeking an alternative pivot), row and/or column interchanges to bring a nonzero element into the pivot position, and rank-revealing approaches that monitor consecutive nulpivots to detect rank deficiency. In some workflows, a small regularization term may be introduced to stabilize the process, though this alters the original problem.

Implications

Nulpivot handling affects numerical stability and accuracy. It can reveal rank deficiency and influence conditioning, especially in ill-posed or near-singular systems. Careful choice of tolerance and pivoting strategy is essential to balance progress of the factorization with solution quality.

See also: pivoting, partial pivoting, complete pivoting, LU decomposition, rank deficiency, regularization.