Pivotolás

Pivotolás, or pivoting, is a technique in numerical linear algebra used during procedures such as Gaussian elimination and LU decomposition to improve numerical stability. By rearranging the rows and, in some cases, the columns of a matrix, pivoting ensures that the chosen pivot element is suitably large in magnitude, reducing the amplification of rounding errors during elimination.

There are several forms of pivotolás. Partial pivotolás selects the largest absolute value in the current column from the current row downward and swaps rows accordingly. Complete pivotolás allows permutation of both rows and columns, locating the overall largest element in the remaining submatrix as the pivot. More specialized variants include rook pivoting and threshold pivoting, each aiming to balance stability and computational effort.

The main purpose of pivotolás is to avoid divisions by very small numbers and to minimize error propagation in finite-precision arithmetic. By ensuring that the pivot is not near zero, the algorithm remains numerically safer, which is particularly important for ill-conditioned systems. Pivoting often comes with a modest additional cost, mainly due to row and possibly column swaps, but this cost is typically outweighed by the gains in accuracy.

Pivotolás is integral to LU factorization with pivoting and to stable implementations of Gaussian elimination. It also influences the computation of determinants, inverses, and, indirectly, certain eigenvalue algorithms that rely on stable factorization steps. While some matrices warrant less or no pivoting (for example, diagonally dominant or symmetric positive definite matrices), pivotolás is a standard tool to enhance robustness in general linear system solving.