# pivoting

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## piv·ot·ing

(pivŏt-ing)
A swinging motion of the hand and arm carried out by balancing on the fulcrum finger during periodontal scaling. The hand pivot is used to assist in maintaining adaptation of the working-end of the scaling instrument.
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The filled boxes indicate diagonal supernodes of different sizes [n.sub.s] where Supernode-Bunch-Kaufman pivoting is performed and the dotted boxes indicate the remaining part of the supernodes.
An interchange among the rows and columns of a supernode of diagonal size [n.sub.s], referred to as Supernode-Bunch-Kaufman pivoting, has no effect on the overall fill-in and this is the mechanism for finding a suitable pivot in our SBK method.
Algorithm 2.1 describes the usual 1 x 1 and 2 x 2 Bunch-Kaufman pivoting strategy within the diagonal block corresponding to a supernode of size [n.sub.s].
Symmetric reorderings to improve the results of pivoting on restricted subsets.
Pivoting Variant 1: Fill-in reduction [P.sub.Fill] based on compressed subgraphs.
Pivoting Variant 2: Use All Preselected 2 x 2 Pivots.
As we have commented above, the triangular factors of the LDU-decomposition appearing when we apply complete pivoting to an n x n nonsingular matrix have off-diagonal elements with absolute value bounded above by 1 and then their condition numbers are bounded in terms of n.
Let us start by showing in the next result a first example, where pivoting is not necessary.
Gaussian elimination with a given pivoting strategy, for nonsingular matrices A = [([alpha].sub.ij]).sub.1[less than or equal to] i,j [less than or equal to] n], consists of a succession of at most n - 1 major steps resulting in a sequence of matrices as follows:
The matrix [[??].sup.(t)] = [([[??].sup.(t).sub.ij]).sub.1 [less than or equal to] i, j [less than or equal to] n] is obtained from the matrix [A.sup.(t)] by reordering the rows and/or columns t, t + 1, ..., n of [A.sup.(t)] according to the given pivoting strategy and satisfying [[??].sup.(t).sub.tt] [not equal to] 0.
In  we defined a symmetric maximal absolute diagonal dominance (m.a.d.d.) pivoting as a symmetric pivoting which chooses as pivot at the tth step (1 [less than or equal to] t [less than or equal to] n - 1) a row [i.sub.t] ([greater than or equal to] t) satisfying
Although Gauss elimination without pivoting of a positive definite symmetric matrix is stable, it does not guarantee the well conditioning of the triangular factors.

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