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.
For positive definite symmetric matrices, it is well known that we can perform symmetric complete pivoting (which increases in O([n.
The following example shows that applying symmetric complete pivoting to a Stieljes matrix does not guarantee that the triangular factors are diagonally dominant.
then the permutation matrix associated to the symmetric complete pivoting is the identity and A = [LDL.
In this section we give an example, discovered in [5, section 2], of a symmetric positive semi-definite matrix for which Cholesky decomposition with complete pivoting does not result in strong RRCh decomposition because condition (2) of the definition in section 3 is violated.
When we perform Cholesky decomposition with complete pivoting on C([theta]) , we will observe that, because of the way this matrix is designed, no pivoting will be necessary.
The practical growth rate is much slower than the theoretical, but it is still super-polynomial, implying that Cholesky decomposition with complete pivoting will not be strong RRCh decomposition because condition (2) in the definition is violated.