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- W1888890775 abstract "Perturbation theory is developed for the Cholesky decomposition of an $n timesn$ symmetric positive semidefinite matrix $A$ of rank~$r$. The matrix$W=All^{-1}A{12}$ is found to play a key role in the perturbation bounds,where $All$ and $A{12}$ are $r times r$ and $r times (n-r)$ submatrices of$A$ respectively.A backward error analysis is given; it shows that the computed Choleskyfactors are the exact ones of a matrix whose distance from $A$ is bounded by$4r(r+1)bigl(norm{W}+1bigr)^2unorm{A}+O(u^2)$, where $u$ is the unitroundoff. Forthe complete pivoting strategy it is shown that $norm{W}^2 le {1 over3}(n-r)(4^r- 1)$, and empirical evidence that $norm{W}$ is usually small ispresented. The overall conclusion is that the Cholesky algorithm with completepivoting is stable for semi-definite matrices.Similar perturbation results are derived for the QR decomposition with columnpivoting and for the LU decomposition with complete pivoting. The results givenew insight into the reliability of these decompositions in rank estimation." @default.
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- W1888890775 date "1990-01-01" @default.
- W1888890775 modified "2023-09-26" @default.
- W1888890775 title "Analysis of the Cholesky Decomposition of a Semi-definite Matrix" @default.
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