Improved Backward Error Bounds for LU and Cholesky Factorizations - Rump, Siegfried M.; Jeannerod, Claude-Pierre (SIAM journal on matrix analysis and applications, 2014)
Assuming standard floating-point arithmetic (in base $β$, precision $p$) and barring underflow and overflow, classical rounding error analysis of the LU or Cholesky factorization of an $n× n$ matrix $A$ provides backward error bounds of the form $|Δ A| \le γ_n |\widehat L| |\widehat U|$ or $|Δ A| \le γ_{n+1} |\widehat R^T| |\widehat R|$. Here, $\widehat L$, $\widehat U$, and $\widehat R$ denote the computed factors, and $γ_n$ is the usual fraction $nu/(1-nu) = nu + {\mathcal O}(u^2)$ with $u$ the unit roundoff. Similarly, when solving an $n× n$ triangular system $Tx = b$ by substitution, the computed solution $\widehat x$ satisfies $(T+Δ T)\widehat x = b$ with $|Δ T| \le γ_n |T|$. All these error bounds contain quadratic terms in $u$ and limit $n$ to satisfy either $nu<1$ or $(n+1)u < 1$. We show in this paper that the constants $γ_n$ and $γ_{n+1}$ can be replaced by $nu$ and $(n+1)u$, respectively, and that the restrictions on $n$ can be removed. To get these new bounds the main ingredient is a general framework for bounding expressions of the form $|ρ-s|$, where $s$ is the exact sum of a floating-point number and $n-1$ real numbers and where $ρ$ is a real number approximating the computed sum $\widehat s$. By instantiating this framework with suitable values of $ρ$, we obtain improved versions of the well-known Lemma 8.4 from [N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002] (used for analyzing triangular system solving and LU factorization) and of its Cholesky variant. All our results hold for rounding to nearest with any tie-breaking strategy and whatever the order of summation. [PUBLICATION ABSTRACT]