Improved Backward Error Bounds for LU and Cholesky Factorizations

Assuming standard floating-point arithmetic (in base $\beta$, precision $p$) and barring underflow and overflow, classical rounding error analysis of the LU or Cholesky factorization of an $n\times n$ matrix $A$ provides backward error bounds of the form $|\Delta A| \le \gamma_n |\widehat L| |\wideh...

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Bibliographic Details
Published in:SIAM journal on matrix analysis and applications Vol. 35; no. 2; pp. 684 - 698
Main Authors: Rump, Siegfried M., Jeannerod, Claude-Pierre
Format: Journal Article
Language:English
Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2014
Subjects:
Algorithms
Arithmetic
Cholesky factorization
Computation
Computer Arithmetic
Computer Science
Error analysis
Errors
Floating point arithmetic
Mathematical models
Numerical Analysis
floating-point summation
rounding error analysis
LU factorization
triangular system solving
Cholesky factorization
backward error
ISSN:0895-4798, 1095-7162
Online Access:Get full text
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Summary:Assuming standard floating-point arithmetic (in base $\beta$, precision $p$) and barring underflow and overflow, classical rounding error analysis of the LU or Cholesky factorization of an $n\times n$ matrix $A$ provides backward error bounds of the form $|\Delta A| \le \gamma_n |\widehat L| |\widehat U|$ or $|\Delta A| \le \gamma_{n+1} |\widehat R^T| |\widehat R|$. Here, $\widehat L$, $\widehat U$, and $\widehat R$ denote the computed factors, and $\gamma_n$ is the usual fraction $nu/(1-nu) = nu + {\mathcal O}(u^2)$ with $u$ the unit roundoff. Similarly, when solving an $n\times n$ triangular system $Tx = b$ by substitution, the computed solution $\widehat x$ satisfies $(T+\Delta T)\widehat x = b$ with $|\Delta T| \le \gamma_n |T|$. 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 $\gamma_n$ and $\gamma_{n+1}$ 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 $|\rho-s|$, where $s$ is the exact sum of a floating-point number and $n-1$ real numbers and where $\rho$ is a real number approximating the computed sum $\widehat s$. By instantiating this framework with suitable values of $\rho$, 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]
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ISSN:0895-4798
1095-7162
DOI:10.1137/130927231