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• W2077855410 abstract "We consider here the linear least squares problem $min_{y in mathbb{R}^n}|Ay-b|_2$, where $b in mathbb{R}^m$ and $A in mathbb{R}^{mtimes n}$ is a matrix of full column rank n, and we denote x its solution. We assume that both A and b can be perturbed and that these perturbations are measured using the Frobenius or the spectral norm for A and the Euclidean norm for b. In this paper, we are concerned with the condition number of a linear function of x ($L^Tx$, where $L in mathbb{R}^{ntimes k}$) for which we provide a sharp estimate that lies within a factor $sqrt{3}$ of the true condition number. Provided the triangular R factor of A from $A^TA=R^TR$ is available, this estimate can be computed in $2kn^2$ flops. We also propose a statistical method that estimates the partial condition number by using the exact condition numbers in random orthogonal directions. If R is available, this statistical approach enables us to obtain a condition estimate at a lower computational cost. In the case of the Frobenius norm, we derive a closed formula for the partial condition number that is based on the singular values and the right singular vectors of the matrix A." @default.
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• W2077855410 date "2007-01-01" @default.
• W2077855410 modified "2023-09-23" @default.
• W2077855410 title "A Partial Condition Number for Linear Least Squares Problems" @default.
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• W2077855410 doi "https://doi.org/10.1137/050643088" @default.
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