Unifying Least Squares, Total Least Squares and Data Least Squares
Unifying Least Squares, Total Least Squares and Data Least Squares
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DOI：
10.1007/978-94-017-3552-0_3
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
C. Paige;Z. Strakoš
中科院分区：
文献类型：
--
作者：
C. Paige;Z. Strakoš
The standard approaches to solving overdetermined linear systemsAx≈bconstruct minimal corrections to the vectorband/or the matrixAsuch that the corrected system is compatible. In ordinary least squares (LS) the correction is restricted tob, while in data least squares (DLS) it is restricted toA. In scaled total least squares (Scaled TLS) [15], corrections to bothbandAare allowed, and their relative sizes depend on a parameter γ. Scaled TLS becomes total least squares (TLS) when γ = 1, and in the limit corresponds to LS when γ → 0, and DLS when γ → ∞.In [13] we presented a particularly useful formulation of the Scaled TLS problem, as well as a new assumption that guarantees the existence and uniqueness of meaningful Scaled TLS solutions for all parameters γ > 0, making the whole Scaled TLS theory consistent. This paper refers to results in [13] and is mainly historical, but it also gives some simpler derivations and some new theory. Here it is shown how any linear systemAx≈bcan be reduced to a minimally dimensioned core system satisfying our assumption. The basics of practical algorithms for both the Scaled TLS and DLS problems are indicated for either dense or large sparse systems.
