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Mathematics of Computation

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A method of virtual displacements for the degenerate discrete $ l\sb{1}$ approximation problem

Authors: W. Fraser and J. M. Bennett
Journal: Math. Comp. 32 (1978), 421-430
MSC: Primary 41A50
MathSciNet review: 0487191
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Abstract: Given the system of equations

$\displaystyle \sum\limits_{j = 1}^n {{a_{ij}}{x_j} = {b_i},\quad i = 1, \ldots ,m,} $

let $ {A_i} = ({a_{i1}}, \ldots ,{a_{in}})$. It is known that if the matrix $ A = ({a_{ij}})$ has rank $ k \leqslant n$, then there is a point X which provides a minimum of

$\displaystyle R(X) = \sum\limits_{i = 1}^m {\vert{r_i}(X)\vert = } \sum\limits_{i = 1}^m {\vert({A_i},X) - {b_i}\vert} $

such that $ {r_i}(X) = 0$ for at least k values of the index i. If $ {r_i}(X) = 0$ for exactly k values of the index i, the point or vertex is called ordinary, while if $ {r_i}(X) = 0$ for more than k values of i, the vertex is termed degenerate.

A necessary and sufficient condition to determine if X minimizes R is valid if X is an ordinary vertex but not if X is degenerate. A degeneracy at X can be removed by applying perturbations to an appropriate number of the $ {b_i}$ so that X becomes an ordinary vertex of a modified problem. By noting that the test uses only values of the $ {A_i}$, it is possible to avoid actual introduction of the perturbations to the $ {b_i}$ with a resulting substantial improvement of the efficiency of the computation.

References [Enhancements On Off] (What's this?)

  • [1] I. BARRODALE & F. D. K. ROBERTS, "Solution of an overdetermined system of equations in the $ {l_1}$ norm," Comm. ACM, v. 17, 1974.

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Article copyright: © Copyright 1978 American Mathematical Society