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The least absolute deviations criterion, or the l1 norm, is frequently used for approximation where the data may contain outliers or wild points'. One of the most popular methods for solving the least absolute deviations data fitting problem is the Barrodale and Roberts (BR) algorithm (1973), which is based on linear programming techniques and the use of a modified simplex method. This algorithm is particularly efficient. However, since it is based upon the simplex method it can be susceptible to the accumulation of unrecoverable rounding errors caused by using an inappropriate pivot. In this paper we shall show how we can extend a numerically stable form of the simplex method to the special case of l1 approximation whilst still maintaining the efficiency of the Barrodale and Roberts algorithm. This extension is achieved by using the l1 characterization to rebuild the relevant parts of the simplex tableau at each iteration. The advantage of this approach is demonstrated most effectively when the observation matrix of the approximation problem is sparse, as in the case when using compactly supported basis functions such as B-splines. Under these circumstances the new method is considerably more efficient than the Barrodale and Roberts algorithm as well as being more robust.

1Woodland Halló

By Robert Bloomfield

LibriVox volunteers bring you 20 recordings of The Woodland Halló by Robert Bloomfield.<br> This was the Weekly Poetry project for March 21, 2021. <br> ------<br> Robert Bloomfield was an English labouring-class poet, whose work is appreciated in the context of other self-educated writers, such as Stephen Duck, Mary Collier and John Clare. One of his early duties, when he was sent to London to work as a shoemaker, was to read the papers aloud while the others in the workshop were working, and he became particularly interested in the poetry section of The London Magazine. - Summary by Wikipedia