Abstract
In a finite dimensional space which has a totally positive basis there exist special bases, called B-bases, such that they generate all totally positive bases by means of totally positive matrices. B-bases are optimal totally positive bases in several senses. From the geometrical point of view, B-bases correspond to the bases with optimal shape preserving properties. From a numerical point of view B-bases are least supported and least conditioned bases among all totally positive bases of space. In order to deal with these questions we introduce a partial order in the set of nonnegative bases: if (v 0,..., v n ) = (u 0,..., u n )H for a nonnegative matrix H, we say that (u 0,..., u n ) ≤ (v 0,..., v n ) . We shall show that B-bases are minimal for this partial order.
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© 1996 Springer Science+Business Media Dordrecht
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Carnicer, J.M., Peña, J.M. (1996). Total positivity and optimal bases. In: Gasca, M., Micchelli, C.A. (eds) Total Positivity and Its Applications. Mathematics and Its Applications, vol 359. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8674-0_8
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DOI: https://doi.org/10.1007/978-94-015-8674-0_8
Publisher Name: Springer, Dordrecht
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