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Backward error analysis for totally positive linear systems

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Summary

Gauss elimination applied to ann×n matrixA in floating point arithmetic produces (if successful) a factorization\(\hat L\hat U\) which differs fromA by no more than\(\gamma |\hat L|{\text{ }}|\hat U|\), for some γ of ordern times the unit roundoff. IfA is totally positive, then both computed factors\(\hat L\) and\(\hat U\) are nonnegative for sufficiently small unit roundoff and one obtains pleasantly small bounds for the perturbation inA which would account for the rounding errors committed in solvingAx=b forx by Gauss eliminationwithout pivoting. It follows that the banded linear system for the B-spline coefficients of an interpolating spline function can be solved safely by Gauss elimination without pivoting.

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Authors and Affiliations

  1. Mathematics Research Center, University of Wisconsin, 610 Walnut Street, 53706, Madison, WI, USA

    Carl de Boor & Allan Pinkus

Authors
  1. Carl de Boor
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  2. Allan Pinkus
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Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the National Science Foundation under Grant No. MPS72-00381 A01.

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de Boor, C., Pinkus, A. Backward error analysis for totally positive linear systems. Numer. Math. 27, 485–490 (1976). https://doi.org/10.1007/BF01399609

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