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Improved componentwise verified error bounds for least squares problems and underdetermined linear systems

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Abstract

Recently Miyajima presented algorithms to compute componentwise verified error bounds for the solution of full-rank least squares problems and underdetermined linear systems. In this paper we derive simpler and improved componentwise error bounds which are based on equalities for the error of a given approximate solution. Equalities are not improvable, and the expressions are formulated in a way that direct evaluation yields componentwise and rigorous estimates of good quality. The computed bounds are correct in a mathematical sense covering all sources of errors, in particular rounding errors. Numerical results show a gain in accuracy compared to previous results.

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References

  1. Arioli, M., Duff, I.S., de Rijk, P.P.M.: On the augmented system approach to sparse least-squares problems. Numer. Math. 55(6), 667–687 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Björck, Å.: Iterative refinement of linear least squares solutions I. BIT 7, 257–278 (1967)

    Article  MATH  Google Scholar 

  3. Davis, T.A., Hu, Y.: The University of Florida Sparse Matrix Collection. ACM Trans. Math. Softw. 38(1), 1:1–1:25 (2011)

    MathSciNet  Google Scholar 

  4. Demmel, J.B., Hida, Y., Kahan, W., Li, X.S., Mukherjee, S., Riedy, E.J.: Error, bounds from extra precise iterative refinement. ACM Trans. Math. Softw. (TOMS) 32(2), 325–351 (2006)

    Article  MathSciNet  Google Scholar 

  5. Frommer, A.: Proving conjectures by use of interval arithmetic. In: Kulisch, U., et al. (eds.) Perspectives on Enclosure Methods. SCAN 2000, GAMM-IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Univ. Karlsruhe, Germany, September 19-22, 2000. Springer, Wien (2001)

    Google Scholar 

  6. Golub, G.H., Van Loan, C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)

    Google Scholar 

  7. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM Publications, Philadelphia (2002)

    Book  Google Scholar 

  8. IEEE Standard for Floating-Point Arithmetic IEEE Std 754-2008. In: IEEE Std 754-2008, pp. 1–58, 29 Aug 2008

  9. MATLAB: Users Guide, Version 7, the MathWorks Inc. (2004)

  10. Miyajima, S.: Fast enclosure for solutions in underdetermined systems. J. Comput. Appl. Math. (JCAM) 234, 3436–3444 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Miyajima, S.: Componentwise enclosure for solutions in least squares problems and underdetermined linear systems. SCAN conference Novosibirsk (2012)

  12. Rump, S.M.: Kleine Fehlerschranken bei Matrixproblemen. PhD thesis. Universität Karlsruhe (1980)

  13. Rump, S.M.: INTLAB - INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp 77–104. Kluwer Academic Publishers, Dordrecht (1999)

    Chapter  Google Scholar 

  14. Rump, S.M.: Verified bounds for least squares problems and underdetermined linear systems. SIAM J. Matrix Anal. Appl. (SIMAX) 33(1), 130–148 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. Sahinidis, N.V., Tawaralani, M.: A polyhedral branch-and-cut approach to global optimization. Math. Program. B103, 225–249 (2005)

    Google Scholar 

  16. Stetter, H.J.: Sequential defect correction in high-accuracy floating-point arithmetics. Num. Anal. 1066, 186-202 (1984). Proceedings, Dundee 1983

    Google Scholar 

  17. Tucker, W.: The Lorenz attractor exists. C. R. Acad. Sci., Paris, Sér. I, Math. 328(12), 1197–1202 (1999)

    Article  MATH  Google Scholar 

  18. Wedin, P.Å.: Perturbation theory for pseudo-inverses. BIT 13, 217–232 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wright, S.J.: A collection of problems for which Gaussian elimination with partial pivoting is unstable. SIAM J. Sci. Comput. (SISC) 14(1), 231–238 (1993)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Reliable Computing, Hamburg University of Technology, Schwarzenbergstraße 95, Hamburg, 21071, Germany

    Siegfried M. Rump

  2. Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku-ku, Tokyo, 169–8555, Japan

    Siegfried M. Rump

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  1. Siegfried M. Rump
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Correspondence to Siegfried M. Rump.

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Rump, S.M. Improved componentwise verified error bounds for least squares problems and underdetermined linear systems. Numer Algor 66, 309–322 (2014). https://doi.org/10.1007/s11075-013-9735-6

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  • DOI: https://doi.org/10.1007/s11075-013-9735-6

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