New floating-point error estimates

Classical floating-point error estimates are based on a factor gamma _n := n eps / (1-n eps) for eps denoting the relative rounding error unit. These Wilkinson-type estimates are used since 50 years. They imply an intrinsic limitation of n eps < 1. We rework classical error estimates by replacing gamma _n by n eps for any order of computation. Moreover, the restriction on n is removed so that the new estimates are the first one valid for any problem size.

Publikationen

• Ozaki, K.; Ogita, T.; Bünger, F.; Oishi, S.: Accelerating interval matrix multiplication by mixed precision arithmetic.. Nonlinear Theory and its Applications, IEICE, 6(3): S. 364-376, 2015. , https://www.jstage.jst.go.jp/article/nolta/6/3/6_364/_article
• S.M. Rump: Computable backward error bounds for basic algorithms in linear algebra.. Nonlinear Theory and Its Applications, IEICE(6): S. 1–4, 2015. , http://www.ti3.tuhh.de/paper/rump/Ru14c.pdf
• Lange, M.; Rump, S.M.: Error estimates for the summation of real numbers with application to floating-point summation. BIT, 57: S. 927-941, 2017. , DOI: 10.1007/s10543-017-0658-9, http://www.ti3.tuhh.de/paper/rump/LaRu2017.pdf
• Rump, S.M.: Error estimation of floating-point summation and dot product.. BIT Numerical Mathematics, 2012(52(1)): S. 201–220, 2012.
• Lange, M.; Rump, S.M.: Faithfully Rounded Floating-point Computations. submitted for publication, 2017. , http://www.ti3.tuhh.de/paper/rump/LaRu2017b.pdf
• Rump, S.M.: Fast Interval Matrix Multiplication. . Numerical Algorithms, 2012(61(1)): S. 1–34, 2012.
• Rump, S.M.; Bünger, F.; Jeannerod,C.-P.: Improved Error Bounds for Floating-Point Products and Horner's Scheme.. BIT Numerical Mathematics, 2016(56(1)): S. 293-307, 2016. , DOI: 10.1007/s10543-015-0555-z, http://www.ti3.tuhh.de/paper/rump/RuBueJea14.pdf
• Rump, S.M.; Jeannerod, C.-P.: Improved backward error bounds for LU and Cholesky factorizations. . SIAM. J. Matrix Anal. & Appl. (SIMAX), 2014(35(2)): S. 684–698, 2014. , DOI: doi:10.1137/130927231, http://www.ti3.tuhh.de/paper/rump/RuJea13.pdf
• Jeannerod, C.-P.; Rump, S.M.: Improved error bounds for inner products in floating-point artihmetic. . SIAM. J. Matrix Anal. & Appl. (SIMAX), 2013(34(2)): S. 338–344, 2013.
• Rump, S.M.: Interval Arithmetic Over Finitely Many Endpoints.. BIT Numerical Mathematics, 2012(52(4)): S. 1059–1075, 2012.
• Rump, S.M.; Ogita, T.; Oishi, S.: Interval Arithmetic without Changing the Rounding Mode.. submitted for publication, 2013, 2013.
• Rump, S.M.; Ogita, T.; Morikura, Y.; Oishi, S.: Interval arithmetic with fixed rounding mode.. Nonlinear Theory and its Aplications (IEICE), 2016(7(3)): S. 362–373, 2016. , DOI: 10.1587/nolta.7.362
• Jeannerod, C.P.; Rump, S.M.: On relative errors of floating-point operations: Optimal bounds and applications. Mathematics of Computation, 2017. , http://www.ti3.tuhh.de/paper/rump/JeaRu17.pdf
• Lange, M.; Rump, S.M.: Sharp estimates for perturbation errors in summations. Math. of Comp., 2017. , http://www.ti3.tuhh.de/paper/rump/LaRu2017a.pdf