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

• Rump, S.M.; Bünger, F; Jeannerod,C.-P.: Improved Error Bounds for Floating-Point Products and Horner's Scheme.. to appear in BIT, 2014. , http://www.ti3.tu-harburg.de/paper/rump/RuBueJea14.pdf
• Rump, S.M.: Computable Error Bounds for Basic Algorithms in Linear Algebra.. submitted for publication, 2014., 2014. , http://www.ti3.tuhh.de/paper/rump/Ru14c.pdf
• 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
• Rump, S.M.: Error estimation of floating-point summation and dot product.. BIT Numerical Mathematics, 2012(52(1)): S. 201–220, 2012.
• Rump, S.M.: Fast Interval Matrix Multiplication. . Numerical Algorithms, 2012(61(1)): S. 1–34, 2012.
• 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.
• Jeannerod, C.-P.; Rump, S.M.: On relative errors of floating-point operations: optimal bounds and applications.. Preprint, 2014, 2014.
• Rump, S.M.; Lange, M.: On the Definition of Unit Roundoff.. submitted for publication, 2014. , http://www.ti3.tuhh.de/paper/rump/RuLa14.pdf
• Ozaki, K; Bünger, F.; Ogita, T.; Oishi, S.; Rump, S.M.: Simple floating-point filters for the two-dimensional orientation problem.. BIT Numerical Mathematics, 2015. , DOI: 10.1007/s10543-015-0574-9, http://www.ti3.tuhh.de/paper/rump/OzBueOgOiRu15.pdf
• Rump, S.M.: The componentwise structured and unstructured backward error can be arbitrarily far apart.. submitted for publication, 2014, 2014. , http://www.ti3.tuhh.de/paper/rump/Ru14b.pdf
• 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