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.
• Rump, S.M.: Computable Error Bounds for Basic Algorithms in Linear Algebra. submitted for publication, 2014., 2014.
• 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.
• 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.
• Rump, S.M.: The componentwise structured and unstructured backward error can be arbitrarily far apart. submitted for publication, 2014, 2014.