- ``Accurate and efficient expression evaluation and linear algebra,’’, by J. Demmel, I. Dumitriu, O. Holtz, P. Koev Acta Numerica, V. 17, May 2008. This paper shows how to exploit the mathematical structure of a problem to get higher accuracy.
- Efficient software for very high precision floating point arithmetic ARPREC MPFR GMP
- Efficient software for high precision and reproducible Basic Linear Algebra Subroutines (BLAS) XBLAS ReproBLAS
- Notes on IEEE Floating Point Arithmetic, by Prof. W. Kahan
- Other notes on arithmetic, error analysis, etc. by Prof. W. Kahan
- Report on arithmetic error that cause the Ariane 5 Rocket Crash
- The IEEE floating point standard was updated in 2008. Look here for a summary.
- The IEEE floating point standard was updated again in 2019. Look here for a summary.
- For a variety of papers on solving linear algebra problems with guaranteed accuracy, by using interval arithmetic, see Siegfried Rump’s web site
- For a paper on using mixed precision arithmetic, doing most work in fast, low precision, and a little in slower, high precision, with the goal of getting the answer to high precision fast, see A Survey of Numerical Methods Utilizing Mixed Precision Arithmetic. There is also a recent Nvidia blog post on this.
- For a paper that shows how one might decrease the factor d = problem size in floating point error bounds, see Stochastic Rounding and its Probabilistic Backward Error Analysis Variable precision floating point
- For a paper on posits, a recently proposed fixed length variable precision arithmetic format, see Beating Floating Point at its Own Game by John Gustavson. I disagree with several claims in this paper. For a paper analyzing posits, see Posits: The good, the bad and the ugly
- For a recorded debate between Gustavson and Kahan on the pros and cons of Gustavson’s earlier proposal for the variable length format called unums, see this youtube video.
- For a (much older) paper showing how to do error analysis with variable precision arithmetic, many versions of which have been proposed before, see On Error Analysis in Arithmetic with Varying Relative Precision
