In lecture 10, we discussed ordinary floating-point arithmetic, conducting an first-order error analysis of the Horner scheme and introducing compensated summation. Extensions of ordinary floating-point arithmetic are multiprecision and interval arithmetic.

To supplement the treatment of the textbook, we point to several excellent sources, mostly available online:

• David Goldberg: "What every computer scientist should know about floating-point arithmetic". We downloaded an edited reprint of ACM Computing Surveys 23(1):5-48, 1991. The title says it all... Look for Kahan's summation formula
• Gregory Tarsy and Neil Toda: Floating-Point Computing: A Comedy of Errors?". 20 January 2004. source for download A practical study of compensated summation.
• J.M. Peņa and T. Sauer: "On the multivariate Horner Scheme II: Running error analysis". Computing 65: 313-322, 2000. An efficient and accurate multivariate Horner scheme

• Chapter 4 of "Algorithms for Computer Algebra" by Keith O. Geddes, Stephen R. Czapor, and George Labahn. Kluwer Academic Publishers, 1992 (4th printing 1995).
• Chapters 8 and 9 of "Modern Computer Algebra" by Joachim von zur Gathen and Jürgen Gerhard. Cambridge University Press, 1999.

• P. Van Hentenryck, D. McAllester, and D. Kapur: "Solving Polynomial Systems Using a Branch and Prune Approach". SIAM Journal on Numerical Analysis 34(2): 797-827, 1997. A successful generalization of the bisection method for polynomial systems.
• Markus Grimmer, Knut Petras, and Nathalie Revol: "Multiple Precision Interval Packages: Comparing Different Approaches". In "Numerical Software with Result Verification", LNCS 2991, edited by R. Alt et al., pages 64-90, 2004. A combination of multiprecision and interval arithmetic.