The Higher Arithmetic : Extended Bibiography
(accompanies "Computing Science: The Higher Arithmetic,"
by Brian Hayes, American Scientist 97:364-368)

[Aaronson undated]
Aaronson, Scott. Undated web page; accessed 2009-07-05. Who can name the bigger number? http://www.scottaaronson.com/writings/bignumbers.html

[Arnold, Bailey, Cowles and Cupal 1989]
Arnold, M. G., T. A. Bailey, J. R. Cowles and J. J. Cupal. 1989. Redundant logarithmic number systems. In Proceedings of the Ninth Symposium on Computer Arithmetic, pp. 144–151. Washington, D.C.: IEEE Computer Society Press.

[Arnold 2005]
Arnold, Mark G. 2005. The residue logarithmic number system: Theory and implementation. In Proceedings of the 17th Symposium on Computer Arithmetic, pp. 196–205. Los Alamitos, Calif.: IEEE Computer Society Press.

[Azmi and Lombardi 1989]
Azmi, Aqil M., and Fabrizio Lombardi. 1989. On a tapered floating point system. In Proceedings of the Ninth Symposium on Computer Arithmetic, pp. 2–9. Washington, D.C.: IEEE Computer Society Press.

[Bailey 2005]
Bailey, David H. 2005. High-precision floating-point arithmetic in scientific computation. Computing in Science and Engineering 7(3):54–61.

[Benouamer, Jaillon, Michelucci and Moreau 1993]
Benouamer, M. O., P. Jaillon, D. Michelucci and J.-M. Moreau. 1993. A lazy exact arithmetic. In Proceedings of the 11th Symposium on Computer Arithmetic, pp. 242–249. Los Alamitos, Calif.: IEEE Computer Society Press.

[Brent 1973]
Brent, Richard P. 1973. On the precision attainable with various floating-point number systems. IEEE Transactions on Computers C-22:601–607.

[Brown 1981]
Brown, W. S. 1981. A simple but realistic model of floating-point computations. ACM Transactions on Mathematical Software 7:445–480.

[Chatelin and Frayssé 1991]
Chatelin, F., and V. Frayssé. 1991. Analysis of arithmetic algorithms: A statistical study. In Proceedings of the 10th Symposium on Computer Arithmetic, pp. 10–16. Los Alamitos, Calif.: IEEE Computer Society Press.

[Clenshaw and Olver 1984]
Clenshaw, C. W., and F. W. J. Olver. 1984. Beyond floating point. Journal of the Association for Computing Machinery 31:319–328.

[Clenshaw and Olver 1987]
Clenshaw, C. W., and F. W. J. Olver. 1987. Level-index arithmetic operations. SIAM Journal on Numerical Analysis 24(2):470–485.

[Clenshaw and Turner 1988]
Clenshaw, C. W., and P. R. Turner. 1988. The symmetric level-index system. IMA Journal of Numerical Analysis 8(4):517–526.

[Clenshaw, Olver and Turner 1989]
Clenshaw, C. W., F. W. J. Olver and P. R. Turner. 1989. Level-index arithmetic: An introductory survey. In Numerical Analysis and Parallel Processing: Lectures Given at Lancaster Numerical Analysis Summer School, 1987, pp. 95–168. Lecture Notes in Mathematics Vol. 1397. Berlin: Springer-Verlag.

[Coleman and Chester 1999]
Coleman, J. N., and E. I. Chester. 1999. A 32 bit logarithmic arithmetic unit and its performance compared to floating-point. In Proceedings of the 14th Symposium on Computer Arithmetic, pp. 142–151. DOI 10.1109/ARITH.1999.762839. Los Alamitos, Calif.: IEEE Computer Society Press.

[Conway and Guy 1996]
Conway, John H., and Richard K. Guy. 1996. The Book of Numbers. New York: Copernicus. (See pp. 61–62).

[Davis 1961]
Davis, Philip J. 1961. The Lore of Large Numbers. Washington: The Mathematical Association of America.

[Demmel 1987]
Demmel, James W. 1987. On error analysis in arithmetic with varying relative precision. In Proceedings of the Eighth Symposium on Computer Arithmetic, pp. 153–157. Washington, D.C.: IEEE Computer Society Press.

[Exoo 2003]
Exoo, Geoffrey. 2003. A Euclidean Ramsey problem. Discrete and Computational Geometry 29:223–227.

[Feldstein and Turner 2006]
Feldstein, Alan, and Peter R. Turner. 2006. Gradual and tapered overflow and underflow: A functional differential equation and its approximation. Applied Numerical Mathematics 56(3):517–532.

[Forsythe 1970]
Forsythe, George E. 1970. Pitfalls in computation, or why a math book isn’t enough. American Mathematical Monthly 77:931–956.

[Fraenkel 1983]
Fraenkel, Aviezri S. 1983. Systems of numeration. In Proceedings of the Sixth Symposium on Computer Arithmetic, pp. 37–42. Washington, D.C.: IEEE Computer Society Press.

[Gabrielian 1975]
Gabrielian, Armen. 1975. Formal systems of numerals. In Proceedings of the Third Symposium on Computer Arithmetic, pp. 76–81. Washington, D.C.: IEEE Computer Society Press.

[Gardner 1977]
Gardner, Martin. 1977. Mathematical games: In which joining sets of points by lines leads into diverse (and diverting) paths. Scientific American 237:18–28.

[Goldberg 1991]
Goldberg, David. 1991. What every computer scientist should know about floating-point arithmetic. ACM Computing Surveys 23(1):5–48.

[Graham and Rothschild 1971]
Graham, R. L., and B. L. Rothschild. 1971. Ramsey’s theorem for n-parameter sets. Transactions of the American Mathematical Society 159:257–292. doi:10.2307/1996010.

[Hamada 1987]
Hamada, Hozumi. 1987. A new real number representation and its operation. In Proceedings of the Eighth Symposium on Computer Arithmetic, pp. 153–157. Washington, D.C.: IEEE Computer Society Press.

[Hamming 1970]
Hamming, R. W. 1970. On the distribution of numbers. The Bell System Technical Journal 49(8):1609–1625.

[Hanrot, Lefèvre, Stehlé and Zimmermann 2007]
Hanrot, Guillaume, Vincent Lefèvre, Damien Stehlé and Paul Zimmermann. 2007. Worst cases of a periodic function for large arguments. In Proceedings of the 18th Symposium on Computer Arithmetic, pp. 133–140. Los Alamitos, Calif.: IEEE Computer Society Press.

[Hauser 1996]
Hauser, John R. 1996. Handling floating-point exceptions in numeric programs. ACM Transactions on Programming Languages and Systems 18(2):139–174.

[Holmes 1997]
Holmes, W. Neville. 1997. Composite arithmetic: proposal for a new standard. Computer 30(3):65–73.

[Hull and Cohen 1987]
Hull, T. E., and M. S. Cohen. 1987. Toward an ideal computer arithmetic. In Proceedings of the Eighth Symposium on Computer Arithmetic, pp. 131–138. Washington, D.C.: IEEE Computer Society Press.

[IEEE 2008]
IEEE Computer Society. 2008. IEEE Std-754-2008 Standard for Floating Point Arithmetic.

[Irwin and Stefanelli 1987]
Irwin, Mary Jane, and Renato Stefanelli (editors). 1987. Proceedings of the 8th Symposium on Computer Arithmetic. Washington, D.C.: IEEE Computer Society Press.

[Jebelean 1993]
Jebelean, T. 1993. Comparing several GCD algorithms. In Proceedings of the 11th Symposium on Computer Arithmetic, pp. 180–185. Los Alamitos, Calif.: IEEE Computer Society Press.

[Kahan and Zuras 2005]
Kahan, William, and Dan Zuras. 2005. An open question to developers of numerical software. Computer 38(5):91–94.

[Knuth 1976a]
Knuth, Donald E. 1976. Mathematics and computer science: Coping with finiteness. Science. 194:1235–1242.

[Knuth 1992]
Knuth, Donald E. 1992. Two notes on notation. American Mathematical Monthly 99(5):403–422.

[Kulisch 2008]
Kulisch, Ulrich. 2008. Computer Arithmetic and Validity: Theory, Implementation, and Applications. New York: Walter de Gruyter.

[Lester 2001]
Lester, David. 2001. Effective continued fractions. In Proceedings of the 15th Symposium on Computer Arithmetic, pp. 163–170. DOI 10.1109/ARITH.2001.930116. Los Alamitos, Calif.: IEEE Computer Society Press.

[Li et al. 2002]
Li, Xiaoye S., James W. Demmel, David H. Bailey, Greg Henry, Yozo Hida, Jimmy Iskandar, William Kahan, Suh Y. Kang, Anil Kapur, Michael C. Martin, Brandon J. Thompson, Teresa Tung and Daniel J. Yoo. 2002. Design, implementation and testing of extended and mixed precision BLAS. ACM Transactions on Mathematical Software 28:152–205.

[Lozier 1993]
Lozier, Daniel W. 1993. An underflow-induced graphics failure solved by SLI arithmetic. In Proceedings of the 11th Symposium on Computer Arithmetic, pp. 10–17. Los Alamitos, Calif.: IEEE Computer Society Press.

[Lozier and Olver 1990]
Lozier, D. W., and F. W. J. Olver. 1990. Closure and precision in level-index arithmetic. SIAM Journal on Numerical Analysis 27:1295–1304.

[Matsui and Iri 1981]
Matsui, Shourichi, and Masao Iri. 1981. An overflow/underflow-free floating-point representation of numbers. Journal of Information Processing 4:123–133.

[Matula and Kornerup 1983]
Matula, David W., and Peter Kornerup. 1983. An order preserving finite binary encoding of the rationals. In Proceedings of the Sixth Symposium on Computer Arithmetic, pp. 201–209. Washington, D.C.: IEEE Computer Society Press.

[Menninger 1958]
Menninger, Karl. 1958, 1969. Number Words and Number Symbols: A Cultural History of Numbers. Translated by Paul Broneer. Cambridge, Mass.: The MIT Press.

[Monniaux 2008]
Monniaux, David. 2008. The pitfalls of verifying floating-point computations. http://hal.archives-ouvertes.fr/docs/00/28/14/29/PDF/floating-point-article.pdf

[Morris 1971]
Morris, Robert. 1971. Tapered floating point: A new floating-point representation. IEEE Transactions on Computers C-20:1578–1579.

[Muller, Tisserand and Scherbyna 1995]
Muller, Jean-Michel, Arnaud Tisserand and Alexandre Scherbyna. 1995. Semi-logarithmic number systems. In Proceedings of the 12th Symposium on Computer Arithmetic, pp. 201–207. DOI 10.1109/ARITH.1995.465358. Los Alamitos, Calif.: IEEE Computer Society Press.

[Olver 1978]
Olver, F. W. J. 1978. A new approach to error arithmetic. SIAM Journal on Numerical Analysis 15:368–393.

[Olver 1987]
Olver, F. W. J. 1987. A closed computer arithmetic. In Proceedings of the Eighth Symposium on Computer Arithmetic, pp. 139–143. Washington, D.C.: IEEE Computer Society Press.

[Olver and Turner 1987]
Olver, F. W. J., and P. R. Turner. 1987. Implementation of level-index arithmetic using partial table look-up. In Proceedings of the Eighth Symposium on Computer Arithmetic, pp. 144–147. Washington, D.C.: IEEE Computer Society Press.

[Paliouras and Stouraitis 2001]
Paliouras, V., and T. Stouraitis. 2001. Low-power properties of the logarithmic number system. In Proceedings of the 15th Symposium on Computer Arithmetic, pp. 229–236. DOI 10.1109/ARITH.2001.930124. Los Alamitos, Calif.: IEEE Computer Society Press.

[Prömel 2002]
Prömel, Hans Jürgen. 2002. Large numbers, Knuth's arrow notation, and Ramsey theory. Synthese 133:87–105.

[Rao 1981]
Rao, T. R. N. 1981. Arithmetic of finite fields. In Proceedings of the Fifth Symposium on Computer Arithmetic, pp. 2–5. Washington, D.C.: IEEE Computer Society Press.

[Richardson 1993]
Richardson, Stephen E. 1993. Exploiting trivial and redundant computation. In Proceedings of the 11th Symposium on Computer Arithmetic, pp. 220–227. Los Alamitos, Calif.: IEEE Computer Society Press.

[Robertson and Trivedi 1973]
Robertson, James E., and Kishor S. Trivedi. 1973. The status of investigations into computer hardware design based on the use of continued fractions. IEEE Transactions on Computers C-22:555–560.

[Rotman 2000]
Rotman, Brian. 2000. Mathematics as Sign: Writing, Imagining, Counting. Stanford, Calif.: Stanford University Press.

[Rump 1985]
Rump, Siegfried M. 1985. Higher order computer arithmetic. In Proceedings of the Seventh Symposium on Computer Arithmetic, pp. 302–308. Washington, D.C.: IEEE Computer Society Press.

[Schwarz 1989]
Schwarz, Jerry. 1989. Implementing infinite precision arithmetic. In Proceedings of the Ninth Symposium on Computer Arithmetic, pp. 10–17. Washington, D.C.: IEEE Computer Society Press.

[Schwartzlander, Irwin and Jullien 1993]
Schwatrzlander, Earl, Jr., Mary Jane Irwin and Graham Jullien (editors). 1993. Proceedings of the 11th Symposium on Computer Arithmetic. Los Alamitos, Calif.: IEEE Computer Society Press.

[Scott 1985]
Scott, Norman R. 1985. Computer Number Systems and Arithmetic. Englewood Cliffs, N.J.: Prentice-Hall.

[Shen and Turner 2006a]
Shen, Xunyang, and Peter R. Turner. 2006. Taylor approximation for symmetric level-index arithmetic processing. IMA Journal of Numerical Analysis 26:584–603.

[Shen and Turner 2006b]
Shen, Xunyang, and Peter R. Turner. 2006. A hybrid number representation scheme based on symmetric level-index arithmetic. In Proceedings of the 2006 International Conference on Scientific Computing, CSC 2006, pp. 118–123. Las Vegas, Nev.: CSREA Press. (PDF available at: http://ww1.ucmss.com/books/LFS/CSREA2006/CSC4292.pdf)

[Shen and Turner 2006c]
Shen, Xunyang, and Peter R. Turner. 2006. Towards a fast and reliable software implementation of SLI-FLP hybrid computer arithmetic. In Proceedings of the 6th WSEAS International Conference on Systems Theory and Scientific Computation, pp.101–108. Stevens Point, Wis.: World Scientific and Engineering Academy and Society.

[Shen 2007]
Shen, Xunyang. 2007. C++ implementation of symmetric level-index arithmetic. http://code.google.com/p/sli-c-library/

[Shen and Turner 2007]
Shen, Xunyang, and Peter R. Turner. 2007. The application of SLI arithmetic on SSQR method for the Hermitian eigenvalue problem. In Proceedings of the 2007 International Conference on Scientific Computing, CSC 2007, pp. 165–170. Las Vegas, Nev.: CSREA Press.

[Shewchuk 1997]
Shewchuk, Jonathan Richard. 1997. Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete and Computational Geometry 18:305–363. Preprint available at http://www.cs.cmu.edu/afs/cs/project/quake/public/papers/robust-arithmetic.ps

[Sterbenz 1975]
Sterbenz, Pat H. 1975. Understandable arithmetic. In Proceedings of the Third Symposium on Computer Arithmetic, pp. 33–35. Washington, D.C.: IEEE Computer Society Press.

[Swartzlander and Alexopoulos 1975]
Swartzlander, Earl E., Jr., and Aristides G. Alexopoulos. 1975. The sign/logarithm number system. IEEE Transactions on Computers C-24:1238–1243.

[Turner 1989]
Turner, P. R. 1989. A software implementation of SLI arithmetic. In Proceedings of the Ninth Symposium on Computer Arithmetic, pp. 18–24. Washington, D.C.: IEEE Computer Society Press.

[Turner 1991]
Turner, Peter R. 1991. Implementation and analysis of extended SLI operations. In Proceedings of the 10th Symposium on Computer Arithmetic, pp. 118–126. Los Alamitos, Calif.: IEEE Computer Society Press.

[Turner 1992]
Turner, Peter R. 1992. A history of the Lords of Number-Crunching. American Mathematical Monthly 99:907–916.

[Turner 1993]
Turner, P. R. 1993. Complex SLI arithmetic: Representation, algorithms and analysis. In Proceedings of the 11th Symposium on Computer Arithmetic, pp. 18–25. Los Alamitos, Calif.: IEEE Computer Society Press.

[Van Wyk 1995]
Van Wyk, Christopher J. 1995. Missing real numbers. American Mathematical Monthly 102:260–265.

[Weaver 1948]
Weaver, Warren. 1948. Size. Atlantic Monthly 182(September):88–90.

[Vardi 1998]
Vardi, Ilan. 1998. Archimedes' cattle problem. The American Mathematical Monthly 105:305–319.

[Vardi undated]
Vardi, Ilan. Undated manuscript. Archimedes, the sand reckoner. http://www.lix.polytechnique.fr/Labo/Ilan.Vardi/sand_reckoner.ps.

[Yates 1997]
Yates, David M. 1997. Turing's Legacy: A History of Computing at the National Physical Laboratory 1945–1995. London: National Museum of Science and Industry.

[Yokoo 1992]
Yokoo, Hidetoshi. 1992. Overflow/underflow-free floating-point number representations with self-delimiting variable-length exponent field. IEEE Transactions on Computers 41(8):1033–1039.

[Zuras 1993]
Zuras, Dan. 1993. On squaring and multiplying large integers. In Proceedings of the 11th Symposium on Computer Arithmetic, pp. 260–271. Los Alamitos, Calif.: IEEE Computer Society Press.