Dear colleagues,In recent mails I have noticed a tendency to separate interval arithmetic from the IEEE P754 standard altogether or to put it in an appendix. I think this would be a big mistake. Interval arithmetic is an integral and fundamental tool of modern scientific computing. As such it must be an integral part of IEEE P754.
On Sept. 13 the IFIP Working Group 2.5 on Numerical Software sent a letter to Bob Davis, Dan Zuras and other members of IEEE 754R. About a month later the IFIP WG 2.5 sent a document showing how the requirements of the letter could be incorporated into the proposed standard IEEE P754 (see the attachments).
Very capable commercial (NAG, IMSL, Visual Numerics, …) and non-commercial software developers are among the members of the IFIP WG 2.5. They would like to develop efficient and reliable software based on the requirements of the IFIP WG 2.5 letter straight away. For this purpose well defined interval operations are needed. These should be a mandatory part of IEEE P754 and not specified as a possibly optional secondary standard.
At a recent meeting at Dagstuhl I became aware that a number of colleagues are still reluctant to include division by an interval which contains zero in a standard. To avoid further delays, therefore, I suggest leaving this operation aside and just specifying the four basic operations for closed and bounded intervals (with zero not contained in the divisor) for the double precision format in the standard. These minimum requirements should be generally acceptable to the interval and scientific computing community. For details see the Kirchner-Kulisch paper.
In addition to this the IFIP WG 2.5 letter requires fast support of multiple precision interval arithmetic. An exact scalar product or complete arithmetic is the basic tool to achieve this (see the letter). Fast multiple precision interval arithmetic opens a large area of new applications. The newly defined functions sum and dot (see P754, Draft 1.6.0) are not suited to replace the exact scalar product. The latter, however, can be used for fast execution of sum and dot. Multiple precision arithmetic benefits fully from a high speed exact scalar product. The exponent range of the double precision format is not a limitation for multiple precision arithmetics. An exponent part can easily be added as a scaling factor (made available in C-XSC by Blomquist, Hofschuster, and Kraemer).
Nearly everything that is needed for fast interval arithmetic is available on existing Intel and AMD processors (see the Kirchner-Kulisch paper). I am sure that the remaining properties would immediately be made available if interval arithmetic were to be specified and required within P754.
Please consider this mail as an early contribution to the newly established Study Group for Interval Arithmetic.
With best regards Ulrich Kulisch -- ************************************************************************************* Ulrich Kulisch Institut für Angewandte Mathematik Universität Karlsruhe 76128 Karlsruhe, Germany Tel.: (+49) 721 608 2680 oder (+49) 721 608 4202 *************************************************************************************
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