Multi-Precision Arithmetic

Background

While the standard 64-bit IEEE arithmetic system is adequate for almost all scientific applications, a small but growing set of applications requires more. These include the following:

Highly sensitive structural analysis eigenvalue/eigenvector computations. Engineers who do these computations have noticed that in some recent large examples, it is increasingly difficult to unambiguously distinguish between a "true" zero and an "approximate" zero. Somewhat more precision, say double-double, would make this distinction unequivocal.
Climate modeling applications. Climate modeling calculations are known to be chaotic, with enormous sensitivity to initial conditions and numerical error. These two effects cannot easily be distinguished. Further, the numerical sensitivity means that it is very difficult to reproduce results when calculations are performed on different systems. The best solution to these difficulties is to employ double-double arithmetic, at least for certain critical global summation operations. Details can be found at http://www.nersc.gov/research/SCG/ocean/NRS.
Vortex roll-up simulations. There is no known way to study certain vortex roll-up phenomena, except by the use of high precision arithmetic. For one set of parameters, 32-digit arithmetic (i.e. double-double arithmetic) is required. In a study that further presses the state-of-the-art in this field, 64- to 80-digit arithmetic is required.
Linear algebra computations. A number of recently discovered linear algebra algorithms have the property that while very efficient, they require extra precision in order to return results correct to say the normal 64-bit IEEE precision (15-16 digits). Details can be found in Jim Demmel's talk, "Recent Progress in High Accuracy and High Performance Linear Algebra Algorithms".
Integer relation detection. Given an n-long set of real numbers (typically given by high-precision floating-point values), the "integer relation" problem is to find integers a_i such that a₁ x₁ + a₂ x₂ + ... + a_n x_n = 0. Integer relation programs have been used to discover new formulas of mathematics and physic, including a new formula for pi, and the identification of constants associated with certain Feynmann diagrams in quantum field theory. These calculations require very high precision arithmetic, typically between 100 to 10,000 digit precision. Details can found in the paper "Parallel Integer Relation: Techniques and Applications".

Project Ideas

There are a number of projects along this line that need to be done, and would be appropriate for CS267. These include:

Design and implement a portable double-double arithmetic package in C++ or Java/Titanium. DHB has a Fortran-90 double-double package, but there is no equivalent in C++ or Java/Titanium. Such a package would be designed to exploit the features of IEEE arithmetic that make it possible to obtain full-accuracy results of arithmetic operations. Such a package would be useful for a wide variety of applications.
Design and implement a portable quad-double arithmetic package, preferably in Fortran-90 or C++. One would exploit the features of IEEE arithmetic that make it possible to obtain full-accuracy results of arithmetic operations. This package would be of use in particular for the vortex roll-up research mentioned above.
Design and implement a new portable arbitrary precision package, preferably in Fortran-90 or C++. It would be based on the features of IEEE arithmetic that permit full-accuracy results to be easily obtained. DHB has an arbitrary precision package, but it was designed for a more general floating-point arithmetic system, and is not nearly as efficient as it could be if designed specifically to take advantage of IEEE arithmetic. Details of DHB's arbitrary precision package can be found in the article "A Fortran-90 Based Multiprecision System", at DHB's web site (see above). A full-blown arbitrary precision package may not be possible in the scope of a CS267 project, but some subset (say the basic arithmetic routines) would be appropriate.

Final project page, main CS267 page, and the TA's CS267 page

E. Jason Riedy
ejr@cs.berkeley.edu