An alternative to interval arithmetic in order to gain confidence in computed results in face of computational uncertainty is high-precision calculation. Consider the computer solution of a linear system of equations where the condition number (see [45] for details) is on the order of 10 ¹⁴. With IEEE double precision arithmetic, one may not have any confidence in the computed result. However, if the underlying floating point arithmetic carries an equivalent of, for instance, 30 decimal digits, then the computer solution of the same linear system would be as trustworthy as that of a well-conditioned linear system carried out in IEEE double precision arithmetic.

The idea of arbitrarily high precision arithmetic is straightforward. As an alternative to native hardware supported floating-point object, one uses the flexibility of software to define and implement computation on an arithmetic object whose precision can be made as high as practically possible (limited basically by memory and/or computational time). For example, one can use an array of 64-bit integers to represent the sign, exponent, and a large number of significant bits, of a numerical value. Depending on the specific representation chosen, suitable algorithms and implementations can be used to support the basic arithmetic operations as well as elementary transcendental functions. Existing software packages that define and implement this arithmetic can be found in [46, 47, 48, 49]. Although the representation here bear some resemblances to that of quad arithmetic discussed in Chapter 9, the algorithms are quite different due to the dynamic...