Using multiple-precision arithmetic

There are many different kinds of problems for which high precision calculation can be useful. The most common situation involves a computation that is numerically unstable, so that using double precision is not sufficient to get the final result as accurately as it is needed. We will use the term “significant digits” to mean the number of equivalent decimal digits of precision, as opposed to the actual number of digits carried when the base is not ten. For most current machines, double precision is the highest accuracy provided in hardware, and this is 53 digits in base 2, or around 16 significant digits. Being able to do a calculation with 30 or 40 significant digits will often overcome the instability and give adequate accuracy for the final result. Even though the term “multiple-precision” may first bring to mind applications such as the computation of billions of digits of π, most actual applications use precisions of a few tens of digits, rather than millions or even hundreds. As an example of such a calculation, let us look at the computation of the Bessel function J1(x) for |x| up to a few hundred. If we actually needed to compute J1(x), there are existing libraries that could be used, but we might have a similar formula that is not one of these standard functions. For small values of x, there is a convergent series,