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Numerical Techniques

5. Numerical Techniques It is not easy to compute numerical values of any of these Euler sums to high precision. Straightforward evaluation using the defining formulas, to some upper limit feasible on present-day computers, yields only about eight digits accuracy. Because integer relation algorithms require much higher precision to obtain reliable results, more advanced techniques must be employed.

Our approach to computing numerical values of these sums involves the compound application of the Euler-Maclaurin summation formula (see [2, p. 289,]), which can be stated as follows. Suppose

has at least 2p + 2 continuous derivatives on

. Let D be the differentiation operator, let

denote the k-th Bernoulli number, and let

denote the k-th Bernoulli polynomial. Then

where the remainder

is given [2, p. 289,] by

We will briefly present a method for computing

. See [4] for more details. Let

and

. By the Euler-Maclaurin summation formula,

Since

for all k and for

(see [1, p. 805,]), it follows that the remainder

has a well-defined limit

as k approaches infinity. Now since Euler's constant

, it follows that

We will use

to denote this particular approximation (i.e., (3) without the error term). Now consider the sum

Let c be a large integer, and let . Applying the Euler-Maclaurin summation formula (1) again, we can write

This formula suggests the following computational scheme. First, explicitly evaluate the sum

for

, using a numeric working precision of 150 digits. Secondly, perform the symbolic integration and differentiation steps indicated in formula (4). Finally, evaluate the resulting expression, again using a working precision of 150 digits. The final result should be equal to

to approximately 135 significant digits.

The difficulty and cost of performing the symbolic integration and differentiation operations indicated in (4) can be greatly reduced by approximating

as follows: first, expand

, the numerator of

, into a sum of individual terms; next, write

; next, expand

using the binomial theorem to 18 terms; next, multiply together the resulting numerator and denominator expressions; finally, omit all terms whose exponent of

is greater than 18. The result is a linear sum of terms of the form

for modest-sized integers p and q.

We have performed many computations of this type. The integration and differentiation operations required in (4) can be handled using a symbolic mathematics package, such as Maple [11] or Mathematica [17]. The explicit summation of the first c terms, as indicated in (4), could be performed by utilizing the multiple precision facility in the Maple or Mathematica packages. However, it was found that the MPFUN multiple precision package and translator developed by one of us [3] was significantly faster for this purpose.

Whatever software is used, this explicit summation is an expensive operation. For example, the evaluation of

terms, using the MPFUN package with 150-digit precision arithmetic, requires 20 hours on a ``Crimson'' workstation manufactured by Silicon Graphics, Inc. Thus while such runs can be made, clearly this is pressing the limits of current workstation technology. Fortunately, it is possible to perform such computations on a highly parallel computer system. The details of this parallel algorithm are given in [4].

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