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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 as ; 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 to
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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