Abstract
Purpose
In recent years attention has been directed to the problem of solving the Poisson equation, either in engineering scenarios (computational) or in regard to crystal structure (theoretical).
Methods
In (Bailey et al. in J. Phys. A, Math. Theor. 46:115201, 2013, doi:10.1088/17518113/46/11/115201) we studied a class of lattice sums that amount to solutions of Poisson’s equation, utilizing some striking connections between these sums and Jacobi ϑfunction values, together with highprecision numerical computations and the PSLQ algorithm to find certain polynomials associated with these sums. We take a similar approach in this study.
Results
We were able to develop new closed forms for certain solutions and to extend such analysis to related lattice sums. We also alluded to results for the compressed sum
where
Conclusions
As in the previous study, we find some surprisingly simple closedform evaluations
of these sums. In particular, we find that in some cases these sums are given by
PACS Codes: 02.30.Lt, 02.30.Mv, 02.30.Nw, 41.20.Cv.
MSC: 06B99, 35J05, 11Y40.
Keywords:
lattice sums; Poisson equation; experimental mathematics; highprecision computation1 Background
In [1], we analyzed various generalized lattice sums [2], which have been studied for many years in the mathematical physics community, for example, in [24]. More recently interest was triggered by some intriguing research in image processing techniques [5]. These developments have underscored the need to better understand the underlying theory behind both lattice sums and the associated Poisson potential functions.
In recent times, attention was directed to the problem of solving the Poisson equation, either in engineering scenarios (computationally, say for image enhancement) or in regard to crystal structure (theoretically). In [1] we thus studied a class of lattice sums that amount to Poisson solutions, namely the ndimensional forms
By virtue of striking connections with Jacobi ϑfunction values, we were able to develop new closed forms for certain values of the
coordinates
In this paper we discuss (2) in more detail. More detailed motivation can be found in [1,5].
Such work is made possible by numbertheoretical analysis, symbolic computation and experimental mathematics, including extensive numerical computations using up to 25,000digit arithmetic.
In Section 1.1, we describe, consistent with [5], the underlying equations along with ‘natural’ Madelung constants and relate them
to the classical Madelung constants. In Section 1.2, we produce the solutions
1.1 Madelungtype sums
In a recent treatment of ‘natural’ Madelung constants [5], it is pointed out that the Poisson equation for an ndimensional pointcharge source,
gives rise to an electrostatic potential  we call it the barecharge potential  of the form
where
where in cases such as this log sum one must infer an analytic continuation [5], as the literal sum is quite nonconvergent. This
only for
A method for gleaning information about
Accordingly  based on the Poisson equation (3)  solutions
1.2 The crystal solutions
ϕ
n
In [5] it is argued that a solution to (8) is
where
To render this representation more explicit and efficient, we could write equivalently
It is also useful that  due to the symmetry inherent in having odd summation indices  we can cavalierly replace the cosine product in (10) with a simple exponential:
This follows from the simple observation that
We observe that the convergence of these conditionally convergent sums is by no means obvious, but that results such as [[2], Thm. 8.3 and Thm. 8.5] ensure that
is convergent and analytic with abscissa
2 Methods
2.1 Fast series for
ϕ
n
From previous work [5] we know a computational series
suitable for any nonzero vector
These series, (13) and (14) are valid, respectively, for
For clarification, we give here the
Though it may not be manifest in this asymmetricallooking series, it turns out that
for any dimension n the
2.2 Computational techniques
In this study, we employ an experimental scheme similar to that used in [1], as well as in numerous other studies that have been performed. In particular, we
compute the key expressions in this study to very high precision, then use the PSLQ
algorithm in an attempt to recognize the computed numerical values in terms of relatively
simple, closedform expressions. Given an nlong input vector (
Numerous experimental evaluations of minimal polynomials satisfied by
This computational scheme is as follows:
1. Given x, y and d, select a conjectured polynomial degree m and a precision level P. We typically set the numeric precision level P somewhat greater than
2. Compute
3. Generate the
4. Apply the PSLQ algorithm (we employed the twolevel multipair variant of PSLQ for
d up to 19, and the threelevel multipair variant for
5. If no numerically significant relation is found, try again with a larger value of m and correspondingly higher precision. If a relation is found, try with somewhat lower m, until the minimal m is found that produces a numerically significant relation vector A. Here ‘numerically significant’ means that the relation holds to at least 100 digits beyond the level of precision required to discover it. To obtain greater assurance that the polynomial produced by this process is in fact the minimal polynomial for α, use the polynomial factorization facilities in Mathematica or Maple to attempt to factor the resulting polynomial.
3 Results and discussion
3.1 Closed forms for the
ϕ
2
potential
Provably we have the following results which were established by factorization of lattice sums after being empirically discovered by the methods described in the next few sections.
Theorem 1 ([1])
We have
Using the integer relation method PSLQ [9] to hunt for results of the form
for α algebraic, we may obtain and further simplify many results as follows.
Conjecture 2 ([1])
We have discovered and subsequently proven
where the notation
Such hunts are made entirely practicable by (14). Note that for general x and y we have
Remark 3 (Algebraicity)
In light of our current evidence, we conjectured that for x, y rational,
for α algebraic. Theorem 5 proved this conjecture. We note that Theorem 5 proves that all values should be algebraic but does not, a priori, establish the precise values we have found. This will be addressed in Section 3.4.
3.2 Madelung and ‘jellium’ crystals, and Jacobi ϑfunctions
We have studied
as a ‘natural’ potential for
where
Now it is explained and illustrated in [5] that this
Briefly, a fast series for
valid on the Delord ncube, i.e., for
In the following, we shall also use the general form of the general Jacobi theta functions, defined as in [10] and [[6], Sec. 2.6]:
One useful relation is
3.3 Closed forms for
ψ
2
and
ϕ
2
Using the series (23) for highprecision numerics, it was discovered (see [[5], Appendix]) that previous latticesum literature had concealed a longtime typographical
error for certain twodimensional sums, and that a valid closed form for
In the Madelung case, it then became possible to cast
for
Theorem 4 ([1])
For
and
where
and
with
We recall the general ϑtransform giving for all z with
for
which directly relates
Then
or
Likewise (28) is unchanged on replacing λ by κ. Hence, it is equivalent to (20) to prove that for all
Theorem 5 (Algebraic values of λ and μ, [1])
For all
This type of analysis also works for any singular value
3.4 Explicit equations for degree 2, 3 and 5
We illustrate the complexity of
Example 6 (
As described in [1],
and so letting
Iteration yields
where
In addition, since we have an algebraic relation
We may perform the same work inductively for
Example 7 (Empirical computation of
Given z and
Example 8
We conclude this subsection by proving the following empirical evaluation:
Let
since
so that
In this particular case, we use
Recall that the length of a polynomial is the sum of the absolute value of the coefficients.
Theorem 9 (Determining a zero)
SupposePis an integral polynomial of degreeDand lengthL. Suppose thatαis algebraic of degreedand lengthℓ. Then either
Example 10 (
Making explicit the recipe for
This shows how generous our estimates were in Example 8.
In similar fashion we can now computationally confirm all of the exact evaluations
in Conjecture 2 and the like. For example, we know that
We now turn to our main focus.
3.5 ‘Compressed’ potentials
Yet another solution to the Poisson equation with crystal charge source, for
This is the potential inside a crystal compressed by
Along the same lines as the analysis of (13), we can posit a fast series
valid on the Delord cuboid, i.e.,
Theorem 11 (ϑrepresentation for compressed potential)
We have
where
This theorem is consistent with
We note that there are various trivial evaluations such as
and have (where
(see [[6], Prop. 2.1]). We observe that (47) can be written for all x, y as
on appealing to (49) and (50). Hence we may now study
Fix an integer
when replacing z by
Example 12 (
As for
This becomes
and
We turn to
Example 13 (
As for
In tandem with (57) this becomes
since
This can also be determined by using the methods of Example 7 to obtain the algebraic
equation linking
We next exhibit several sample compressedsum evaluations, of which (62) is the most striking.
Theorem 14 (Some explicit compressed sums)
Proof Let us first sketch a proof of (62). This requires showing
However, 3z is in the compressed lattice generated by 1,
From this, we may deduce that
To prove (61), we similarly consider (51) and (60) with
and the result again follows.
To prove (63) requires an application of this method with
In general, the polynomial for
3.6 Further results for
ϕ
2
(
a
/
c
,
b
/
c
d
,
d
)
As noted, we anticipated similarly algebraic evaluations for
Example 15 (Some explicit compressed polynomials for
We present here the polynomials
Note that reversing the order of the variables leads to a more intractable computation
since
3.7 Computational results
Results that we have obtained for the special set of cases
Table 1. PSLQ runs to recover minimal polynomials satisfied by
The degree
We were not able to obtain relations for all cases
In the earlier study [1], we included Jason Kimberley’s observation that the degree
Unfortunately, we have not yet been able to find a similar formula for the minimal polynomial degree in the compressed cases under study in Table 1. We are actively investigating this at the present time.
With regards to the run times listed in Table 1 (given to 0.01 second accuracy), it should be recognized that like all computer run times, particularly in a multicore or multiprocessor environment, they are only repeatable to two or three significant digits. They are listed here only to emphasize the extremely rapid increase in computational cost as the degree m and corresponding precision level P increase.
Remark 16 (A refined approach)
A referee has suggested making more more explicit the relations with elliptic functions. We had opted to limit the theory needed so as to better understand what experimental computation could produce for a wider audience.
(a) The basic quantity
Notice that
(b) Similarly, for fixed
Then, in terms of the elliptic function cn[6,10], we can write
and the corresponding duplication formula for f is
[[2], Appendix] which makes the minimal polynomial for
Let
4 Conclusions
In this study, we have obtained some explicit closedform evaluations of certain summations
that arise as solutions of the Poisson equation, which in turn arises in numerous
arenas ranging from engineering to studies of crystal structures. Many of these evaluations
were first found experimentally by means of a process we have employed in other settings:
computing the relevant expressions to high precision, then applying the PSLQ algorithm
to recognize the resulting numerical values in closed form. In this case, we computed
the constants
For further details of the computation of lattice sums and related theta functions the reader is referred to [1322].
Competing interests
Neither author has any personal, financial or other competing interests relevant to the topics studied in this paper.
Authors’ contributions
Both authors participated in the research work. DHB performed many of the numerical computations; JMB did much of the theoretical work.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors thank many colleagues and the referees for fruitful discussions about lattice sums and theta functions. The first author supported in part by the Director, Office of Computational and Technology Research, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy, under contract number DEAC0205CH11231. The second author is supported in part by the Australian Research Council.
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