4. *The Pleiades*

Finally, we considered the problem "Pleiades" as given in ([22], pg. 245).

$$\mu^i y^{\prime\prime} = \sum\_{i \neq j} \frac{\mu\_j \left(^j y - ^i y\right)}{\rho\_{ij}}, \; ^i z^{\prime\prime} = \sum\_{i \neq j} \frac{\mu\_j \left(^j z - ^i z\right)}{\rho\_{ij}}, \;$$

with

$$\rho\_{ij} = \sqrt{\left(^{i}y - {^{j}y}\right)^{2} + \left(^{i}z - {^{j}z}\right)^{2}}, i, j = 1, \cdots, 7.$$

The initial values are

$$\,\_2^1y(0) = 3,\,\,\_2^2y(0) = 3,\,\,\_3^3y(0) = -1,\,\,\_4^4y(0) = -3,\,\,\_5^5y(0) = 2,\,\,\_6^6y(0) = -2,\,\,\_7^7y(0) = 2,$$

$$\,\_1^1z(0) = 3,\,\,\_2^2z(0) = -3,\,\,\_2^3z(0) = 2,\,\,\_4^4z(0) = 0,\,\,\_5^5z(0) = 0,\,\,\_6^6z(0) = -4,\,\,\_7^7z(0) = 4,$$

$$\begin{aligned} \,^1y'(0) &= 0, \,^2y'(0) = 0, \,^3y'(0) = 0, \,^4y'(0) = 0, \,^5y'(0) = 0, \,^6y'(0) = 1.75, \,^7y'(0) = -1.5, \\\,^1z'(0) &= 0, \,^2z'(0) = 0, \,^3z'(0) = 0, \,^4z'(0) = -1.25, \,^5z'(0) = 1, \,^6z'(0) = 0, \,^7z'(0) = 0, \,^7z'(0) = 0, \,^8z'(0) = 0. \end{aligned}$$

We again transformed this problem to a system of fourteen first-order equations and solved it to *xend* = 3 and 4. We recorded the endpoint errors after we estimated the solution there by a very accurate integration using Mathematica and quadruple precision. The efficiency measure ratios of DLMP6(5) vs. NEW6(5) for Pleiades can be found in Table 8.

**Table 8.** Efficiency measures ratios of DLMP6(5) vs. NEW6(5) for Pleiades.


We estimated 168 (i.e., 12 problems times 7 tolerances times two end points) efficiency measures for each pair. In average we observed a ratio of 1.98, meaning that DLMP6(5) is about 98% more expensive! This is quite remarkable since a great deal of effort has been put over the years towards achieving 10–20% efficiency [23,24]. In reverse, this means that about log10 1.98<sup>6</sup> <sup>≈</sup> 1.8 digits were gained on average at the same costs.

#### **4. Conclusions**

This paper is concerned with training the coefficients of a Runge–Kutta pair for addressing a certain kind of problem. We concentrated on problems with Kepler-type orbits and an extensively studied family of Runge–Kutta pairs of orders six and five. After optimizing the free parameters (coefficients) in a couple of runs on Kepler orbits, we concluded to a certain pair. This pair was found to outperform other representatives from this family in a wide range of relevant problems.

**Author Contributions:** All authors contributed equally. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflicts of interest.
