Email: zfyang@sei.ecnu.edu.cn; lzhi@mmrc.iss.ac.cn

VerifyRealRoots

A MATLAB package for computing real verified solutions of polynomial systems of equations and inequalities

Zhengfeng Yang and Lihong Zhi

Examples of kissing number problems

The kissing number problems seeks the maximal number of unit n−dimensional spheres such that they each can touch another given unit n−dimensional sphere. The kissing number problems can be transformed as verifying the existence of real roots of the semi-algebraic systems. Here kissing_n_k is to test whether there exist k unit n−dimensional spheres which each can touch another given unit n−dimensional sphere.

Performance summary on kissing number problems

problem #var #eq #ineq degree MMCRSolver Hom4ps
timing(s) #solution timing(s) #solution
kissing_2_1 2 1 0 2 1.935 2 0.088 2
kissing_2_2 4 2 1 2 6.130 6 0.161 4
kissing_2_3 6 3 3 2 11.520 8 0.263 2
kissing_2_4 8 4 6 2 21.982 4 13.729 1
kissing_2_5 10 5 10 2 117.345 19 - -
kissing_2_6 12 6 15 2 1898.403 13 - -
kissing_3_1 3 1 0 2 0.630 4 0.129 2
kissing_3_2 6 2 1 2 4.250 10 0.298 4
kissing_3_3 9 3 3 2 15.640 16 25.695 4
kissing_3_4 12 4 6 2 30.307 2 - -
kissing_3_5 15 5 10 2 482.830 6 - -
kissing_3_6 18 6 15 2 68762.261181 14 - -
kissing_3_7 21 7 21 2 6623.484375 30 - -
kissing_3_8 24 8 28 2 28078.484375 49 - -
kissing_3_9 27 9 36 2 74293.921875 50 - -
kissing_4_1 4 1 0 2 2.160 4 0.227 2
kissing_4_2 8 2 1 2 7.840 7 0.484 2
kissing_4_3 12 3 3 2 252.71 8 1194.98 2
kissing_4_4 16 4 6 2 100.52 10 - -
kissing_4_5 20 5 10 2 1011.590 36 - -
kissing_4_6 24 6 15 2 4974.03125 34 - -
kissing_4_7 28 7 21 2 20987 38 - -
kissing_5_1 5 1 0 2 1.820 4 0.556 2
kissing_5_2 10 2 1 2 9.391 8 1.103 4
kissing_5_3 15 3 3 2 297.130 26 - -
kissing_5_4 20 4 6 2 644.370 34 - -
kissing_5_5 25 5 10 2 24573.7 46 - -
kissing_5_6 30 6 15 2 82169.14 53 - -
kissing_6_1 6 1 0 2 4.332 4 0.446 2
kissing_6_2 12 2 1 2 9.682 11 0.812 2
kissing_6_3 18 3 3 2 445.390 25 - -
kissing_6_4 24 4 6 2 1401.440 34 - -
kissing_6_5 30 5 10 2 54679.32 45 - -
kissing_7_1 7 1 0 2 3.310 7 0.802 2
kissing_7_2 14 2 1 2 24.023 18 1.186 2
kissing_7_3 21 3 3 2 684.789 29 - -
kissing_7_4 28 4 6 2 9131.4375 43 - -
kissing_7_5 35 5 10 2 165529.56 54 - -
kissing_8_1 8 1 0 2 0.885 2 1.603 2
kissing_8_2 16 2 1 2 15.354 15 1.800 2
kissing_8_3 24 3 3 2 1559.370 37 - -
kissing_8_4 32 4 6 2 81729.6 43 - -
kissing_9_1 9 1 0 2 50.046875 10 479.40625 2
kissing_9_2 18 2 1 2 590.4375 23 - -
kissing_9_3 27 3 3 2 5713.640625 35 - -
kissing_10_1 10 1 0 2 54.484375 8 - -
kissing_10_2 20 2 1 2 950.484375 25 - -
kissing_10_3 30 3 3 2 12253.984375 34 - -
kissing_11_1 11 1 0 2 65.75 10 - -
kissing_11_2 22 2 1 2 1345 27 - -
kissing_12_1 12 1 0 2 85.25 11 - -
kissing_12_2 24 2 1 2 2662.421875 40 - -
kissing_13_1 13 1 0 2 90.1875 11 - -