Additional file 2 of On the rank-distance median of 3 permutations

Example of numerical unstability of “triple-null” formula in the Implementation section. This Additional file, in text format suitable to be read by Octave or MATLAB, contains three 100×100 matrices A,B, and C for which the triple-null formula does not work, and the alternative single-null formula for the intersection im ( A − B ) ∩ im ( C − B ) $\text {im}(A-B) \cap \text {im}(C-B)$ was used. The file can be checked in Octave as follows: Save it as triple.mat, Call Octave, and then type in its prompt: > load(“triple.mat”) > meet = null([(null(A’-B’))’;(null(C’-B’))’]); > size(meet) ans = 100 0 > meet = orth((A-B)*null([A-B C-B])(1:100,:)); > size(meet) ans = 100 2 This shows that, while the first formula produced a zero-dimensional subspace, the second one actually produced a 2-dimensional subspace. (MAT 222 kb)

本附加文件为实现章节（Implementation section）中三重零化公式（triple-null formula）数值不稳定性的示例。该文件采用Octave或MATLAB可读的文本格式，内含三个100×100的矩阵A、B与C，此时三重零化公式无法正常工作，需改用用于求解像空间交集$ ext{im}(A-B) cap ext{im}(C-B)$的替代单零化公式（single-null formula）。可在Octave中按如下步骤验证该文件：将其保存为triple.mat，启动Octave后在命令提示符中输入：
> load("triple.mat")
> meet = null([(null(A'-B'))';(null(C'-B'))']);
> size(meet)
ans = 100 0
> meet = orth((A-B)*null([A-B C-B])(1:100,:));
> size(meet)
ans = 100 2
上述结果表明，第一种公式得到了零维子空间，而第二种公式实际得到了二维子空间。（MAT格式文件，大小222 KB）