In \code{pcoa}, when negative eigenvalues are present in the decomposition results, the distance matrix D can be modified using either the Lingoes or the Cailliez procedure to produce results without negative eigenvalues.
In the Lingoes (1971) procedure, a constant c1, equal to twice absolute value of the largest negative value of the original principal coordinate analysis, is added to each original squared distance in the distance matrix, except the diagonal values. A newe principal coordinate analysis, performed on the modified distances, has at most (n-2) positive eigenvalues, at least 2 null eigenvalues, and no negative eigenvalue.
In \code{pcoa}, when negative eigenvalues are present in the decomposition results, the distance matrix D can be modified using either the Lingoes or the Cailliez procedure to produce results without negative eigenvalues.
In the Lingoes (1971) procedure, a constant c1, equal to twice absolute value of the largest negative value of the original principal coordinate analysis, is added to each original squared distance in the distance matrix, except the diagonal values. A newe principal coordinate analysis, performed on the modified distances, has at most (n-2) positive eigenvalues, at least 2 null eigenvalues, and no negative eigenvalue.