SJSU Singular Matrix Database

The calculated gap in the singular values

If the calculated numerical rank of A is numrank and if the calculated singular values are svals(1), svals(2), . . . then

gap = svals(numrank) / svals(numrank+1).

For an m by n matrix A note that svals(numrank+1) is smaller than our default tolerance max(m,n) * eps(norm(A)) and, as discussed here, svals(numrank+1) may have little relative accuracy. Consequently the calculated gap may also have little relative accuracy. Even so, the calculated gap is often useful to help determine if the numerical rank is well defined. See this link.