Accurate Computations with Totally Nonnegative Matrices

We consider the problem of performing accurate computations with rectangular $(m\times n)$ totally nonnegative matrices. The matrices under consideration have the property of having a unique representation as products of nonnegative bidiagonal matrices. Given that representation, one can compute the inverse, LDU decomposition, eigenvalues, and SVD of a totally nonnegative matrix to high relative accuracy in $\mathcal{O}(\max(m^3,n^3))$ time—much more accurately than conventional algorithms that ignore that structure. The contribution of this paper is to show that the high relative accuracy is preserved by operations that preserve the total nonnegativity—taking a product, re-signed inverse (when $m=n$), converse, Schur complement, or submatrix of a totally nonnegative matrix, any of which costs at most $\mathcal{O}(\max(m^3,n^3))$. In other words, the class of totally nonnegative matrices for which we can do numerical linear algebra very accurately in $\mathcal{O}(\max(m^3,n^3))$ time (namely, those for which we have a product representation via nonnegative bidiagonals) is closed under the operations listed above.