Accurate Computations with Totally Nonnegative Matrices

Is it possible to perform numerical linear algebra with structured matrices to high relative accuracy at a reasonable cost? In our talk we answer this question affirmatively for a class of structured matrices whose applications range from approximation theory to combinatorics to multivariate statistical analysis [1, 2, 4]—the Totally Nonnegative (TN) matrices, i.e. matrices all of whose minors are nonnegative. The trick in performing accurate computations with TN matrices is to choose a representation that reveals their TN structure. The matrix entries are a poor choice of parameters and it is difficult to tell from them if all 4n minors are nonnegative (and thus the matrix is TN). Instead, we represent any TN matrix A as a product of nonnegative bidiagonal matrices: