Computing eigenvalues for products of two classes of sign regular matrices to high relative accuracy

In this paper, we consider how to accurately solve the product eigenvalue problem for the class of sign regular (SR) matrices with signature ( 1 , ⋯ , 1 , − 1 ) and the class of totally nonnegative (TN) matrices, which tend to be extremely ill-conditioned. We present algorithms with O ( n 3 ) complexity to accurately compute the parameter matrices of products of TN matrices and SR matrices with signature ( 1 , … , 1 , − 1 ) . Based on the accurate parameter matrices, all eigenvalues of the product matrix are computed to high relative accuracy. Numerical experiments are provided to confirm the claimed high relative accuracy.