Computing singular values of diagonally dominant matrices to high relative accuracy

For a (row) diagonally dominant matrix, if all of its off-diagonal entries and its diagonally dominant parts (which are defined for each row as the absolute value of the diagonal entry subtracted by the sum of the absolute values of off-diagonal entries in that row) are accurately known, we develop an algorithm that computes all the singular values, including zero ones if any, with relative errors in the order of the machine precision. When the matrix is also symmetric with positive diagonals (i.e. a symmetric positive semidefinite diagonally dominant matrix), our algorithm computes all eigenvalues to high relative accuracy. Rounding error analysis will be given and numerical examples will be presented to demonstrate the high relative accuracy of the algorithm.