Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices

n fi When computing eigenvalues of symmetric matrices and singular values of general matrices i nite precision arithmetic we in general only expect to compute them with an error bound pron portional to the product of machine precision and the norm of the matrix. In particular , we do ot expect to compute tiny eigenvalues and singular values to high relative accuracy. There are l m some importan t classes of matrices where we can do much better, including bidiagona atrices, scaled diagonally dominan t matrices, and scaled diagonally dominan t defin ite pencils. c These classes include many graded matrices, and all symmetric positive defin ite matrices which an be consisten tly ordered ( and thus all symmetric positive defin ite tridiagonal matrices) . In n i particular , the singular values and eigenvalues are determined to high relative precisio ndependent of their magn itudes, and there are algorithms to compute them this accurately. c The eigenvectors are also determined more accura..