When computing eigenvalues of sym metric matrices and singular values of general matrices in finite precision arithmetic we in general only expect to compute them with an error bound pro-portional to the product of machine precision and the norm of the matrix. In particular, we do not expect to compute tiny eigenvalues and singular values to high relative accuracy. There are some important classes of matrices where we can do much better, including bidiagonal matrices, scaled diagonally dominant matrices, and scaled diagonally dominant definite pencils. These classes include many graded matrices, and all sym metric positive definite matrices which can be consistently ordered (and thus all symmetric positive definite tridiagonal matrices). In particular, the singular values and eigenvalues are determined to high relative precision independent of their magnitudes, and there are algorithms to compute them this accurately. The eigenvectors are also determined more accurately than for general matrices, and may be computed more accurately as well. This work extends results of Kahan and Demmel for bidiagonal and tridiagonal matrices.