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We consider the class of totally nonnegative (TN) matrices---matrices all of whose minors are nonnegative. Any nonsingular TN matrix factors as a product of nonnegative bidiagonal matrices. The entries of the bidiagonal factors parameterize the set of nonsingular TN matrices.We present new O(n3) algorithms that, given the bidiagonal factors of a nonsingular TN matrix, compute its eigenvalues and SVD to high relative accuracy in floating point arithmetic, independent of the conventional condition number. All eigenvalues are guaranteed to be computed to high relative accuracy despite arbitrary nonnormality in the TN matrix.We prove that the entries of the bidiagonal factors of a TN matrix determine its eigenvalues and SVD to high relative accuracy.We establish necessary and sufficient conditions for computing the entries of the bidiagonal factors of a TN matrix to high relative accuracy, given the matrix entries.In particular, our algorithms compute all eigenvalues and the SVD of TN Cauchy, Vandermonde, Cauchy--Vandermonde, and generalized Vandermonde matrices to high relative accuracy.