We present new O(n 3 ) algorithms to compute very accurate singular value decompositions of Cauchy matrices, Vandermonde matrices, and related "unit-displacement-rank" matrices. These algorithms compute all the singular values with guaranteed relative accuracy, independent of their dynamic range. In contrast, previous O(n 3 ) algorithms can potentially lose all relative accuracy in the tiniest singular values.
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James Demmel (2000) studied this question.
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