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A new subtraction-free formula for lower bounds of the minimal singular value of an upper bidiagonal matrix. (English) Zbl 1329.65079
Traces of inverse powers of a matrix \(BB^T\) determine lower bounds of the smallest singular value of an upper bidiagonal matrix \(B\) with positive entries on both diagonals. Several approaches to the computation of these traces have been studied previously, including one subtraction-free formula. This paper derives another subtraction-free formula different from the previous one. An algorithm for its computation is presented. A comparison of computational costs shows that the evaluation of the new formula requires less operations than the previously proposed one. An efficient implementation of the algorithm for the special case of the second power is included. Numerical experiments conclude the paper.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A42 Inequalities involving eigenvalues and eigenvectors
65F50 Computational methods for sparse matrices
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