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by P. I. Davies, N. J. Higham, Philip I. Davies, Nicholas J. Higham
http://www.maths.man.ac.uk/~nareports/narep338.ps.gz
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Abstract:
For a symmetric positive definite matrix the relative error in the eigenvalues computed by the two-sided Jacobi method is bounded in terms of the condition number of the matrix scaled to have unit diagonal. Similarly, for a general matrix the relative error in the singular values computed by the one-sided Jacobi method is bounded in terms of the condition number of the matrix scaled to have rows or columns of unit-2 norm. We show how to generate random matrices having these scalings and given eigenvalues and singular values, respectively. For the two-sided Jacobi method we apply an algorithm of Bendel and Mickey for generating random correlation matrices, with an improved formula for the rotations. We show how to modify the algorithm to generate matrices for the one-sided Jacobi method. Using these test matrices we show empirically that the forward error bounds for the oneand two-sided Jacobi methods are sharp. Key words. Jacobi method, correlation matrix, eigenvalues, singular value
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