Implicit Standard Jacobi Gives High Relative Accuracy

We prove that the Jacobi algorithm applied implicitly on a decompositionA = XDXT of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(Ekappa(X)), where E is the machine precision and kappa(X) ≡ ||X||2 · ||X−1||2 is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense.We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite)matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.

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