Implicit Standard Jacobi Gives High Relative Accuracy Articles uri icon

publication date

  • October 2009

start page

  • 519

end page

  • 553


  • 4


  • 113

International Standard Serial Number (ISSN)

  • 0029-599X

Electronic International Standard Serial Number (EISSN)

  • 0945-3245


  • 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 ยท ||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.