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Implicit standard Jacobi gives high relative accuracy

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Abstract

We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX T 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 \({O(\epsilon \kappa (X))}\) , where \({\epsilon}\) is the machine precision and \({\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2}\) 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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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM and Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911, Leganés, Spain

    Froilán M. Dopico

  2. Department of Mathematics, San Jose State University, One Washington Square, San Jose, CA, 95192, USA

    Plamen Koev

  3. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avda. Universidad 30, 28911, Leganés, Spain

    Juan M. Molera

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  1. Froilán M. Dopico
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  2. Plamen Koev
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  3. Juan M. Molera
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Correspondence to Froilán M. Dopico.

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Dopico, F.M., Koev, P. & Molera, J.M. Implicit standard Jacobi gives high relative accuracy. Numer. Math. 113, 519–553 (2009). https://doi.org/10.1007/s00211-009-0240-8

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  • DOI: https://doi.org/10.1007/s00211-009-0240-8

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