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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Anda A., Park H.: Fast plane rotations with dynamic scaling. . SIAM J. Matrix Anal. Appl. 15, 162–174 (1994)
Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., Sorensen, D.: LAPACK Users’ Guide, 3rd edn, Software Environ. Tools 9. SIAM, Philadelphia (1999)
Barlow J., Demmel J.: Computing accurate eigensystems of scaled diagonally dominant matrices. SIAM J. Num. Anal. 27(3), 762–791 (1990)
Businger P., Golub G.H.: Linear least squares solutions by householder transformations. Numer. Math. 7, 269–276 (1965)
Demmel J.: Accurate singular value decompositions of structured matrices. SIAM J. Matrix Anal. Appl. 21(2), 562–580 (1999)
Demmel J., Gragg W.: On computing accurate singular values and eigenvalues of acyclic matrices. Linear Algebra Appl. 185, 203–218 (1993)
Demmel J., Gu M., Eisenstat S., Slapničar I., Veselić K., Drmač Z.: Computing the singular value decomposition with high relative accuracy. Linear Algebra Appl. 299(1–3), 21–80 (1999)
Demmel J., Kahan W.: Accurate singular values of bidiagonal matrices. SIAM J. Sci. Statist. Comput. 11(5), 873–912 (1990)
Demmel, J., Koev, P.: Necessary and sufficient conditions for accurate and efficient rational function evaluation and factorizations of rational matrices. In: Structured matrices in mathematics, computer science, and engineering, II (Boulder, CO, 1999), Contemp. Math., vol 281, pp. 117–143. Amer. Math. Soc., Providence, RI (2001)
Demmel J., Koev P.: Accurate SVDs of weakly diagonally dominant M-matrices. Numer. Math. 98(1), 99–104 (2004)
Demmel J., Koev P.: Accurate SVDs of polynomial Vandermonde matrices involving orthonormal polynomials. Linear Algebra Appl. 417(2–3), 382–396 (2006)
Demmel J., Veselić K.: Jacobi’s method is more accurate than QR. SIAM J. Matrix Anal. Appl. 13(4), 1204–1246 (1992)
Demmel J.W.: Applied Numerical Linear Algebra. SIAM, Philadelphia (1997)
Dopico F.M., Koev P.: Accurate symmetric rank revealing and eigendecompositions of symmetric structured matrices. SIAM J. Matrix Anal. Appl. 28(4), 1126–1156 (2006)
Dopico F.M., Molera J.M.: Perturbation theory for factorizations of LU type through series expansions. SIAM J. Matrix Anal. Appl. 27(2), 561–581 (2005)
Dopico F.M., Molera J.M., Moro J.: An orthogonal high relative accuracy algorithm for the symmetric eigenproblem. SIAM J. Matrix Anal. Appl. 25(2), 301–351 (2003)
Drmač Z.: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic. SIAM J. Sci. Comput. 18(4), 1200–1222 (1997)
Drmač Z.: Accurate computation of the product induced singular value decomposition with applications. SIAM J. Num. Anal. 35(5), 1969–1994 (1998)
Drmač Z.: A posteriori computation of the singular vectors in a preconditioned Jacobi SVD algorithm. IMA J. Num. Anal. 19, 191–213 (1999)
Drmač Z., Veselić K.: New fast and accurate Jacobi SVD algorithm. I. SIAM J. Matrix Anal. Appl. 29(4), 1322–1342 (2008)
Drmač Z., Veselić K.: New fast and accurate Jacobi SVD algorithm. II. SIAM J. Matrix Anal. Appl. 29(4), 1343–1362 (2008)
Eisenstat S., Ipsen I.: Relative perturbation techniques for singular value problems. SIAM J. Numer. Anal. 32(6), 1972–1988 (1995)
Fernando K., Parlett B.: Accurate singular values and differential qd algorithms. Numer. Math. 67, 191–229 (1994)
Gentleman W.M.: Error analysis of QR decompositions by Givens transformations. Linear Algebra and Appl. 10, 189–197 (1975)
Golub G., Van Loan C.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)
Hari V.: Accelerating the SVD block-Jacobi method. Computing 75(1), 27–53 (2005)
Hari V.: Convergence of a block-oriented quasi-cyclic Jacobi method. SIAM J. Matrix Anal. Appl. 29(2), 349–369 (2007)
Higham, N.J.: The Matrix computation toolbox. http://www.ma.man.ac.uk/~higham/mctoolbox
Higham N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)
Horn, R.A., Johnson, C.R.: Topics in matrix analysis. Cambridge University Press, Cambridge (1994). (Corrected reprint of the 1991 original)
Koev P.: Accurate eigenvalues and SVDs of totally nonnegative matrices. SIAM J. Matrix Anal. Appl. 27(1), 1–23 (2005)
Koev P., Dopico F.: Accurate eigenvalues of certain sign regular matrices. Linear Algebra Appl. 424(2–3), 435–447 (2007)
Li R.C.: Relative perturbation theory. II. Eigenspace and singular subspace variations. SIAM J. Matrix Anal. Appl. 20(2), 471–492 (1999)
Li R.C.: Relative perturbation theory. IV. sin 2θ theorems. Linear Algebra Appl. 311(1–3), 45–60 (2000)
Mathias R.: Accurate eigensystem computations by Jacobi methods. SIAM J. Mat. Anal. Appl. 16(3), 977–1003 (1995)
The MathWorks, Inc., Natick, MA: MATLAB Reference guide (1992)
Parlett B.N.: The symmetric eigenvalue problem. SIAM, Philadelphia (1998)
Peláez M.J., Moro J.: Accurate factorization and eigenvalue algorithms for symmetric DSTU and TSC matrices. SIAM J. Matrix Anal. Appl. 28(4), 1173–1198 (2006)
Rutishauser H.: The Jacobi method for real symmetric matrices. Numer. Math. 9(1), 1–10 (1966)
Slapničar I.: Componentwise analysis of direct factorization of real symmetric and Hermitian matrices. Linear Algebra Appl. 272, 227–275 (1998)
Slapničar I.: Highly accurate symmetric eigenvalue decomposition and hyperbolic SVD. Linear Algebra Appl. 358, 387–424 (2003) Special issue on accurate solution of eigenvalue problems (Hagen, 2000)
Slapničar, I.: Accurate symmetric eigenreduction by a Jacobi method. Ph.D. thesis, Fernuniversität - Hagen, Hagen, Germany (1992)
Stewart G.W., Sun J.G.: Matrix perturbation theory. Academic Press, New York (1990)
Veselić K.: A Jacobi eigenreduction algorithm for definite matrix pairs. Num. Math. 64, 241–269 (1993)
Ye Q.: Computing singular values of diagonally dominant matrices to high relative accuracy. Math. Comp. 77(264), 2195–2230 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-009-0240-8