The Ehrlich-aberth method for the nonsymmetric tridiagonal eigenvalue problem

SIAM Journal on Matrix Analysis and Applications

Volumn 27, Issue 1, 2006, Pages 153-175

  Bini, Dario A ^a   Gemignani, Luca ^a   Tisseur, Françoise ^b  

^a UNIVERSITY OF PISA   (Italy)

^b UNIVERSITY OF MANCHESTER   (United Kingdom)

Author keywords

Divide and conquer; Nonsyrnmetric eigenvalue problem; QR decomposition; Root finder; Symmetric indefinite generalized eigenvalue problem; Tridiagonal matrix

Indexed keywords

DIVIDE AND CONQUER; NONSYRNMETRIC EIGENVALUE PROBLEM; QR DECOMPOSITION; ROOT FINDER; SYMMETRIC INDEFINITE GENERALIZED EIGENVALUE PROBLEM; TRIDIAGONAL MATRIX;

ALGORITHMS; APPROXIMATION THEORY; COMPUTATION THEORY; MATRIX ALGEBRA; NUMERICAL ANALYSIS; PROBLEM SOLVING;

EIGENVALUES AND EIGENFUNCTIONS;

EID: 33144484860     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/S0895479803429788     Document Type: Article

References (40)

1. O. ABERTH, Iteration methods for finding all zeros of a polynomial simultaneously, Math. Comp., 27 (1973), pp. 339-334.
2. L. ADAMS AND P. ARBENZ, Towards a divide and conquer algorithm for the real nonsymmetric eigenvalue problem, SIAM J. Matrix Anal. Appl., 15 (1994), pp. 1333-1353.
3. D. BINDEL, J. DEMMEL, W. KAHAN, AND O. MARQUES, On computing Givens rotations reliably and efficiently, ACM Trans. Math. Software, 28 (2002), pp. 206-238.
4. D. A. BINI, Numerical computation of polynomial zeros by means of Aberth's method, Numer. Algorithms, 13 (1996), pp. 179-200.
5. D. A. BINI AND G. FIORENTINO, MPSsolve: Numerical computation of polynomial roots, v. 2.2, FRISCO report, 1999. Software and papers available online from ftp://ftp.dm.unipi.it/pub/mpsolve.
6. D. A. BINI AND G. FIORENTINO, Design, analysis, and implementation of a multiprecision polynomial rootfinder, Numer. Algorithms, 23 (2000), pp. 127-173.
7. M. A. BREBNER AND J. GRAD, Eigenvalues of Ax = λBx for real symmetric matrices A and B computed by reduction to a pseudosymmetric form and the HR process, Linear Algebra Appl., 43 (1982), pp. 99-118.
8. A. BUNSE-GERSTNER, An analysis of the HR algorithm for computing the eigenvalues of a matrix, Linear Algebra Appl., 35 (1981), pp. 155-173.
9. C. CARSTENSEN, Inclusion of the roots of a polynomial based on Gerschgorin's theorem, Numer. Math., 59 (1991), pp. 349-360.
10. P. A. CLEMENT, A class of triple-diagonal matrices for test purposes, SIAM Rev., 1 (1959), pp. 50-52.
11. I. S. DHILLON, Reliable computation of the condition number of a tridiagonal matrix in O(n) time, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 776-796.
12. 2) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem, Ph.D. thesis, UCB Tech. Report UCB//CSD-97-971, University of California, Berkeley, CA, 1997.
13. J. J. DONGARRA, G. A. GEIST, AND C. H. ROMINE, Algorithm 710: FORTRAN subroutines for computing the eigenvalues and eigenvectors of a general matrix by reduction to general tridiagonal form, ACM Trans. Math. Software, 18 (1992), pp. 392-400.
14. Q. DU, M. JIN, T. Y. Li, AND Z. ZENG, The quasi-Laguerre iteration, Math. Comp., 66 (1997), pp. 345-361.
15. L. W. EHRLICH, A modified Newton method for polynomials, Comm. ACM, 10 (1967), pp. 107-108.
16. K. V. FERNANDO, On computing an eigenvector of a tridiagonal matrix. Part I: Basic results, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 1013-1034.
17. I. GARGANTINI, Parallel Laguerre iterations: The complex case, Numer. Math., 26 (1976), pp. 317-323.
18. G. A. GEIST, Reduction of a general matrix to tridiagonal form, SIAM J. Matrix Anal. Appl., 12 (1991), pp. 362-373.
19. P. E. GILL, G. H. GOLUB, W. MURRAY, AND M. A. SAUNDERS, Methods for modifying matrix factorizations, Math. Comp., 28 (1974), pp. 505-535.
20. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
21. H. W. GUGGENHEIMER, Initial approximations in Durand-Kerner's root finding method, BIT, 26 (1986), pp. 537-539.
22. P. HENRICI, Applied and Computational Complex Analysis, Vol. 1, Wiley, New York, 1974.
23. D. J. HIGHAM AND N. J. HIGHAM, Structured backward error and condition of generalized eigenvalue problems, SIAM J. Matrix Anal. Appl., 20 (1998), pp. 493-512.
24. N.J. HIGHAM, Efficient algorithms for computing the condition number of a tridiagonal matrix, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 150-165.
25. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, 2002.
26. Y. IKEBE, On inverses of Hessenberg matrices, Linear Algebra Appl., 24 (1979), pp. 93-97.
27. E. R. JESSUP, A case against a divide and conquer approach to the nonsymmetric eigenvalue problem, Appl. Numer. Math., 12 (1993), pp. 403-420.
28. W. KAHAN, Accurate Eigenvalues of a Symmetric Tri-diagonal Matrix, Tech. Report CS-TR-66-41, Stanford University, Stanford, CA, 1966.
29. Z. S. LIU, On the Extended HR Algorithm, Tech. Report PAM-564, Center for Pure and Applied Mathematics, University of California, Berkeley, CA, 1992.
30. B. N. PARLETT, The Symmetric Eigenvalue Problem, SIAM, Philadelphia, 1998.
31. B. N. PARLETT AND H. C. CHEN, Use of indefinite pencils for computing damped natural modes, Linear Algebra Appl., 140 (1990), pp. 53-88.
32. B. N. PARLETT AND I. S. DHILLON, Fernando's solution to Wilkinson's problem: An application of double factorization, Linear Algebra Appl., 267 (1997), pp. 247-279.
33. L. PASQUINI, Accurate computation of the zeros of the generalized Bessel polynomials, Numer. Math., 86 (2000), pp. 507-538.
34. M.S. PETKOVIĆ, Iterative Methods for Simultaneous Inclusion of Polynomial Zeros, Springer-Verlag, Berlin, Heidelberg, New York, 1989.
35. M. S. PETKOVIĆ, L. D. PETKOVIĆ, AND L. RANČIĆ, Higher-order simultaneous methods for the determination of polynomial multiple zeros, Int. J. Comput. Math., 80 (2000), pp. 1407-1427.
36. M. S. PETKOVIĆ, L. D. PETKOVIĆ, AND D. ŽIVKOVIC, Laguerre-like methods for the simultaneous approximation of polynomial zeros, in Topics in Numerical Analysis, Comput. Suppl. 15, Springer-Verlag, Vienna, 2001, pp. 189-209.
37. M. S. PETKOVIĆ AND D. V. VRANIĆ, The convergence of Euler-like method for the simultaneous inclusion of polynomial zeros, Comput. Math. Appl., 39 (2000), pp. 95-105.
38. H. RUTISHAUSER, Solution of eigenvalue problems with the LR-transformation, Nat. Bur. Standards Appl. Math. Ser., 1958 (1958), pp. 47-81.
39. F. TISSEUR, Tridiagonal-diagonal reduction of symmetric indefinite pairs, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 215-232.
40. F. TISSEUR AND K. MEERBERGEN, The quadratic eigenvalue problem, SIAM Rev., 43 (2001), pp. 235-286.