Parallel computation of polynomials with minimal uniform norm and its application to large eigenproblems

Journal of Computational and Applied Mathematics

Volumn 82, Issue 1-2, 1997, Pages 185-198

  Heuveline, Vincent ^a   Sadkane, Miloud ^a  

^a CAMPUS DE BEAULIEU   (France)

Author keywords

Arnoldi's method; Best uniform approximation; Chebyshev iterations; Faber iterations; Krylov subspace; Parallel implementation

Indexed keywords

APPROXIMATION THEORY; EIGENVALUES AND EIGENFUNCTIONS; ITERATIVE METHODS; PARALLEL ALGORITHMS; PARALLEL PROCESSING SYSTEMS;

ARNOLDI'S METHOD; CHEBYSHEV ITERATION; FABER ITERATION; KRYLOV SUBSPACE;

POLYNOMIALS;

EID: 0031235218     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(97)00046-0     Document Type: Article

References (39)

1. E. Anderson et al., LAPACK Users' Guide, SIAM, Philadelphia, 1992.
2. I. Barrodale, L.M. Delves, J.C. Mason, Linear Chebyshev approximation of complex-valued functions, Math. Comput. 32 (143) (1978) 853-863.
3. E.W. Cheney, Introduction to Approximation Theory, 2nd ed., Chelsea, New York, 1986.
4. A.J. Clayton, Further results on polynomials having least maximum modulus over an ellipse in the complex plane, UKAEA Memorandum, AEEW, 1963.
5. I.S. Duff, R.G. Grimes, J.G. Lewis, Sparse matrix test problems, ACM Trans. Math. Soft. 15 (1989) 1-14.
6. S. Ellacot, J. Williams, Linear Chebyshev approximation in the complex plane using Lawson's algorithm, Math. Comput. 30 (133) (1976) 35-44.
7. B. Fischer, R. Freund, Chebyshev polynomials are not always optimal, J. Approx. Theory, 65 (1991) 261-273.
8. B. Fischer, J. Modersitzki, An algorithm for complex linear approximation based on semi-infinite programming, Numer. Algorithm 5 (1993) 287-299.
9. K. Glashoff, K. Roleff, A new method for Chebyshev approximation of complex-valued functions, Math. Comput. 36 (1981) 233-239.
10. G.H. Golub, C.F. Van Loan, Matrix Computations, 2nd ed., The Johns Hopkins University Press, Baltimore, 1989.
11. U. Grothkopf, G. Opfer, Complex Chebyshev polynomials on circular sectors with degree six or less, Math. Comput. 39 (1982) 599-615.
12. V. Heuveline, M. Sadkane, Chebyshev acceleration techniques for large complex non hermitian eigenvalue problems, Reliable Computing 2 (1996) 111-117.
13. V. Heuveline, M. Sadkane, Arnoldi-Faber method for large non hermitian eigenvalue problems, Technical Report no. 1042, IRISA, 1996.
14. D. Ho, Tchebychev acceleration technique for large scale nonsymmetric matrices, Numer. Math. 56 (1990) 721-734.
15. D. Ho, F. Chatelin, M. Bennani, Arnoldi Tchebychev procedure for large scale nonsymmetric matrices, Mathematical Modelling Numer. Analysis 1 (1990) 53-65.
16. W. Kerner, Large-scale complex eigenvalue problems, J. Comput. Phys. 85 (1989) 1-85.
17. W. Krabs, G. Opfer, Eine Methode zur Lösung des komplexen Approximationsproblem mit einer Anwendung auf konforme Abbildung, Zamm 55 (1975) 208-211.
18. R.B. Lehoucq, Analysis and implementation of an implicitly estarted Arnoldi iteration Ph.D. Thesis, Department of Computational and Applied Mathematics, Rice University, 1995.
19. D.G. Luenberger, Linear and nonlinear programming, 2nd ed., Addison-Wesley, Reading, MA, 1984.
20. T.A. Manteuffel, The Tchebychev iteration for nonsymmetric linear system, Numer. Math. 28 (1977) 307-327.
21. T.A. Manteuffel, Adaptative procedure for estimation of parameter for the nonsymmetric Tchebychev iteration, Numer. Math. 31 (1978) 187-208.
22. R.B. Morgan, On restarting the Arnoldi method for large nonsymmetric eigenvalue problems, Math. Comput. 65 (1996) 1213-1230.
23. M.J.D Powell, Approximation theory and methods, Cambridge University Press, Cambridge, 1981.
24. R. Reemtsen, Some outer approximation methods for semi-infinite optimization problems, Technical Report, Technische Universität Berlin, 1992.
25. T.J. Rivlin, H.S. Shapiro, A unified approach to certain problems of approximation and minimization, J. Soc. Ind. Appl. Math. 9(4) (1961) 670-699.
26. Y. Saad, Variations on Arnoldi's method for computin eigenelements of large unsymmetric matrices, Linear Algebra Appl. 34 (1980) 269-295.
27. Y. Saad, Chebyshev acceleration techniques for solving nonsymmetric eigenvalue problems, Math. Comput. 42 (1984) 567-588.
28. Y. Saad, Least squares polynomials in the complex plane and their use for solving nonsymmetric linear systems, SIAM J. Numer. Anal. 24 (1987) 155-169.
29. Y. Saad, Numerical Methods for Large Eigenvalue Problems, Manchester University Press, Manchester, 1992.
30. M. Sadkane, A block Arnoldi-Chebyshev method for computing the leading eigenpairs of large sparse unsymmetric matrices, Numer. Math. 64 (1993) 181-193.
31. D.C. Smolarski, Optimum semi-iterative methods for the solution of any linear algebraic system with a square matrix, Ph.D. Thesis, Department of Computer Science, University of Illinois at Urbana-Champaign, 1981.
32. D.C. Smolarski, P.E. Saylor, An optimum iterative method for solving any linear system with a square matrix, BIT 28 (1988) 163-178.
33. D.C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl. 13 (1992) 357-385.
34. C. Spagl, Charakterisierung und Numerik in der linearen Tschebyscheff Approximation, Ph.D. Thesis, Katolische Universität Eichstät (in German), 1988.
35. G. Starke, R.S. Varga, A hybrid Arnoldi-Faber method for nonsymmetric systems of linear equations, Numer. Math. 64 (1993) 213-240.
36. P.T.P Tang, Chebyshev approximation on the complex plane, Ph.D. Thesis, University of California at Berkeley, 1987.
37. M.B. van Gijzen, Iterative solution methods for linear equations in finite element, Ph.D. Thesis, University of Delft, 1994.
38. Y. Yamamoto, H. Ohtsubo, Subspace iteration accelerated by using Chebyshev polynomials for eigenvalue problems with symmetric matrices, Int. J. Numer. Methods Eng. 10 (1976) 935-944.
39. ∞ of overdetermined linear systems, Technical Report MD 21228, University of Maryland, Baltimore, 1990.