Stability and sensitivity of Darboux transformation without parameter

Electronic Transactions on Numerical Analysis

Volumn 18, Issue , 2004, Pages 101-136

  Bueno, M Isabel ^a   Dopico, Froilán M ^a  

^a UNIVERSIDAD CARLOS III DE MADRID   (Spain)

Author keywords

Darboux transformation; LR algorithm; LU factorization; Orthogonal polynomials; Sensitivity; Stability; Tridiagonal matrices

Indexed keywords

EID: 22744441233     PISSN: 10689613     EISSN: 10689613     Source Type: Journal    
DOI: None     Document Type: Article

References (19)

1. R. P. BOAS, The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc., 45 (1939), pp. 399-404.
2. M. I. BUENO AND F. M. DOPICO, Stability and sensitivity of tridiagonal LU factorization without pivoting, to appear in BIT.
3. M. I. BUENO AND F. MARCELLÁN, Darboux Transformation and Perturbation of linear functional, Linear Algebra Appl., 384 (2004), pp. 215-242.
4. X. CHANG AND C. C. PAIGE, On the sensitivity of the LU factorization, BIT, 38 (1998), pp. 486-501.
5. T. S. CHIHARA, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
6. G. DARBOUX, Leçons sur la Théorie Générale des Surfaces, 2éme partie, Gauthiers-Villars, Paris, 1889.
7. A. J. DURÁN, The Stieltjes moment problem for rapidly decreasing functions, Proc. Amer. Math. Soc., 107 (1989), pp. 731-741.
8. D. GALANT, An implementation of Christoffel's Theorem in the Theory of Orthogonal Polynomials, Math. Comp., 25 (1971), pp. 111-113.
9. D. GALANT, Algebraic Methods for modified orthogonal polynomials, Math. Comp., 59 (1992), pp. 541-546.
10. W. GAUTSCHI, An algorithmic implementation of the generalized Christoffel theorem, Numerical Integration, G. Hämmerlin, ed., Internat. Ser. Numer. Math., 57 (1982), pp. 89-106.
11. W. GAUTSCHI, The interplay between classical analysis and (numerical) linear algebra - A tribute to Gene H. Golub, Electron. Trans. Numer. Anal., 13 (2002), pp. 119-147. http://etna.mcs.kent.edu/vol.13.2002/pp119-147.dir/pp119-147.pdf.
12. G. H. GOLUB AND J. KAUTSKY, Calculation of Gauss Quadratures with Multiple Free and Fixed Knots, Numer. Math., 41 (1983), pp. 147-163.
13. F. A. GRÜNBAUM AND L. HAINE, Orthogonal Polynomials satisfying Differential Equations: the Role of the Darboux Transformation, CRM Proceedings and Lecture Notes, vol. 9. Amer. Math. Soc., Providence, Rhode Island, 1996, pp. 143-154.
14. F. A. GRÜNBAUM AND L. HAINE, Bispectral Darboux Transformations:An Extension of the Krall Polynomials, Int. Math. Res. Not., 8 (1997), pp. 359-392.
15. F. A. GRÜNBAUM, L. HAINE, AND E. HOROZOV, Some functions that generalize the Krall-Laguerre polynomials, J. Comput. Appl. Math., 106 (1999), pp. 271-297.
16. N. J. HIGHAM, Accuracy and Stability of Numerical Algorithms, 2nd. ed., SIAM, Philadelphia, 2002.
17. J. KAUTSKY AND G. H. GOLUB, On the calculation of Jacobi Matrices, Linear Algebra Appl., 52/53 (1983), pp. 439-455.
18. V. B. MATVEEV AND M. A. SALLE, Differential-Difference Evolution Equations. II (Darboux Transformation for the Toda Lattice), Lett. Math. Phys., 3 (1979), pp. 425-429.
19. G. W. STEWART, On the perturbation of LU and Cholesky factors, IMA J. Numer. Anal., 17 (1997), pp. 1-6.