Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems

Journal of Computational and Applied Mathematics

Volumn 125, Issue 1-2, 2000, Pages 41-56

  Borovykh, N ^a   Spijker, M N ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

DISCRETIZATION; ERROR GROWTH; TRAPEZOIDAL RULE;

BOUNDARY VALUE PROBLEMS;

EID: 0034512163     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00457-X     Document Type: Article

References (32)

1. V.K. Barwell, Special stability problems for functional differential equations, BIT 15 (1975) 130-135.
2. F.F. Bonsall, J. Duncan, Numerical ranges, in: R.G. Bartle (Ed.), Studies in Functional Analysis, The Mathematical Association of America, 1980, pp. 1-49.
3. N. Borovykh, D. Drissi, M.N. Spijker, A note about Ritt's condition, related resolvent conditions and power bounded operators, Numer. Funct. Anal. Optim., in press.
4. N. Borovykh, M.N. Spijker, The sharpness of stability estimates corresponding to a general resolvent condition, Linear Algebra Appl. 311 (2000) 161-175.
5. J.L.M. van Dorsselaer, J.F.B.M. Kraaijevanger, M.N. Spijker, Linear stability analysis in the numerical solution of initial value problems, Acta Numer. 1993 (1993) 199-237.
6. S. Friedland, A generalization of the Kreiss matrix theorem, SIAM J. Math. Anal. 12 (1981) 826-832.
7. E. Hairer, G. Wanner, Solving Ordinary Differential Equations, Springer, Berlin, 1991.
8. R.A. Horn, C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1990.
9. J.F.B.M. Kraaijevanger, Two counterexamples related to the Kreiss matrix theorem, BIT 34 (1994) 113-119.
10. H.-O. Kreiss, Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren, BIT 2 (1962) 153-181.
11. H.W.J. Lenferink, M.N. Spijker, A generalization of the numerical range of a matrix, Linear Algebra Appl. 140 (1990) 251-266.
12. R.J. LeVeque, L.N. Trefethen, On the resolvent condition in the Kreiss matrix theorem, BIT 24 (1984) 584-591.
13. Ch. Lubich, O. Nevanlinna, On resolvent conditions and stability estimates, BIT 31 (1991) 293-313.
14. O. Nevanlinna, On the growth of the resolvent operators for power bounded operators, in: J. Janas, F.H. Szafraniec, J. Zemánek (Eds.), Linear Operators, Banach Center Publications, vol. 38, Inst. Math. Pol. Acad. Sciences, Warszawa, 1997, pp. 247-264.
15. C. Palencia, Stability of rational multistep approximations of holomorphic semigroups, Math. Comp. 64 (1995) 591-599.
16. S.V. Parter, Stability, convergence, and pseudo-stability of finite-difference equations for an over-determined problem, Numer. Math. 4 (1962) 277-292.
17. S.C. Reddy, L.N. Trefethen, Stability of the method of lines, Numer. Math. 62 (1992) 235-267.
18. R.D. Richtmyer, K.W. Morton, Difference Methods for Initial-Value Problems, 2nd Edition, Wiley, New York, 1967.
19. W. Rudin, Functional Analysis, McGraw-Hill, New York, 1973.
20. M.N. Spijker, On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem, BIT 31 (1991) 551-555.
21. M.N. Spijker, Numerical stability, resolvent conditions and delay differential equations, Appl. Numer. Math. 24 (1997) 233-246.
22. M.N. Spijker, F.A.J. Straetemans, Stability estimates for families of matrices of nonuniformly bounded order, Linear Algebra Appl. 239 (1996) 77-102.
23. M.N. Spijker, F.A.J. Straetemans, Error growth analysis, via stability regions, for discretizations of initial value problems, BIT 37 (1997) 442-464.
24. M.N. Spijker, S. Tracogna, B.D. Welfert, About the sharpness of the stability estimates in the Kreiss matrix theorem, Math. Comp., in press.
25. J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth, Belmont, 1989.
26. J.C. Strikwerda, B.A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, in: J. Janas, F.H. Szafraniec, J. Zemánek (Eds.), Linear Operators, Banach Center Publications, vol. 38, Inst. Math. Pol. Acad. Sciences, Warszawa, 1997, pp. 329-360.
27. 2-stability, the resolvent condition, and strict H-stability, Linear Algebra Appl. 41 (1981) 151-159.
28. E. Tadmor, The resolvent condition and uniform power boundedness, Linear Algebra Appl. 80 (1986) 250-252.
29. K.C. Toh, L.N. Trefethen, The Kreiss matrix theorem on a general complex domain, Report no. 97/13, Oxford Univ. Comp. Lab, 1997.
30. D.S. Watanabe, M.G. Roth, The stability of difference formulas for delay differential equations, SIAM J. Numer. Anal. 22 (1985) 132-145.
31. E. Wegert, L.N. Trefethen, From the Buffon needle problem to the Kreiss matrix theorem, Amer. Math. Monthly 101 (1994) 132-139.
32. M. Zennaro, P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math. 49 (1986) 305-318.