The sharpness of stability estimates corresponding to a general resolvent condition

Linear Algebra and Its Applications

Volumn 311, Issue 1-3, 2000, Pages 161-175

  Borovykh, N ^a   Spijker, M N ^a  

^a LEIDEN UNIVERSITY   (Netherlands)

Author keywords

Kreiss matrix theorem; Numerical processes; Powers of matrices; Resolvent condition; Sharpness; Stability analysis; Stability estimates; Upper bounds

Indexed keywords

EID: 0034419033     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(00)00085-9     Document Type: Article

References (18)

1. N. Borovykh, D. Drissi, M.N. Spijker, A note about Ritt's condition, related resolvent conditions and power bounded operators, Numer. Funct. Anal. Optim., to appear.
2. N. Borovykh, M.N. Spijker, Resolvent conditions and bounds on the powers of matrices, with relevance to numerical stability of initial value problems, J. Comput. Appl. Math., submitted.
3. van Dorsselaer J.L.M., Kraaijevanger J.F.B.M., Spijker M.N. Linear stability analysis in the numerical solution of initial value problems. Acta Numerica. 1993;199-237.
4. Giles M.B. On the stability and convergence of discretizations of initial value pdes. IMA J. Numer. Anal. 17:1997;563-576.
5. J.F.B.M. Kraaijevanger, private communication.
6. Kraaijevanger J.F.B.M. Two counterexamples related to the Kreiss matrix theorem. BIT. 34:1994;113-119.
7. Kreiss H.-O. Über die Stabilitätsdefinition für Differenzengleichungen die partielle Differentialgleichungen approximieren. BIT. 2:1962;153-181.
8. LeVeque R.J, Trefethen L.N. On the resolvent condition in the Kreiss matrix theorem. BIT. 24:1984;584-591.
9. McCarthy C.A., Schwartz J. On the norm of a finite Boolean algebra of projections and applications to theorems of Kreiss and Morton. Comm. Pure Appl. Math. 18:1965;191-201.
10. O. Nevanlinna, On the growth of the resolvent operators for power bounded operators, in: Linear Operators, Banach Center Publications, Inst. Math. Pol. Acad. Sciences, Warszawa, vol. 38 (1997) pp. 247-264.
11. Reddy S.C., Trefethen L.N. Stability of the method of lines. Numer. Math. 62:1992;235-267.
12. Spijker M.N. On a conjecture by LeVeque and Trefethen related to the Kreiss matrix theorem. BIT. 31:1991;551-555.
13. Spijker M.N. Numerical stability resolvent conditions and delay differential equations. Appl.Numer. Math. 24:1997;233-246.
14. Spijker M.N., Straetemans F.A.J. Stability estimates for families of matrices of nonuniformly bounded order. Linear Algebra Appl. 239:1996;77-102.
15. J.C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth, Belmont, 1989.
16. J.C. Strikwerda, B.A. Wade, A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions, in: Linear Operators, Banach Center Publications, Inst. Math. Pol. Acad. Sciences, Warszawa, vol. 38, 1997, pp. 329-360.
17. Toh K.C., Trefethen L.N. The kreiss matrix theorem on a general complex domain. SIAM J. Matrix Anal. Appl. 21:1999;145-165.
18. Wegert E., Trefethen L.N. From the Buffon needle problem to the Kreiss matrix theorem. Amer. Math. Monthly. 101:1994;132-139.