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On the existence of solutions to the algebraic equations in implicit Runge-Kutta methods. (English) Zbl 0506.65030

65L05 Numerical methods for initial value problems involving ordinary differential equations
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[1] K. Burrage and J. C. Butcher,Stability criteria for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 16 (1979), 46–57. · Zbl 0396.65043
[2] M. Crouzeix, Sur la B-stabilité des méthodes de Runge-Kutta, Numer. Math. 32 (1979), 75–82. · Zbl 0431.65052
[3] M. Crouzeix and P. A. Raviart, Unpublished Lecture Notes, Université de Rennes 1980.
[4] G. Dahlquist,Error analysis for a class of methods for stiff nonlinear initial value problems, Lecture Notes in Mathematics 506, Berlin, Springer-Verlag (1976). · Zbl 0352.65042
[5] G. Dahlquist and R. Jeltsch,Generalized disks of contractivity for explicit and implicit Runge-Kutta methods. Report TRITA-NA-7906, Dept. Comp. Sci., Roy. Inst. of Techn., Stockholm (1979). · Zbl 1106.65062
[6] C. A. Desoer and H. Haneda,The measure of a matrix as a tool to analyse computer algorithms for circuit analysis. IEEE Trans. Circuit Theory 19 (1972), 480–486.
[7] H. Gajewski, K. Gröger and K. Zacharias,Nichtlineare Operatorgleichungen und Operator-differentialgleichungen, Berlin, Akademie-Verlag (1974). · Zbl 0289.47029
[8] J. M. Ortega and W. C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables, New York, Academic Press (1970). · Zbl 0241.65046
[9] I. W. Sandberg,Theorems on the computation of transient response of nonlinear networks containing transistors and diodes, Bell System Tech. J. 49 (1970), 1739–1776. · Zbl 0221.94047
[10] I. W. Sandberg and H. Shichman,Numerical integration of systems of stiff nonlinear differential equations, Bell System Tech. J. 47 (1968), 511–527. · Zbl 0221.65128
[11] J. Williams,The problem of implicit formulas in numerical methods for stiff differential equations, Numer. Anal. Report No. 40, Depth. of Math., University of Manchester, Manchester (1979).
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