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Some relationships between implicit Runge-Kutta, collocation and Lanczos \(\tau\) methods, and their stability properties. (English) Zbl 0208.41602

65L20 Stability and convergence of numerical methods for ordinary differential equations
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
Full Text: DOI
[1] O. Axelsson,A Class of A-Stable Methods, BIT 9 (1969), 185–199. · Zbl 0208.41504 · doi:10.1007/BF01946812
[2] J. C. Butcher,Implicit Runge-Kutta Processes, Math. Comp. 18 (1964), 50–64. · Zbl 0123.11701 · doi:10.1090/S0025-5718-1964-0159424-9
[3] J. C. Butcher,Integration Processes Based on Radau Quadrature Formulas, Math. Comp. 18 (1964), 233–244. · Zbl 0123.11702 · doi:10.1090/S0025-5718-1964-0165693-1
[4] L. Collatz,The Numerical Treatment of Differential Equations, (2nd English Ed.) Springer, Berlin (1960). · Zbl 0086.32601
[5] G. J. Cooper,Interpolation and Quadrature Methods for Ordinary Differential Equations, Math. Comp. 22 (1968), 69–73. · Zbl 0155.47201 · doi:10.1090/S0025-5718-1968-0224289-7
[6] C. F. Curtiss and J. O. Hirschfelder,Integration of Stiff Equations, Proc. Nat. Acad. Sci. U.S. (1952), 235–243. · Zbl 0046.13602
[7] G. G. Dahlquist,A special stability problem for linear multistep methods, BIT 3 (1963), 27–43. · Zbl 0123.11703 · doi:10.1007/BF01963532
[8] B. L. Ehle,High Order A-Stable Methods for the Numerical Solution of Systems of Differential Equations, BIT 8 (1968), 276–278. · Zbl 0176.14604 · doi:10.1007/BF01933437
[9] F. R. Gantmacher,Matrix Theory, Vol. II, Chelsea, New York (1959). · Zbl 0085.01001
[10] C. W. Gear,The Automatic Integration of Stiff Ordinary Differential Equations, Proc. I.F.I.P. Congress (preprint) (1968), A81–A85.
[11] C. Lanczos,Trigonometric Interpolation of Empirical and Analytical Functions, J. Math. Phys. 17 (1938), 123–199. · Zbl 0020.01301 · doi:10.1002/sapm1938171123
[12] C. Lanczos,Tables of Chebyshev Polynomials (Introduction), Nat. Bur. Stand. Appl. Math. Ser. 9 (1952).
[13] G. J. Makinson,Stable High Order Implicit Methods for the Numerical Solution of Systems of Differential Equations, Comp. J. 11 (1968), 305–310. · Zbl 0167.15704 · doi:10.1093/comjnl/11.3.305
[14] H. Mineur,Techniques de Calcul Numerique, Beranger, Paris (1952).
[15] M. R. Osborne,A New Method for the Integration of Stiff Systems of Ordinary Differential Equations, Proc. IFIP Congress (preprint) (1968), A86–A90.
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