Two-step methods and bi-orthogonality

Authors:
A. Iserles and S. P. Nørsett

Journal:
Math. Comp. **49** (1987), 543-552

MSC:
Primary 65L05; Secondary 33A65

MathSciNet review:
906187

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Abstract: We study order and zero-stability of two-step methods of Obrechkoff type for ordinary differential equations. A relation between order and properties of *m*th degree polynomials orthogonal to , , where , is established. These polynomials are investigated, focusing on their explicit form, Rodrigues-type formulae and loci of their zeros.

**[1]**A. Iserles,*Two-step numerical methods for parabolic differential equations*, BIT**21**(1981), no. 1, 80–96. MR**616702**, 10.1007/BF01934073**[2]**C. Brezinski, A. Draux, A. P. Magnus, P. Maroni, and A. Ronveaux (eds.),*Polynômes orthogonaux et applications*, Lecture Notes in Mathematics, vol. 1171, Springer-Verlag, Berlin, 1985. MR**838964****[3]**A. Iserles & S. P. Nørsett,*On the Theory of Bi-Orthogonal Polynomials*, Tech. Rep. NA1, DAMTP, University of Cambridge, 1986.**[4]**Syvert P. Nørsett,*One-step methods of Hermite type for numerical integration of stiff systems*, Nordisk Tidskr. Informationsbehandling**14**(1974), 63–77. MR**0337014****[5]**Syvert P. Nørsett,*Splines and collocation for ordinary initial value problems*, Approximation theory and spline functions (St. John’s, Nfld., 1983), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 136, Reidel, Dordrecht, 1984, pp. 397–417. MR**786857****[6]**R. E. Shafer,*On quadratic approximation*, SIAM J. Numer. Anal.**11**(1974), 447–460. MR**0358161****[7]**G. Wanner, E. Hairer, and S. P. Nørsett,*Order stars and stability theorems*, BIT**18**(1978), no. 4, 475–489. MR**520756**, 10.1007/BF01932026

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1987-0906187-8

Article copyright:
© Copyright 1987
American Mathematical Society