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

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

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*, v. 21, 1981, pp. 80-96. MR**616702 (83i:65065)****[2]**A. Iserles & S. P. Nørsett, "Bi-orthogonal polynomials," in*Orthogonal Polynomials and Their Applications*(A. Draux, A. Magnus and P. Maroni, eds.), Lecture Notes in Math., vol. 1171, Springer-Verlag, Berlin, 1985, pp. 92-100. MR**838964 (87f:00023)****[3]**A. Iserles & S. P. Nørsett,*On the Theory of Bi-Orthogonal Polynomials*, Tech. Rep. NA1, DAMTP, University of Cambridge, 1986.**[4]**S. P. Nørsett, "One-step methods of Hermite type for numerical integration of stiff systems,"*BIT*, v. 14, 1974, pp. 63-77. MR**0337014 (49:1787)****[5]**S. P. Nørsett, "Splines and collocation for ordinary initial value problems," in*Approximation Theory and Spline Functions*(S. P. Singh, J. H. W. Burry and B. Watson, eds.), NATO ASI Series, Vol. C136, 1983, pp. 397-417. MR**786857 (86j:41008)****[6]**R. E. Shafer, "On quadratic approximation,"*SIAM J. Numer. Anal.*, v. 7, 1974, pp. 447-460. MR**0358161 (50:10626)****[7]**G. Wanner, E. Hairer & S. P. Nørsett, "Order stars and stability theorems,"*BIT*, v. 18, 1978, pp. 475-489. MR**520756 (81b:65070)**

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

Article copyright:
© Copyright 1987
American Mathematical Society