Operations on oscillatory functions. (English) Zbl 0930.65150

Summary: We introduce a procedure which allows modifying some standard algorithms with the aim of making them tuned on oscillatory functions. We consider the case when the involved functions are of form \(y(x)=f_1(x) \sin (\omega x)+ f_2(x) \cos(\omega x)\) where \(f_1(x)\) and \(f_2(x)\) are smooth functions, to obtain tuned formulae for the first and second derivatives, for the Simpson quadrature formula and for the Numerov algorithm for differential equations. The expressions of the parameters of the new formulae are written in a way which makes them tuned also for functions of form \(y(x)= f_1(x)\text{sinh} (\lambda x)+ f_2(x)\text{cosh} (\lambda x)\). The proposed procedure is general enough to allow amending many other classical algorithms.


65T40 Numerical methods for trigonometric approximation and interpolation
42A10 Trigonometric approximation
65D32 Numerical quadrature and cubature formulas
65L12 Finite difference and finite volume methods for ordinary differential equations


Full Text: DOI


[1] Phillips, J. M.; Taylor, P. J., Theory and Applications of Numerical Analysis (1973), Academic Press: Academic Press London, New York · Zbl 0312.65002
[2] Davis, P. J.; Rabinowitz, P., Methods of Numerical Integration (1984), Academic Press: Academic Press New York · Zbl 0154.17802
[3] Ehrenmark, U. T., J. Comput. Appl. Math., 21, 87 (1988) · Zbl 0632.65017
[4] Van Daele, M.; De Meyer, H.; Vanden Berghe, G., Int. J. Comput. Math., 42, 83 (1992) · Zbl 0765.65025
[5] Coleman, J. P.; Ixaru, L. Gr., IMA J. Numer. Anal., 16, 179 (1996) · Zbl 0847.65052
[6] Ixaru, L. Gr., Numerical Methods for Differential Equations and Applications (1984), Reidel: Reidel Dordrecht, Boston, Lancaster · Zbl 0301.34010
[7] Lyche, T., Numer. Math., 19, 65 (1972) · Zbl 0221.65123
[8] Ixaru, L. Gr.; Rizea, M., J. Comput. Phys., 73, 306 (1987) · Zbl 0633.65131
[9] Pryce, J. D., Numerical Solution of Sturm-Liouville Problems (1993), Clarendon Press: Clarendon Press Oxford, New York, Tokyo · Zbl 0795.65053
[10] Ixaru, L. Gr.; De Meyer, H.; Vanden Berghe, G.; Van Daele, M., Comput. Phys. Commun., 100, 71 (1997) · Zbl 0927.65099
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.