The coefficients of numerical methods derived by exponential and trigonometric fittings are functions of parameter z = ωh where ω and h are the involved frequency and the step size, respectively. The problem is that, for the versions described until now in the literature, the accurate computation of each coefficient asks for four different formulas (an analytic formula valid for big z and a power series expansion for small z, in each of the two fittings). In this paper, we describe an algorithm-like technique which allows replacing the set of the four by a single formula. The latter is universally valid, in the sense that it can be successfully applied irrespective of whether z is small or big, or of whether the fitting is trigonometric or exponential. Two sets of special functions, sets C and S, are introduced for this purpose, and their mathematical properties are established. Examples and applications are also presented.
Ixaru, L.Gr., Rizea, M.: A Numerov-like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies. Comput. Phys. Commun. 19, 23–27 (1980)CrossRefGoogle Scholar
Paternoster, B.: Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70-th anniversary. Comput. Phys. Commun. 183, 2499–2512 (2012)MathSciNetCrossRefGoogle Scholar
Monovasilis, Th., Simos, T.E.: New second-order exponentially and trigonometrically fitted symplectic integrators for the numerical solution of the time-independent Schrödinger equation. J. Math. Chem. 42, 535–545 (2007)MathSciNetCrossRefGoogle Scholar
Monovasilis, T., Kalogiratou, Z., Ramos, H., Simos, T.E: Modified two-step hybrid methods for the numerical integration of oscillatory problems. Math. Meth. Appl. Sci. 40, 5286–5294 (2017)MathSciNetCrossRefGoogle Scholar
Ndukum, P.L., Biala, T.A., Jator, S.N., Adeniyi, R.B.: On a family of trigonometrically fitted extended backward differentiation formulas for stiff and oscillatory initial value problems. Numer. Algor. 74, 267–287 (2017)MathSciNetCrossRefGoogle Scholar