Quadratic splines interpolating derivatives. (English) Zbl 0758.41005

Kobza, Jiří

Acta Univ. Palacki. Olomuc., Fac. Rerum Nat. 100, Math. 30, 219-233 (1991).

Let \(\Delta_ x=\{x_ i: i=o(1)n+1\}\) be an increasing set of knots and let \(S(2,\Delta_ x)\) be the set of spline functions \(s\) satisfying the conditions: \(s\in C^ 1[x_ 0,x_{n+1}]\), \(s\) is a quadratic polynomial on each interval \([x_ i,x_{i+1}]\), \(i=o(1)n\). The author proves that there exists a unique spline function \(s\in S(2,\Delta_ x)\) verifying the conditions \(s'(x_ i)=m_ i\), \(i=o(1)n+1\) (interpolation conditions) and \(s(x_ 0)=s_ 0\) (the initial condition). One studies the existence and the uniqueness of a spline function \(s\in S(2,\Delta_ x)\) taking given values for the first and the second derivatives at the interpolation points \(t_ i\), \(i=o(1)n\), i.e. on the mesh \[ (\Delta_ x\Delta_ t): x_ 0\leq t_ 0<x_ 1<t_ 1<\cdots< t_ n\leq x_{n+1}, \] are verified conditions of the form \(s'(t_ i)=m_ i\) and \(s''(t_ i)=M_ i\), \(i=o(1)n\). The author gives algorithms for determining the parameters involved in the calculations of these splines.

Reviewer: C.Mustăţa (Cluj-Napoca)

MSC:

41A05	Interpolation in approximation theory
41A15	Spline approximation

Keywords:

uniqueness

References:

[1]	Anwar M.N.: A direct cubic spline approach for IVP’s. in [9] , 9-18.
[2]	Berg L.: Differenzengleichungen zweiter Ordnung mit Anwendungen. DVW Berlin, 1979. · Zbl 0406.39001
[3]	Boor C.de: A Practical Guide to Splines. Springer Verlag, N.Y. 1978. · Zbl 0406.41003
[4]	Hřebíček J., Mikulík M.: Cubic splines preserving monotonicity and convexity. (in Czech), Num. Math. Phys. Metalurgy, Blansko 198E.
[5]	Kobza J.: On algorithms for parabolic splines. Acta UPO, Fac.rer.nat., Math. XXIV, V. 88 (1987), 169-185. · Zbl 0693.65005
[6]	Kobza J.: Some properties of interpolating quadratic spline. Acta UPO, Fac.rer.nat. 97 (1990) · Zbl 0748.41006
[7]	Makarov V.L., Chlobystov V.V.: Spline-Approximation of Functions. (in Russian), Nauka, Moscow, 1983.
[8]	Sallam S., El-Tarazi M.N.: Quadratic spline interpolation on uniform meshes. in [9], 145-150. · Zbl 0765.41014 · doi:10.1016/0168-9274(93)90063-W
[9]	Schmidt J.W., Späth H. (eds.): Splines in Numerical Analysis. Akademie-Verlag, Berlin, 1989. · Zbl 0664.00022
[10]	St\?čkin S.B., Subbotin J.N.: Splines in Numerical Analysis. (in Russian), Nauka, Moscow 1976.
[11]	Usmani R.A.: On quadratic spline interpolation. BIT 27 (1987), 615-622. · Zbl 0631.41009 · doi:10.1007/BF01937280