Optimal properties for deficient quartic splines of Marsden type

Abstract

In this work, we obtain an improved error estimate in the interpolation with the Hermite \(C^{2}\)-smooth deficient complete quartic spline that has the distribution of nodes following the Marsden type scheme and investigate the possibilities to compute the derivatives on the knots such that the obtained spline \(S\in C^{1}[a,b]\) has minimal curvature and minimal \(L^{2}\)-norm of \(S^{\prime }\) and \(S^{\prime \prime \prime }\). In each case, the interpolation error estimate is performed in terms of the modulus of continuity.%MCEPASTEBIN%

The Akima's fitting method for quartic splines

“…In the case of the Akima quartic spline with natural type endpoint conditions we obtain the estimate (7) |m ] and [x i−1/2 , x i ], the estimate of |S (x) − y (x)| will be performed on each half-subinterval [x i−1 , x i−1/2 ] and [x i−1/2 , x i ], i = 1, n, similarly as in the proof of Corollary 7 from [6], obtaining In contrast with the case of cubic splines, by comparing ( 7)-( 8) and ( 12)-( 13), we see that for the quartic splines (3) the condition of minimal curvature on the intervals [x 0 , x 1 ] and [x n−1 , x n ], and the condition s ′′ (a) = s ′′ (b) = 0, lead to different spline interpolants.…”

“…These values of the local derivatives m i , i = 0, 5 are presented below and the obtained quartic splines are represented in Figs. 6 In Fig. 6.1 we represent the Akima's quartic spline with natural type endpoint conditions S ′′ (a) = S ′′ (b) = 0 (denoted by (EN) and drawn as solid line curve) and the Akima's quartic spline with minimal derivative oscillation J 1 (m 0 , m 5 ) near endpoints (denoted by (AD), the dots-line curve).…”

“…6.1 we represent the Akima's quartic spline with natural type endpoint conditions S ′′ (a) = S ′′ (b) = 0 (denoted by (EN) and drawn as solid line curve) and the Akima's quartic spline with minimal derivative oscillation J 1 (m 0 , m 5 ) near endpoints (denoted by (AD), the dots-line curve). Differences are observed in the first and in the last interval [0, 2] and [6,7], respectively, where the curve (AD) has smaller oscillation. In Fig.…”

“…In Fig. 6.2 are represented by comparison the Akima's quartic spline with minimal curvature J 2 (m 0 , m 5 ) on the intervals [x 0 , x 1 ] = [0, 2] and [x 4 , x 5 ] = [6,7] (denoted by (CM) and plotted with dots), and the classical Akima's cubic spline (denoted by (AK) and drawn as solid line curve) interpolating the points (x 0 , y 0 ) , . .…”

“…Error estimates in the interpolation with the C 2 quartic splines (4) were established in [12], [10] and [16]. In [6], the values of the derivatives m i , i = 0, n, were determined in order to minimize the L 2 -norm of S ′ , S ′′ , and S ′′′ , respectively. Here, the values of m i , i = 0, n, will be obtained by using a new Akima's type method.…”