\(^\ast \)Department of Mathematics and Computer Science, University of Oradea, str. Universităţii no. 1, 410087 Oradea, Romania, e-mail: abica@uoradea.ro

\(^{\ast \ast }\)Department of Mathematics and Computer Science, University of Oradea, str. Universităţii no. 1, 410087 Oradea, Romania, e-mail: curila_diana@yahoo.com

1 Introduction

Before the fundamental work of Schoenberg (see [ 15 ] ) where the notion of B-spline is introduced in explicit way, according to de Boor and Pinkus [ 7 ] , the first apparition of spline functions can be found in the pioneering works of Popoviciu (see [ 14 ] ) and Chakalov (see [ 9 ] ). Through polynomial spline functions, the widely used are cubic splines which can be expressed both in terms of the moments (second order derivatives of the spline on knots) and in terms of the local first order derivatives \(m_{i},\) \(i=\overline{0,n}\), as in the case of Hermite type cubic splines, that are presented in the following. On a partition \(\Delta :a=x_{0}{\lt}x_{1}{\lt}\ldots {\lt}x_{n-1}{\lt}x_{n}=b\) of \([a,b]\), the Hermite type cubic spline \(s\in C^{1}[a,b]\) has the expression

where \(h_{i}=x_{i}-x_{i-1},\) \(i=\overline{1,n},\) and \(y_{i}=s\left( x_{i}\right) ,\) \(i=\overline{0,n}\). For the computation of the local derivatives \(m_{i},\) \(i=\overline{0,n},\) were proposed several procedures. Imposing the smallest deficiency, that is \(s\in C^{2}[a,b],\) and considering two endpoint conditions, various types of cubic splines are obtained such as complete cubic splines, not-a-knot splines, periodic cubic splines, \(E\left( \alpha \right) \) cubic splines, natural cubic splines (see [ 3 ] and [ 13 ] ). For instance, the natural cubic spline that minimizes the L\(_{2}\)-norm of \(s^{\prime \prime }\) is generated by the endpoint conditions \(s^{\prime \prime }\left( a\right) =s^{\prime \prime }\left( b\right) =0.\) Another idea is to determine the derivatives \(m_{i},\) \(i=\overline{0,n}\) under the smoothness property \(s\in C^{1}[a,b]\) and to consider some geometric type procedures such as in [ 1 ] and [ 8 ] , or by minimizing a functional related to the data polygon (see [ 4 ] and [ 11 ] ).

The derivatives \(m_{i},\) \(i=\overline{0,n},\) are computed in [ 1 ] by using geometric reasoning based on the slopes \(p_{i}=\tfrac {y_{i+1}-y_{i}}{x_{i+1}-x_{i}}\), \(i=\overline{0,n-1},\) and are given as,

In order to extend formula (2) for \(i=\overline{0,n},\) the previously computed slopes are not enough and therefore, Akima proposes the construction of four new supplementary slopes \(p_{-1},p_{-2},p_{n},p_{n+1}\), as follows: \(p_{-1}=2p_{0}-p_{1},\) \(p_{-2}=3p_{0}-2p_{1},\) \(p_{n}=2p_{n-1}-p_{n-2},\) \(p_{n+1}=3p_{n-1}-2p_{n-2}.\) As it is shown in [ 4 ] and [ 5 ] , sometimes, this treatment near endpoints could generate significant oscillations. Therefore, in [ 5 ] , the values of the derivatives on the first two and last two knots are computed by using optimal procedures.

In this work we focus our attention to quartic splines and propose an Akima’s type procedure for computing the derivatives \(m_{i},\) \(i=\overline{0,n},\) of the deficient \(C^{1}\)-smooth quartic spline \(S\in C^{1}[a,b]\) proposed in [ 12 ] , which has the following expression on the intervals \(\left[ x_{i-1},x_{i}\right] ,\) \(i=\overline{1,n}\):

where \(m_{i}=S^{\prime }\left( x_{i}\right) ,\) \(y_{i}=S\left( x_{i}\right) ,\) \(i=\overline{0,n},\) and \(y_{i-1/2}=S\left( \tfrac {x_{i-1}+x_{i}}{2}\right) , \) \(i=\overline{1,n}.\)

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 \(m_{i},\) \(i=\overline{0,n},\) were determined in order to minimize the L\(_{2}\)-norm of \(S^{\prime }\), \(S^{\prime \prime }\), and \(S^{\prime \prime \prime }\), respectively. Here, the values of \(m_{i},\) \(i=\overline{0,n},\) will be obtained by using a new Akima’s type method.

As will be viewed in the following, while for the Akima’s method a special treatment is required on four knots (the first two and the last two), in our method the special treatment is involved only on endpoints. The reason is in the fact that on each interval \(\left[ x_{i-1},x_{i}\right] ,\) \(i=\overline{1,n},\) the derivatives \(m_{i-1}\) and \(m_{i}\) are computed on the points \(x_{i-1}\) and \(x_{i}\) by using the values on the midpoints \(x_{i-1/2}=\tfrac {x_{i-1}+x_{i}}{2}\), \(i=\overline{1,n}\), too. Therefore, the knowledge at midpoints is an advantage. In the treatment of endpoints, in order to avoid the introduction of supplementary slopes, we develop three special variants for computing the values \(m_{0}\) and \(m_{n}\). The interpolation error estimates of this Akima’s variant quartic spline is given in terms of the modulus of continuity for less smooth class of functions. When smooth functions are interpolated we obtain the corresponding error estimates and prove that the order of approximation is \({\mathcal O}\left( h^{3}\right) .\) Finally, a numerical experiment is presented in order to illustrate the behaviour of the proposed interpolation procedure, including a comparison with the classical Akima’s cubic spline interpolation method.

2 The construction of the Akima’s type procedure for quartic splines

Consider two neighbouring intervals \(\left[ x_{i-1},x_{i}\right] \) and \(\left[ x_{i},x_{i+1}\right] \) and the midpoints \(x_{i-1/2},\) \(x_{i+1/2}\) in each of these intervals. Suppose that the points to be interpolated are \(\left( x_{0},y_{0}\right) \), \(\left( x_{1},y_{1}\right) \), …, \(\left( x_{n},y_{n}\right) \) and define the slopes

for \(i=\overline{1,n-1}.\) Let \(p_{i}\) and \(p_{i+1}\) be the quadratic Lagrange polynomials interpolating the points \(x_{i-1},\) \(x_{i-1/2},\) \(x_{i}\) and respectively, \(x_{i},\) \(x_{i+1/2},\) \(x_{i+1}\), on the intervals \(\left[ x_{i-1},x_{i}\right] \) and \(\left[ x_{i},x_{i+1}\right] ,\)

Computing the derivatives of \(p_{i}\) and \(p_{i+1}\) on the point \(x_{i}\) we get

Let

be the three-point difference approximation formula of the derivative in the point \(x_{i}\) computed on the interval \([x_{i-1/2},x_{i+1/2}]\). Now, we compute the left tangent and the right tangent in \(x_{i}\) by

If \(\vert d_{i-1/2}-d_{i}\vert +\vert d_{i+1}-d_{i+1/2}\vert \neq 0\) we propose the value of the expected derivative \(m_{i}\) to be

and if \(\vert d_{i-1/2}-d_{i}\vert +\vert d_{i+1}-d_{i+1/2}\vert =0,\) then this value will be

We see that the values \(m_{0}\) and \(m_{n}\) at the endpoints remain free. If the values \(y^{\prime }\left( a\right) \) and \(y^{\prime }\left( b\right) \) are known, then we put \(m_{0}=y^{\prime }\left( a\right) \) and \(m_{n}=y^{\prime }\left( b\right) \), but if these values \(y^{\prime }\left( a\right) \) and \(y^{\prime }\left( b\right) \) are not available we will compute the values \(m_{0}\) and \(m_{n}\) by using three proposed variants that will be presented in what follows.

3 The treatment of the endpoints

Firstly, we can consider the endpoint type conditions \(S^{\prime \prime }(a)=S^{\prime \prime }(b)=0\), that usually appears at natural cubic splines. These conditions lead to the equations

obtaining in this way the Akima’s quartic spline with natural endpoint conditions with \(m_{0}=\tfrac {m_{1}}{4}+\tfrac {-11y_{0}+16y_{1-1/2}-5y_{1}}{4h_{1}},\) \(m_{n}=\tfrac {m_{n-1}}{4}+\tfrac {5y_{n-1}-16y_{n-1/2}+11y_{n}}{4h_{n}}.\)

Another variant is to consider the local optimal condition involving the minimization of the integrals

near endpoints, resulting minimal local curvature on the first and on the last subinterval \([x_{0},x_{1}]\) and \([x_{n-1},x_{n}].\) In this purpose we consider the functionals

the system of normal equations \(J_{2}^{\prime }\left( m_{0}\right) =0,\) \(J_{2}^{\prime }\left( m_{n}\right) =0\) being

Then one obtains

The following variant is inspired by the idea of the work [ 5 ] but uses the technique from [ 11 ] , minimizing the derivative oscillation on the intervals \([x_{0},x_{1}]\) and \([x_{n-1},x_{n}]\). For this purpose we consider the functionals

By the normal equations \(J_{1}^{\prime }\left( m_{0}\right) =0\), \(J_{1}^{\prime }\left( m_{n}\right) =0\) we get the values

4 The interpolation error estimates

For the derivatives computed by the Akima’s variant (5) we have the estimate

and so, \(\left\vert m_{i}\right\vert \leq \max \{ \left\vert T_{-}\left( x_{i}\right) \right\vert ,\left\vert T_{+}\left( x_{i}\right) \right\vert \} \). Now, the estimates

are obtained in terms of the modulus of continuity, where \(\underline{h}=\min \{ h_{i}:i=\overline{1,n}\} \) and \(h=\max \{ h_{i}:i=\overline{1,n}\} \). By (4) we infer that

and thus

Consequently, it obtains the estimate

In the case of the Akima quartic spline with natural type endpoint conditions we obtain the estimate

and analogous, \(\left\vert m_{n}\right\vert \leq \tfrac {19}{4\underline{h}}\omega \left( y,\tfrac {h}{2}\right) .\) In the case of minimal local curvature on the intervals \([x_{0},x_{1}]\) and \([x_{n-1},x_{n}]\), we get the estimates

and \(\left\vert m_{0}\right\vert \leq \tfrac {73}{18\underline{h}} \omega \left( y,\tfrac {h}{2}\right) .\) For the Akima quartic spline with minimal derivative oscillation on the intervals \([x_{0},x_{1}]\) and \([x_{n-1},x_{n}]\) one obtains,

and \(\left\vert m_{n}\right\vert \leq \tfrac {57}{16\underline{h}} \omega \left( y,\tfrac {h}{2}\right) .\)

Now, observing that \(D_{i}\) and \(E_{i}\) have the same sign separately on the intervals \([x_{i-1},x_{i-1/2}]\) and \([x_{i-1/2},x_{i}]\), the estimate of \(\left\vert S\left( x\right) -y\left( x\right) \right\vert \) will be performed on each half-subinterval \([x_{i-1},x_{i-1/2}]\) and \([x_{i-1/2},x_{i}]\), \(i=\overline{1,n}\), similarly as in the proof of Corollary 7 from [ 6 ] , obtaining

for \(x\in \lbrack x_{i-1},x_{i-1/2}].\) Analogous, we have similar estimate for \(x\in \lbrack x_{i-1/2},x_{i}].\) Then we get

for \(x\in [ x_{i-1},x_{i}],\) \(i=\overline{1,n}.\) Thus, concerning the interpolation error estimate, we obtain the following result.

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 \(s^{\prime \prime }(a)=s^{\prime \prime }(b)=0,\) lead to different spline interpolants.

5 Error estimates for smooth functions

In this section we provide the error estimates for \(\left\Vert S-f\right\Vert _{\infty }\) and \(\left\Vert S^{\prime }-f^{\prime }\right\Vert _{\infty }\) when the Akima’s variant quartic spline interpolates a smooth function \(f\in C^{4}[a,b]\). First of all we prove a lemma related to Hermite quartic polynomial interpolation.

and for arbitrary fixed \(x\in \lbrack a,b]\) we define \(\varphi _{x}:[a,b]\rightarrow \mathbb {R},\) by

where \(R=f-H_{4}\left( f\right) \) is the remainder. Because \(f\in C^{4}[a,b]\) we infer that \(R\in C^{4}[a,b]\) and \(\varphi _{x}\in C^{4}[a,b]\). Since \(\varphi _{x}\left( a\right) =\varphi _{x}\left( b\right) =\varphi _{x}\left( \tfrac {a+b}{2}\right) =\varphi _{x}\left( x\right) =0\) and \(\varphi _{x}^{\prime }\left( a\right) =\varphi _{x}^{\prime }\left( b\right) =0\) after successive four times applications of the Rolle’s mean value theorem we get \(\varphi _{x}^{\left( 4\right) }\left( v\right) =\varphi _{x}^{\left( 4\right) }\left( w\right) =0\) for some \(v,w\in \left( a,b\right) \), \(v\neq w. \) Based on the fact that \(f^{\left( 4\right) }-R^{\left( 4\right) }\) is constant we get

and thus,

obtaining (14).

Of course, if \(f\in C^{5}[a,b]\), then \(L=\Vert f^{\left( 5\right) }\Vert _{\infty }\) in (14). The error estimate for \(\vert H_{4}\left( f\right) ^{\prime }\left( x\right) -f^{\prime }\left( x\right) \vert \) can be obtained too. Since \(\max \limits _{x\in \lbrack a,b]}\left\vert u^{\prime }\left( x\right) \right\vert =\tfrac {\left( b-a\right) ^{4}}{16}\) and according to the proof of this lemma, we have

Consequently, it obtains

In the case \(f\in C^{4}[a,b],\) if \(f^{\prime }\left( a\right) \) and \(f^{\prime }\left( b\right) \) are unknown we put

where \(y_{i}=f\left( x_{i}\right) \), \(i=\overline{0,n}\), \(y_{i-1/2}=f\left( \tfrac {x_{i-1}+x_{i}}{2}\right) \), \(i=\overline{1,n}\), inspired by the technique from [ 2 ] . When \(f^{\prime }\left( a\right) \) and \(f^{\prime }\left( b\right) \) are known it is natural to consider \(m_{0}=f^{\prime }\left( a\right) \), \(m_{n}=f^{\prime }\left( b\right) \). Concerning the interpolation error estimate of the Akima’s variant quartic spline in the case of smooth functions we obtain the following main result.

Now, by (4) and by the Lagrange numerical differentiation formula will be \(\eta _{i}\in \left( x_{i-1},x_{i}\right) ,\) \(\theta _{i}\in \left( x_{i},x_{i+1}\right) ,\) \(c_{i}\in \left( x_{i-1/2},x_{i+1/2}\right) \) such that

Consequently,

for all \(i=\overline{1,n-1}\), and by (5) we obtain

On the other hand, at endpoints by the same Lagrange differentiation formula it obtains

Since \(S\) and \(H\left( f\right) \) interpolates the same values \(y_{i-1},\) \(y_{i-1/2},\) \(y_{i}\) on \([x_{i-1},x_{i}],\) \(i=\overline{1,n}\), having the same structure, by (18) and (19) we get

and

for \(x\in \lbrack x_{0},x_{1}]\cup \lbrack x_{n-1},x_{n}]\), obtaining (16). Finally, by (15) we get

and since

by (18) and (19) we obtain

and the estimate (17) follows.

In (16) we see that the order of approximation by the Akima’s variant quartic spline is \(O\left( h^{3}\right) \) and, as in [ 12 ] and [ 16 ] , we have obtained the corresponding estimate of \(\left\Vert S^{\prime }-f^{\prime }\right\Vert _{\infty }\) for the derivative \(S^{\prime }\), too.

6 Numerical experiment

In order to illustrate this method we present a numerical example considering the points \(\left( x_{i},y_{i}\right) ,\) \(i=\overline{0,5}:\) \(\left( 0,16\right) ,\) \(\left( 2,20\right) \), \(\left( 3,28\right) \), \(\left( 5,21\right) ,\) \(\left( 6,24\right) ,\) \(\left( 7,28\right) \), while the values on midpoints are \(y_{1-1/2}=12,\) \(y_{2-1/2}=23,\) \(y_{3-1/2}=32,\) \(y_{4-1/2}=18,\) \(y_{5-1/2}=30\). The local derivatives \(m_{i}\), \(i=\overline{1,4}\) are computed by using the Akima’s type procedure (5), while the values at endpoints \(m_{0},\) \(m_{5}\) are computed by using the alternatives presented in Section 3 such as, natural endpoint conditions \(S^{\prime \prime }\left( a\right) =S^{\prime \prime }\left( b\right) =0\), minimal curvature \(J_{2}\left( m_{0},m_{5}\right) \) near endpoints, and minimal derivative oscillation \(J_{1}\left( m_{0},m_{5}\right) \) near endpoints, respectively. These values of the local derivatives \(m_{i},\) \(i=\overline{0,5}\) are presented below and the obtained quartic splines are represented in ??.

In figure 1 we represent the Akima’s quartic spline with natural type endpoint conditions \(S^{\prime \prime }\left( a\right) =S^{\prime \prime }\left( b\right) =0\) (denoted by (EN) and drawn as solid line curve) and the Akima’s quartic spline with minimal derivative oscillation \(J_{1}\left( m_{0},m_{5}\right) \) 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 figure 2 are represented by comparison the Akima’s quartic spline with minimal curvature \(J_{2}\left( m_{0},m_{5}\right) \) 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 \(\left( x_{0},y_{0}\right) ,\ldots ,\left( x_{5},y_{5}\right) \).