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

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

\(^{\ast \ast \ast }\)Department of Environmental Engineering, str. G. Magheru, no. 26, Oradea, Romania, e-mail: mirceacurila@yahoo.com

1 Introduction

Motivated by the nice properties of complete cubic splines, and trying to increase the degree of \(C^{2}\)-smooth complete spline interpolant, but preserving the bandwidth of the diagonally dominant system, Howell introduced in [ 9 ] the deficient complete quartic spline that match the interpolated function \(f\in C[a,b]\) on the interpolation nodes and on midpoints of a grid following the Marsden scheme (see [ 17 ] and [ 25 ] ). More precisely, considering a grid

and the midpoints \(z_{i}=x_{i-1/2}=\tfrac {x_{i-1}+x_{i}}{2},\) \(i=\overline{1,n},\) Howell proved the existence and uniqueness of the deficient quartic spline \(S\in C^{2}[a,b]\) taking prescribed values on \(x_{i},\) \(i=\overline{0,n}\) and on \(x_{i-1/2},\) \(i=\overline{1,n},\) with the endpoint interpolation conditions \(S^{\prime }\left( a\right) =f^{\prime }\left(a\right) ,\) \(S^{\prime }\left( b\right) =f^{\prime }\left( b\right) .\) Moreover, by the continuity condition \(S\in C^{2}[a,b]\) the spline must satisfy the tridiagonal system of equations

with known \(m_{0}=f^{\prime }\left( a\right) ,\) \(m_{n}=f^{\prime }\left(b\right) \), where \(h_{i}=x_{i}-x_{i-1},\) \(i=\overline{1,n},\) and \(m_{i}=S^{\prime }\left( x_{i}\right) ,\) \(i=\overline{0,n}.\) With the notation \(y_{i}=f\left( x_{i}\right) ,\) \(i=\overline{0,n},\) and \(t=\tfrac {x-x_{i-1}}{h_{i}}\in \lbrack 0,1]\) for \(x\in \lbrack x_{i-1},x_{i}],\) this quartic spline has the following expression on each interval \([x_{i-1},x_{i}],\) \(i=\overline{1,n}:\)

Solving this system, the local derivatives \(m_{i},\) \(i=\overline{0,n}\) are uniquely determined. Concerning the interpolation error estimate, in [ 10 ] was established the following result.

In [ 9 ] Howell conjectured that \(C_{1}=1\). Volkov [ 24 ] proved that this conjecture holds true. For much less smooth class of functions \(f\in C[a,b]\) and in the case of uniform partition, considering a simplified endpoint condition \(S^{\prime }\left(a\right)=S^{\prime }\left( b\right) =0\), the corresponding error estimate is obtained in terms of the modulus of continuity as follows:

In [ 9 ] the author had extended this result for nonuniform partitions considering the constant \(\beta =\max \{ h_{i}:i=\overline{1,n}\} / \min \{ h_{i}:i=\overline{1,n}\} \) and \(h=\max \{ h_{i}:i=\overline{1,n}\} \), obtaining:

Error estimates for C\(^{3}\)-smooth quartic splines and for quartic splines that on midpoints matches with the first derivative were established in [ 4 ] , [ 5 ] , [ 12 ] , [ 18 ] , [ 20 ] and [ 21 ] . In [ 20 ] , theorem 1 is generalized by replacing the position of midpoints to a general type of interior points of the form \(z_{i}=x_{i-1}+\theta h_{i},\) \(i=\overline{1,n}\) with \(\theta \in \lbrack \frac{1}{4},\frac{3}{4}]\). In this paper we improve the error estimates (6)–(7) obtaining a smaller constant in terms of the modulus of continuity in the case \(f\in C[a,b]\), and investigate the corresponding error estimates for the situations when the endpoint condition \(S^{\prime }\left( a\right) =S^{\prime }\left( b\right)=0 \) is replaced by other classical ones such as \(S^{\prime \prime }\left(a\right) =S^{\prime \prime }\left( b\right) =0\), \(S^{\prime }\left(a\right)=f^{\prime }\left( a\right) \) and \(S^{\prime }\left( b\right) =f^{\prime } \left( b\right) \), or \(S^{\prime \prime }\left( a\right)=f^{\prime \prime }\left( a\right) \) and \(S^{\prime \prime }\left( b\right) =f^{\prime \prime }\left( b\right)\) .

On the other hand, in the present work we investigate the posibilities to determine the local derivatives \(m_{i},\) \(i=\overline{0,n}\), on the nodes of the deficient quartic spline \(S\in C^{1}[a,b]\) given in (2), such that the mean curvature

and the functionals

for \(k\in \{ 1,2,3\} ,\) are minimized.

Shape preserving properties for quartic splines were investigated in [ 7 ] , [ 22 ] and [ 26 ] . Optimal properties for quartic splines were obtained in [ 6 ] , [ 14 ] , [ 15 ] , and [ 18 ] . In [ 14 ] and [ 15 ] the properties of \(C^{3}\)-smooth quartic splines having interpolation points \(t_{i},\) \(i=\overline{1,n}\) with \(x_{i-1}{\lt}t_{i}{\lt}x_{i}\), different by the grid nodes, were investigated. Natural, complete and periodic \(C^{3}\)-smooth quartic splines in connection with the minimization of the curvature are considered in [ 14 ] , while the parameters of spline are determined in [ 15 ] under the minimization of the functionals \(J_{k}\left( S\right) \), \(k\in \{ 0,1,2,3\} \), by using quadratic programming and the technique of pseudoinverse solution of linear systems in a similar way as was performed in [ 13 ] for cubic splines.

The paper is organized as follows. In section 2 we obtain some improvements of the error estimates presented in [ 10 ] and in [ 9 ] , regarding the interpolation of a continuous function \(f\in C[a,b]\) by the deficient complete quartic spline \(S\in C^{2}[a,b]\) given in (2). section 3 is devoted to the optimal properties of the deficient quartic spline \(S\in C^{1}[a,b]\) in connection with the minimization of the functionals \(J_{k}\left( S\right) \) above presented, of a special interest being the minimal curvature and the minimal slope of the graph of \(S\). In order to illustrate the obtained theoretical results, a numerical example is presented in section 4 and some concluding remarks are pointed out in the last section.

2 Error estimates for deficient quartic splines

Consider the quartic spline \(S\in C^{2}[a,b]\) given in (2) under the endpoint condition \(S^{\prime }\left( a\right) =S^{\prime }\left( b\right)=0 \) interpolating a continuous function \(f\in C[a,b]\) on a uniform parition \(\{ x_{i}\} _{i=0}^{n}\) and matching \(f\) on the midpoints \(x_{i-1/2},\) \(i= \overline{1,n}.\) In that follows, we get an improvement of the estimates (6)–(7) considering both the cases \(f\in C[a,b]\) and \(f\in \operatorname {Lip}[a,b],\) where \(\operatorname {Lip}[a,b]=\{ f\in C[a,b]:\exists L{\gt}0\) with \(\left\vert f\left( x\right) -f\left( y\right) \right\vert \leq L\left\vert x-y\right\vert ,\) \(\forall x,y\in \lbrack a,b]\} \).

Firstly, we observe that \(A_{i}\left( x\right) \geq 0,\) \(B_{i}\left( x\right) \geq 0,\) \(C_{i}\left( x\right) \leq 0\), \(D_{i}\left( x\right) \geq 0,\) \(E_{i}\left( x\right) \geq 0\) for \(x\in \lbrack x_{i-1},x_{i-1/2}]\) and \(A_{i}\left( x\right) \leq 0,\) \(B_{i}\left( x\right) \geq 0,\) \(C_{i}\left( x\right) \geq 0\), \(D_{i}\left( x\right) \leq 0,\) \(E_{i}\left( x\right) \leq 0 \) for \(x\in \lbrack x_{i-1/2},x_{i}]\). Since \(A_{i}\left( x\right) +B_{i}\left( x\right) +C_{i}\left( x\right) =1,\) \(\forall x\in \lbrack x_{i-1},x_{i}]\) we infer that on the interval \([x_{i-1},x_{i-1/2}]\) we have

and on \([x_{i-1/2},x_{i}]\) we get

with \(\left\vert D_{i}\left( x\right) +E_{i}\left( x\right) \right\vert =t\left( 1-t\right) \left\vert 1-2t\right\vert \cdot h\), where \(t=\frac{x-x_{i-1}}{h}\in \lbrack 0,1].\) For estimating \(\max \{ \left\vert m_{i}\right\vert :i=\overline{0,n}\} \) we see that the tridiagonal system (1) becomes in the case of equally spaced knots:

for \(i=\overline{1,n-1},\) with \(m_{0}=m_{n}=0.\) Intending to estimate \(\left\vert d_{i}\right\vert \), \(i=\overline{1,n-1}\), we get

and since the matrix \(A\) of the tridiagonal system is diagonally dominant, we infer that \(\left\Vert A^{-1}\right\Vert \leq \frac{4}{3}\) and then,

Consequently, for the last term in (11) and (12) we get

for all \(x\in \lbrack x_{i-1},x_{i}],\) \(i=\overline{1,n}\). Now, by (11) we obtain

for \(x\in \lbrack x_{i-1},x_{i-1/2}]\). Analogously, by (12) it obtains

when \(x\in \lbrack x_{i-1/2},x_{i}],\) and the estimate (9) follows. After elementary calculus, we see that

and when \(f\in \operatorname {Lip}[a,b]\), the estimate (9) becomes

that is the estimate (10).

Concerning the estimate from theorem 3, in the case of nonuniform partition, we can state the following.

The matrix \(A\) of the tridiagonal, diagonally dominant system (1) has \(\left\Vert A^{-1}\right\Vert \leq \frac{4}{3}\) and when estimate \(\left\vert d_{i}\right\vert ,\) \(i=\overline{1,n-1}\), we get

for all \(i=\overline{1,n-1}\). Then, \(\max \{ \left\vert m_{i-1}\right\vert ,\left\vert m_{i}\right\vert \} \leq \frac{8}{3\underline{h}}\left( \beta +\frac{1}{\beta }\right) \cdot \omega \left( f,\frac{h}{2}\right) \) and since

and \(\max \limits _{t\in \lbrack 0,\frac{1}{2}]}\left\vert C_{i}\left( x\right) \right\vert =\max \limits _{t\in \lbrack \frac{1}{2},1]}\left\vert A_{i}\left( x\right) \right\vert =\tfrac {1125}{8192}\), the maximum values being attained in \(t=\tfrac {5}{16}\) and \(t=\tfrac {11}{16}\), respectively, by (11) and (12) we obtain,

for all \(x\in \lbrack a,b].\)

If we compare the estimate (13) with the result from Theorem 4.1.3 in [ 9 ] , since \(\max _{t\in \lbrack 0,\frac{1}{2}]}\left\vert t\left( 1-2t\right) \left( 1-t\right) \right\vert =\tfrac {\sqrt{3}}{18}\) and \(\max _{t\in \lbrack 0,\frac{1}{2}]}\left\vert 16t^{4}-28t^{3}+10t^{2}+1\right\vert \) \(=\tfrac {9317}{8192}+\tfrac {1125}{8192}=1.2747\), we see that the presence in (13) of the factor \(\omega ( f,\tfrac {h}{2}) \) instead of \(\omega \left( f,h\right) \) represents an improvement of the result from Theorem 4.1.3 in [ 9 ] .

The endpoint condition \(S^{\prime }\left( a\right) =S^{\prime }\left(b\right) =0\) imposed in [ 10 ] in order to simplify the study of the error estimate in the case \(f\in C[a,b]\) can be replaced by other classical ones. For instance, in that follows we investigate the modification of the error estimate when other two supplementary endpoint conditions are included. In the first case, mentioned in [ 10 ] , we can consider \(S^{\prime }\left( a\right) =f^{\prime }\left( a\right) ,\) \(S^{\prime }\left(b\right) =f^{\prime }\left( b\right) \) and the linear system (1) has the central lines

and the first and the last equations becomes

and

obtaining the same inequality \(\left\vert d_{i}\right\vert \leq \tfrac {4h}{ \underline{h}^{2}}\cdot \omega \left( f,\tfrac {h}{2}\right) ,\) \(i=\overline{2,n-2}\) and

Then, \(\left\Vert m\right\Vert _{\infty }=\max \{ \left\vert m_{i}\right\vert :i=\overline{0,n}\} \leq \tfrac {4}{3}\cdot \left( \tfrac {4h}{\underline{h}^{2}} \cdot \omega \left( f,\tfrac {h}{2}\right) +\tfrac {h\cdot m^{\prime }}{8 \underline{h}^{2}}\right) \) and the error estimate becomes

for all \(x\in \lbrack a,b]\), where \(m^{\prime }=\max \{ \left\vert f^{\prime }\left( a\right) \right\vert ,\left\vert f^{\prime }\left( b\right)\right\vert \} .\)

Taking the natural type endpoint condition \(S^{\prime \prime }\left(a\right) =S^{\prime \prime }\left( b\right) =0\) the linear system (1) receives two supplementary equations

obtaining the estimates \(\left\vert d_{0}\right\vert \leq \tfrac {4}{ \underline{h}}\cdot \omega \left( f,\tfrac {h}{2}\right) \), \(\left\vert d_{n}\right\vert \leq \tfrac {4}{\underline{h}}\cdot \omega \left( f,\frac{h}{2}\right) \) and the interpolation error estimate is the same as in (13).

We can consider now, the second type of complete endpoint conditions \(S^{\prime \prime }\left( a\right) =f^{\prime \prime }\left( a\right) ,\) \(S^{\prime \prime }\left( b\right) =f^{\prime \prime }\left( b\right) \) when the values \(f^{\prime \prime }\left( a\right) \) and \(f^{\prime \prime }\left( b\right) \) are given, the supplementary equations becoming

with \(\left\vert d_{0}\right\vert \leq \frac{4}{\underline{h}}\cdot \omega \left( f,\frac{h}{2}\right) +\frac{h}{8}\left\vert f^{\prime \prime }\left( a\right) \right\vert ,\) and \(\left\vert d_{n}\right\vert \leq \frac{4}{ \underline{h}}\cdot \omega \left(f,\frac{h}{2}\right) +\frac{h}{8} \left\vert f^{\prime \prime }\left( b\right) \right\vert .\) In this case, the error estimate is

where \(m^{\prime \prime }=\max \{ \left\vert f^{\prime \prime }\left(a\right) \right\vert ,\left\vert f^{\prime \prime }\left( b\right) \right\vert \} .\)

3 Optimal properties for deficient quartic splines

3.1 Minimal mean curvature

In this section we consider the deficient quartic spline (2) \(S\in C^{1}[a,b]\) and in this case the spline local derivatives \(m_{i},\) \(i=\overline{0,n}\) remain free. We will determine \(m_{i},\) \(i=\overline{0,n},\) in order to minimize the functionals

According to [ 2 ] , \(\sqrt{J_{1}\left( S\right) }\) is an average of the slope of the graph of \(S\) and \(\sqrt{J_{2}\left( S\right) }\) represents the mean curvature of the graph of \(S\), while \(\sqrt{J_{3}\left( S\right) }\) is related to the mean curvature of the graph of \(S^{\prime }\). Since \(S^{\left( 4\right) }\) is piecewise constant discontinuous function, the minimization of \(J_{4}\left( S\right) \) is without of interest.

By (2) we see that in each interval \([x_{i-1},x_{i}],\) \(i=\overline{1,n}\) we have

and therefore

In order to minimize \(J_{2}\left( S\right) \) the system of normal equations \(\frac{\partial J_{2}}{\partial m_{i}}=0,\) \(i=\overline{0,n},\) is

and after elementary calculus becomes

We see that the matrix \(A^{\prime \prime }\) of this system is strictly diagonally dominant with the index of diagonally dominance 1/6, better than for the system (1), and \(\left\Vert \left( A^{\prime \prime }\right) ^{-1}\right\Vert \leq \frac{6}{5}\)(see [ 19 ] ). Moreover, since \(a_{ii}^{\prime \prime }{\gt}0\) and \(a_{ij}^{\prime \prime }{\lt}0\) for \(i\neq j,\) \(i,j=\overline{0,n}\), we infer that all elements of the matrix \(\left( A^{\prime \prime }\right) ^{-1} \) are nonnegative. The strictly diagonally dominance ensures the existence and uniqueness of the solution of system (16) and the numerical stability of the LU factorization method for solving this system. In this way we obtain the following result.

As above, the system of normal equations associated to the functional \(J_{2}\left( S\right) \) has unique solution due to the strictly diagonally dominance of its matrix and this solution can be obtained by applying the iterative algorithm presented in [ 1 , pp. 14–15 ] , for tridiagonal linear systems. Moreover, by the strictly diagonally dominance of the matrix \(A^{\prime \prime }\) we infer that all the diagonal minors of the Hessian matrix \(\left( \tfrac {\partial ^{2}J_{2}}{\partial m_{i}\partial m_{j}}\right) _{i,j=\overline{0,n}}\) are strictly positive and thus \(\left(m_{0},m_{1},\ldots ,m_{n}\right) \) is a real minimum point of \(J_{2} \left( S\right)\). For obtaining the estimate (17) we observe by (16) that

and

for all \(i=\overline{1,n-1},\) and consequently, \(\left\Vert m\right\Vert _{\infty }=\max \{ \left\vert m_{i}\right\vert :i=\overline{0,n}\} \leq \tfrac {64h}{15\underline{h}^{2}}\cdot \omega \left( f,\tfrac {h}{2}\right)\). Similarly as in the proof of corollary 7 we obtain

Since the solution of (16) minimize \(J_{2}\left( S\right) \) we infer that it minimize the mean curvature \(\sqrt{J_{2}\left( S\right) }\) of the graph of \(S\), too.

Concerning the minimal mean curvature \(\sqrt{J_{3}\left( S\right) }\) of the graph of \(S^{\prime }\), since

the minimization of \(\sqrt{J_{3}\left( S\right) }\) can be obtained by solving the system of normal equations \(\frac{\partial J_{3}}{\partial m_{i}},\) \(i=\overline{0,n}\). After elementary calculus, this system becomes

being diagonally dominant and thus has unique solution that minimizes \(J_{3}\left( S\right) \). Since

\(\forall i=\overline{1,n-1}\) and since the matrix \(A^{\prime \prime \prime }\) of the linear system (18) has the inverse with \(\Vert \left( A^{\prime \prime \prime }\right) ^{-1}\Vert \leq \frac{19}{6}\), for the solution \(m=\left( m_{0},m_{1},\ldots ,m_{n}\right) \) of (18) we obtain the estimate

In this way it obtains the following result.

3.2 Minimal average slope of the graph

In order to establish the paramaters \(m_{i}\), \(i=\overline{0,n}\), of the quartic spline with minimal average slope of the graph \(\sqrt{J_{1}\left( S\right) }\), we minimize the functional \(J_{1}\left( S\right) \) which has the expression

The corresponding system of normal equations \(\frac{\partial J_{1}}{\partial m_{i}}\), \(i=\overline{0,n}\), will be after some computation,

and since the matrix of this system is diagonally dominant, the system (20) has unique solution. In this way we obtain the following result.

Based on the strictly diagonal dominance of the matrix \(A^{\prime }\) of the system (20) we infer that \(\Vert \left(A^{\prime } \right)^{-1}\Vert _{\infty } \leq \tfrac {16}{11}\), this system has unique solution and all the diagonal minors of the Hessian matrix \(\left( \tfrac {\partial ^{2}J_{1}}{\partial m_{i}\partial m_{j}}\right) _{i,j=\overline{0,n}} \) are strictly positive. Then, this solution is the unique extremal point of the functional \(J_{1}\left( S\right) \) and it is a minimum point. In order to obtain the error estimate, firstly we see that

and consequently,

It follows that \(\left\Vert m\right\Vert _{\infty }=\max \{ \left\vert m_{i}\right\vert :i=\overline{0,n}\} \leq \tfrac {42} {11\underline{h}} \cdot \omega ( f,\tfrac {h}{2}) \) obtaining,

4 Numerical experiment

In order to illustrate the theoretical results consider \(n=5\) and the following data presented in section 4.

According to (1), (15) and (16), let \(SC\in C^{2}[0,10]\) be the deficient quartic spline with the endpoint conditions as in (15) and \(SMC\in C^{1}[0,10]\) be the deficient quartic spline with minimal mean curvature of the graph \(\sqrt{J_{2}}\) obtained according to theorem 10. With \(SD\) we denote the deficient quartic spline with minimal average slope of the graph \(\sqrt{J_{1}}\) obtained in theorem 13 after solving the linear system (18), and with \(SDC\) we denote the quartic spline with minimal mean curvature of the graph of the first derivative \(\sqrt{J_{3}}\), according to (20). By solving the linear systems (1)+(15), (16), (18) and (20) we obtain the corresponding local derivatives \(m_{i},\) \(i=\overline{0,5}\), for each of the above mentioned quartic splines, and the results are presented in table 1.

These quartic splines are represented in figure 1 and figure 2 illustrating their interpolation properties. In figure 1 we represent with solid line the quartic spline \(SD\) having minimal average slope of the graph and in dots is plotted the quartic spline \(SC\in C^{2}[0,10]\) obtained by solving the system (1)+(15). The quartic spline \(SMC\) with minimal mean curvature of the graph is represented under solid line in figure 2, while the quartic spline \(SDC\) with minimal mean curvature of the graph of the first derivative is plotted in dots. Investigating ?? we see that smaller oscillation can be observed at the quartic spline \(SD\) with minimal average slope of the graph and at the quartic spline \(SMC\) with minimal mean curvature of the graph, respectively. In order to illustrate this geometric property observed in ?? we compute the length \(L\left( S\right) \) of the graph for this four quartic splines and the results are sumarized in figure 2. As was expected, the deficient quartic spline \(SD\) with minimal average slope of the graph has the smallest graph length.

The graphs and figures were obtained by using the Matlab application.

5 Conclusions

Observing the possibility to express the error estimate of the deficient quartic spline interpolant \(S\in C^{2}[a,b]\) in terms of both \(\omega \left( f,\frac{h}{2}\right) \) and \(\omega \left( f,h\right) \), in the case of interpolated functions \(f\in C[a,b]\), in this work we improve the error estimates from [ 10 ] and [ 9 ] , the results being obtained in theorem 4 and corollary 7. This fact is revealed in remark 5 by observing a smaller constant for equally spaced knots in the case of Lipschitzian interpolated functions. The possibility to determine the local derivatives \(m_{i}\), \(i=\overline{0,n}\), for obtaining certain optimal properties of the deficient quartic spline \(S\in C^{1}[a,b]\), is investigated. In this context, the deficient quartic spline with minimal mean curvature of the graph of \(S\) and \(S^{\prime }\), respectively, are obtained in theorem 10 and corollary 12. Related to the error estimate in terms of the modulus of continuity, a better bound is observed at the deficient quartic spline with minimal average slope of the graph, which is obtained in theorem 13.

The numerical example shows the quality of the interpolation properties of the above presented four quartic splines. The technique of minimizing the mean curvature and the average slope of the graph, presented in this work, can be extended to parametric quartic spline curves, too, and this is the subject of a future work, investigating the use of both the chordal and centripetal parametrization.