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<title>The Akima’s Fitting Method for Quartic Splines: The Akima’s Fitting Method for Quartic Splines</title>
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<h1>The Akima’s Fitting Method for Quartic Splines</h1>
<p class="authors">
<span class="author">Alexandru Mihai Bica\(^\ast \) Diana Curilă-Popescu\({^\ast \ast }\)</span>
</p>
<p class="date">October 2, 2022; accepted: November 15, 2022; published online: December 31, 2022.</p>
</div>
<div class="abstract"><p> For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima’s type. For computing the derivatives on endpoints are also considered alternatives that request optimal properties near the endpoints. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation \(\mathcal{O}(h^{3})\). A numerical experiment is presented for making the comparison between the Akima’s cubic spline and the Akima’s variant quartic spline having deficiency 2 and natural endpoint conditions. </p>
<p><b class="bf">MSC.</b> 65D07, 65D10. </p>
<p><b class="bf">Keywords.</b> Quartic splines, Akima’s fitting spline interpolation procedure, error estimates. </p>
</div>
<p>\(^\ast \)Department of Mathematics and Computer Science, University of Oradea, str. Universităţii no. 1, 410087 Oradea, Romania, e-mail: <span class="ttfamily">abica@uoradea.ro</span> </p>
<p>\(^{\ast \ast }\)Department of Mathematics and Computer Science, University of Oradea, str. Universităţii no. 1, 410087 Oradea, Romania, e-mail: <span class="ttfamily">curila_diana@yahoo.com</span> </p>
<h1 id="a0000000002">1 Introduction</h1>
<p>Before the fundamental work of Schoenberg (see <span class="cite">
	[
	<a href="#sch" >15</a>
	]
</span>) where the notion of B-spline is introduced in explicit way, according to de Boor and Pinkus <span class="cite">
	[
	<a href="#boor" >7</a>
	]
</span>, the first apparition of spline functions can be found in the pioneering works of Popoviciu (see <span class="cite">
	[
	<a href="#popoviciu" >14</a>
	]
</span>) and Chakalov (see <span class="cite">
	[
	<a href="#chakalov" >9</a>
	]
</span>). 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 </p>
<div class="displaymath" id="eq1">
  \begin{align}  \label{eq1} s\left( x\right) = &  \tfrac {\left( x_{i}-x\right) ^{2}[2\left( x-x_{i-1}\right) +h_{i}]}{h_{i}^{3}} y_{i-1}+\tfrac {\left( x-x_{i-1}\right) ^{2}[2\left( x_{i}-x\right) +h_{i}]}{h_{i}^{3}} y_{i} \\ & +\tfrac {\left( x_{i}-x\right) ^{2}\left( x-x_{i-1}\right) }{h_{i}^{2}} m_{i-1}-\tfrac {\left( x-x_{i-1}\right) ^{2}\left( x_{i}-x\right) }{h_{i}^{2}}m_{i}, \qquad x\in \lbrack x_{i-1},x_{i}], \quad i=\overline{1,n} \nonumber \end{align}
</div>
<p>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 <span class="cite">
	[
	<a href="#behalfa" >3</a>
	]
</span> and <span class="cite">
	[
	<a href="#micula" >13</a>
	]
</span>). 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 <span class="cite">
	[
	<a href="#akima" >1</a>
	]
</span> and <span class="cite">
	[
	<a href="#catmul" >8</a>
	]
</span>, or by minimizing a functional related to the data polygon (see <span class="cite">
	[
	<a href="#bica2012" >4</a>
	]
</span> and <span class="cite">
	[
	<a href="#han2018" >11</a>
	]
</span>). </p>
<p>The derivatives \(m_{i},\) \(i=\overline{0,n},\) are computed in <span class="cite">
	[
	<a href="#akima" >1</a>
	]
</span> 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,</p>
<div class="equation" id="eq2">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq2} m_{i}=\tfrac {\left\vert p_{i+1}-p_{i}\right\vert \cdot p_{i-1}+\left\vert p_{i-1}-p_{i-2}\right\vert \cdot p_{i}}{\left\vert p_{i+1}-p_{i}\right\vert +\left\vert p_{i-1}-p_{i-2}\right\vert },\quad i=\overline{2,n-2}. \end{equation}
  </div>
  <span class="equation_label">2</span>
</p>
</div>
<p>In order to extend formula (<a href="#eq2">2</a>) 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 <span class="cite">
	[
	<a href="#bica2012" >4</a>
	]
</span> and <span class="cite">
	[
	<a href="#bica2014" >5</a>
	]
</span>, sometimes, this treatment near endpoints could generate significant oscillations. Therefore, in <span class="cite">
	[
	<a href="#bica2014" >5</a>
	]
</span>, the values of the derivatives on the first two and last two knots are computed by using optimal procedures. </p>
<p>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 <span class="cite">
	[
	<a href="#japprox1989" >12</a>
	]
</span>, which has the following expression on the intervals \(\left[ x_{i-1},x_{i}\right] ,\) \(i=\overline{1,n}\): </p>
<div class="displaymath" id="eq3">
  \begin{align}  \label{eq3} S_{i}\left( x\right) =& \tfrac {\left( x_{i}-x\right) ^{2}\left[ \left( x_{i}-x\right) ^{2}+4\left( x_{i}-x\right) \left( x-x_{i-1}\right) -5\left( x-x_{i-1}\right) ^{2}\right] }{h_{i}^{4}} y_{i-1} \\ & \quad +\tfrac {16\left( x-x_{i-1}\right) ^{2}\left( x_{i}-x\right) ^{2}}{h_{i}^{4}}y_{i-1/2}\nonumber \\ & \quad +\tfrac {\left( x-x_{i-1}\right) ^{2}\left[ \left( x-x_{i-1}\right) ^{2}+4\left( x_{i}-x\right) \left( x-x_{i-1}\right) -5\left( x_{i}-x\right) ^{2}\right] }{h_{i}^{4}} y_{i} \nonumber \\ & \quad +\tfrac {\left( x_{i}-x\right) \left( x-x_{i-1}\right) \left( x_{i-1}+x_{i}-2x\right) \left[ \left( x_{i}-x\right) m_{i-1}+\left( x-x_{i-1}\right) m_{i}\right] }{h_{i}^{3}}\nonumber \\ =& A_{i}\left( x\right) y_{i-1}+B_{i}\left( x\right) y_{i-1/2}+C_{i}\left( x\right) y_{i}+D_{i}\left( x\right) m_{i-1}+E_{i}\left( x\right) m_{i} \nonumber \end{align}
</div>
<p> where \(m_{i}=S^{\prime }\left( x_{i}\right) ,\) \(y_{i}=S\left( x_{i}\right) ,\) \(i=\overline{0,n},\) and \(y_{i-1/2}=S\left( \tfrac {x_{i-1}+x_{i}}{2}\right) , \) \(i=\overline{1,n}.\) </p>
<p>Error estimates in the interpolation with the \(C^{2}\) quartic splines (<a href="#eq4">4</a>) were established in <span class="cite">
	[
	<a href="#japprox1989" >12</a>
	]
</span>, <span class="cite">
	[
	<a href="#dubey2017" >10</a>
	]
</span> and <span class="cite">
	[
	<a href="#volkov" >16</a>
	]
</span>. In <span class="cite">
	[
	<a href="#icata2019" >6</a>
	]
</span>, 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. </p>
<p>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. </p>
<h1 id="a0000000003">2 The construction of the Akima’s type procedure for quartic splines</h1>
<p>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 </p>
<div class="displaymath" id="a0000000004">
  \[  d_{i}=\tfrac {y_{i-1/2}-y_{i-1}}{h_{i}/2}, \quad d_{i-1/2}=\tfrac {y_{i}-y_{i-1/2}}{h_{i}/2}, \quad d_{i+1/2}=\tfrac {y_{i+1/2}-y_{i}}{h_{i+1}/2}, \quad d_{i+1}=\tfrac {y_{i+1}-y_{i+1/2}}{h_{i+1}/2}  \]
</div>
<p> 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 \(\left[ x_{i},x_{i+1}\right] ,\) </p>
<div class="displaymath" id="a0000000005">
  \begin{align*}  p_{i}\left( x\right) = &  \tfrac {2\left( y_{i-1}+y_{i}-2y_{i-1/2}\right) }{h_{i}^{2}} \left( x-x_{i-1}\right) ^{2}+\tfrac {4y_{i-1/2}-y_{i}-3y_{i-1}}{h_{i}} \left( x-x_{i-1}\right) +y_{i-1}, \\ p_{i+1}\left( x\right) = &  \tfrac {2\left( y_{i+1}+y_{i}-2y_{i+1/2}\right) }{h_{i+1}^{2}} \left( x-x_{i}\right) ^{2}+\tfrac {4y_{i+1/2}-y_{i+1}-3y_{i}}{h_{i+1}} \left( x-x_{i}\right) +y_{i}. \end{align*}
</div>
<p>Computing the derivatives of \(p_{i}\) and \(p_{i+1}\) on the point \(x_{i}\) we get </p>
<div class="displaymath" id="a0000000006">
  \begin{align*}  p_{i}^{\prime }\left( x_{i}\right) =&  \tfrac {3y_{i}+y_{i-1}-4y_{i-1/2}}{h_{i}}, \\ p_{i+1}^{\prime }\left( x_{i}\right) =& \tfrac {4y_{i+1/2}-y_{i+1}-3y_{i}}{h_{i+1}}. \end{align*}
</div>
<p>Let</p>
<div class="equation" id="eq4">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq4} \widetilde{y_{i}^{\prime }}=\tfrac {-2h_{i+1}}{h_{i}\left( h_{i}+h_{i+1}\right) } y_{i-1/2}+\tfrac {2\left( h_{i+1}-h_{i}\right) }{h_{i}h_{i+1}} y_{i}+\tfrac {2h_{i}}{h_{i+1}\left( h_{i}+h_{i+1}\right) }y_{i+1/2} \end{equation}
  </div>
  <span class="equation_label">4</span>
</p>
</div>
<p>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 </p>
<div class="displaymath" id="a0000000007">
  \[  T_{-}\left( x_{i}\right) =\tfrac {1}{2}\left( p_{i}^{\prime }\left( x_{i}\right) +\widetilde{y_{i}^{\prime }}\right) \quad {\rm and} \quad T_{+}\left( x_{i}\right) =\tfrac {1}{2}\left( p_{i+1}^{\prime }\left( x_{i}\right) +\widetilde{y_{i}^{\prime }}\right) .  \]
</div>
<p>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</p>
<div class="equation" id="eq5">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq5} m_{i}=\frac{\vert d_{i+1}-d_{i+1/2}\vert \cdot T_{-}\left( x_{i}\right) + \vert d_{i-1/2}-d_{i}\vert \cdot T_{+}\left( x_{i}\right) }{\vert d_{i-1/2}-d_{i}\vert +\vert d_{i+1}-d_{i+1/2}\vert },\quad i=\overline{1,n-1} \end{equation}
  </div>
  <span class="equation_label">5</span>
</p>
</div>
<p>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 </p>
<div class="displaymath" id="a0000000008">
  \[  m_{i}=\tfrac {1}{2}\left( T_{-}\left( x_{i}\right) +T_{+}\left( x_{i}\right) \right) .  \]
</div>
<p>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. </p>
<h1 id="a0000000009">3 The treatment of the endpoints</h1>
<p>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</p>
<div class="equation" id="eq6">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq6} \left\{  \begin{array}{c} 4m_{0}-m_{1}=\frac{1}{h_{1}}\left( -11y_{0}+16y_{1-1/2}-5y_{1}\right) \\ -m_{n-1}+4m_{n}=\frac{1}{h_{n}}\left( 5y_{n-1}-16y_{n-1/2}+11y_{n}\right)\end{array}\right. \end{equation}
  </div>
  <span class="equation_label">6</span>
</p>
</div>
<p>obtaining in this way the Akima’s quartic spline with natural endpoint conditions with \(m_{0}=\tfrac {m_{1}}{4}+\tfrac {-11y_{0}+16y_{1-1/2}-5y_{1}}{4h_{1}},\) \(m_{n}=\tfrac {m_{n-1}}{4}+\tfrac {5y_{n-1}-16y_{n-1/2}+11y_{n}}{4h_{n}}.\) </p>
<p>Another variant is to consider the local optimal condition involving the minimization of the integrals </p>
<div class="displaymath" id="a0000000010">
  \[  \int \limits _{x_{0}}^{x_{1}}\left( S^{\prime \prime }\left( x\right) \right) ^{2}dx \quad {\rm and } \quad \int \limits _{x_{n-1}}^{x_{n}}\left( S^{\prime \prime }\left( x\right) \right) ^{2}dx  \]
</div>
<p> 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 </p>
<div class="displaymath" id="a0000000011">
  \begin{align*} & J_{2}\left( m_{0}\right) =\int \limits _{x_{0}}^{x_{1}}\left( S^{\prime \prime }\left( x\right) \right) ^{2}dx \quad {\rm and }\quad J_{2}\left( m_{n}\right) =\int \limits _{x_{n-1}}^{x_{n}}\left( S^{\prime \prime }\left( x\right) \right) ^{2}dx, \\ & J_{2}\left( m_{0}\right) =\int \limits _{x_{0}}^{x_{1}}\! \Big[A_{1}^{\prime \prime }(x) y_0 \! +\!  B_1^{\prime \prime }(x) y_{1-1/2} \! +\!  C_1^{\prime \prime }(x) y_1 \! +\!  D_1^{\prime \prime }(x) m_0 \! +\!  E_1^{\prime \prime }(x) m_{1}\Big]^{2}dx, \\ & J_{2}\left( m_{n}\right) = \\ & =\int \limits _{x_{n-1}}^{x_{n}}\! \! \Big[A_{n}^{\prime \prime }(x) y_{n-1}\! +\! B_{n}^{\prime \prime }(x) y_{n-1/2}\! +\! C_{n}^{\prime \prime }(x) y_{n}\! +\! D_{n}^{\prime \prime }(x) m_{n-1}\! +\!  E_{n}^{\prime \prime }(x) m_{n}\Big]^{2}dx, \end{align*}
</div>
<p> the system of normal equations \(J_{2}^{\prime }\left( m_{0}\right) =0,\) \(J_{2}^{\prime }\left( m_{n}\right) =0\) being</p>
<div class="displaymath" id="a0000000012">
  \[  \left\{  \begin{array}{c} \frac{36}{5h_{1}}m_{0}-\frac{6}{5h_{1}}m_{1}=-\frac{94}{5h_{1}^{2}} y_{0}+\frac{128}{5h_{1}^{2}} y_{1-1/2}-\tfrac {34}{5h_{1}^{2}} y_{1} \\ -\frac{6}{5h_{n}} m_{n-1}+\frac{36}{5h_{n}} m_{n}=\frac{34}{5h_{n}^{2}} y_{n-1}-\frac{128}{5h_{n}^{2}} y_{n-1/2}+\tfrac {94}{5h_{n}^{2}} y_{n}\end{array}\right.  \]
</div>
<p>Then one obtains </p>
<div class="displaymath" id="a0000000013">
  \begin{align*}  m_{0}=& \tfrac {m_{1}}{6}-\tfrac {47}{18h_{1}}y_{0}+\tfrac {32} {9h_{1}}y_{1-1/2}-\tfrac {17}{18h_{1}}y_{1}, \\ m_{n}=& \tfrac {m_{n-1}}{6}+ \tfrac {17}{18h_{n}}y_{n-1}-\tfrac {32}{9h_{n}}y_{n-1/2}+\tfrac {47}{18h_{n}}y_{n}. \end{align*}
</div>
<p>The following variant is inspired by the idea of the work <span class="cite">
	[
	<a href="#bica2014" >5</a>
	]
</span> but uses the technique from <span class="cite">
	[
	<a href="#han2018" >11</a>
	]
</span>, minimizing the derivative oscillation on the intervals \([x_{0},x_{1}]\) and \([x_{n-1},x_{n}]\). For this purpose we consider the functionals</p>
<div class="displaymath" id="a0000000014">
  \begin{align*} & J_{1}\left( m_{0}\right)= \\ & =\int \limits _{x_{0}}^{x_{1}}\left( S^{\prime }\left( x\right) -\tfrac {y_{1}-y_{0}}{h_{1}}\right) ^{2}dx \\ & =\int \limits _{x_{0}}^{x_{1}}\Big[A_{1}^{\prime }(x) y_{0}\! +\! B_{1}^{\prime }(x) y_{1-1/2}\! +\!  C_{1}^{\prime }(x) y_{1}\! +\! D_{1}^{\prime }\! (x) m_{0}\! +\! E_{1}^{\prime }(x) m_{1}\! -\! \tfrac {y_{1}-y_{0}}{h_{1}}\Big]^{2}dx, \end{align*}
</div>
<div class="displaymath" id="a0000000015">
  \begin{align*} & J_{1}\left( m_{n}\right) = \\ & =\int \limits _{x_{n-1}}^{x_{n}}\left( S^{\prime }\left( x\right) -\tfrac {y_{n}-y_{n-1}}{h_{n}}\right) ^{2}dx \\ & =\! \int \limits _{x_{n-1}}^{x_{n}}\! \! \! \Big[A_{n}^{\prime }\! (x) y_{n-1}\! +\! B_{n}^{\prime }\! (x) y_{n-\frac12}\! +\!  C_{n}^{\prime }\! (x) y_{n}\! +\! D_{n}^{\prime }\! (x) m_{n-1}\! +\! E_{n}^{\prime }\! (x) m_{n}\! -\! \tfrac {y_{n}-y_{n-1}}{h_{n}}\Big]^{2}\! dx. \end{align*}
</div>
<p> By the normal equations \(J_{1}^{\prime }\left( m_{0}\right) =0\), \(J_{1}^{\prime }\left( m_{n}\right) =0\) we get the values </p>
<div class="displaymath" id="a0000000016">
  \begin{align*}  m_{0}=& -\tfrac {5}{16}m_{1}-\tfrac {29}{16h_{1}} y_{0}+\tfrac {1}{h_{1}} y_{1-1/2}+\tfrac {13}{16h_{1}} y_{1} , \\ m_{n}=& -\tfrac {5}{16} m_{n-1}-\tfrac {13}{16h_{n}} y_{n-1}-\tfrac {1}{h_{n}} y_{n-1/2}+\tfrac {29}{16h_{n}} y_{n}. \end{align*}
</div>
<h1 id="a0000000017">4 The interpolation error estimates</h1>
<p>For the derivatives computed by the Akima’s variant (<a href="#eq5">5</a>) we have the estimate </p>
<div class="displaymath" id="a0000000018">
  \[  \left\vert m_{i}\right\vert \leq \tfrac {\left\vert d_{i-1/2}-d_{i}\right\vert \max \{ \left\vert T_{-}\left( x_{i}\right) \right\vert ,\left\vert T_{+}\left( x_{i}\right) \right\vert \} +\left\vert d_{i+1}-d_{i+1/2}\right\vert \max \{ \left\vert T_{-}\left( x_{i}\right) \right\vert ,\left\vert T_{+}\left( x_{i}\right) \right\vert \} }{\left\vert d_{i-1/2}-d_{i}\right\vert +\left\vert d_{i+1}-d_{i+1/2}\right\vert }  \]
</div>
<p> 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</p>
<div class="displaymath" id="a0000000019">
  \begin{align*}  \left\vert p_{i}^{\prime }\left( x_{i}\right) \right\vert \leq & \tfrac {1}{h_{i}}\left( 3\vert y_{i}-y_{i-1/2}\vert +\vert y_{i-1}-y_{i-1/2}\vert \right) \leq \tfrac {4}{\underline{h}} \omega \left( y,\tfrac {h}{2}\right) \\ \left\vert p_{i+1}^{\prime }\left( x_{i}\right) \right\vert \leq & \tfrac {1}{h_{i+1}}\left( 3\vert y_{i+1/2}-y_{i}\vert +\vert y_{i+1/2}-y_{i+1}\vert \right) \leq \tfrac {4}{\underline{h}} \omega \left( y,\tfrac {h}{2}\right) \end{align*}
</div>
<p> 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 (<a href="#eq4">4</a>) we infer that </p>
<div class="displaymath" id="a0000000020">
  \[  \vert \widetilde{y_{i}^{\prime }}\vert \leq \tfrac {-2h_{i+1}}{h_{i}\left( h_{i}+h_{i+1}\right) }\vert y_{i}-y_{i-1/2}\vert +\tfrac {2h_{i}}{h_{i+1}\left( h_{i}+h_{i+1}\right) }\vert y_{i+1/2}-y_{i}\vert \leq \tfrac {2h}{\underline{h}^{2}} \omega \left( y,\tfrac {h}{2}\right)  \]
</div>
<p>and thus </p>
<div class="displaymath" id="a0000000021">
  \[  \left\vert T_{-}\left( x_{i}\right) \right\vert \leq \tfrac {3h}{\underline{h}^{2}} \omega \left( y,\tfrac {h}{2}\right) , \qquad \left\vert T_{+}\left( x_{i}\right) \right\vert \leq \tfrac {3h}{\underline{h}^{2}} \omega \left( y,\tfrac {h}{2}\right) .  \]
</div>
<p> Consequently, it obtains the estimate </p>
<div class="displaymath" id="a0000000022">
  \[  \left\vert m_{i}\right\vert \leq \tfrac {3h}{\underline{h}^{2}} \omega \left( y,\tfrac {h}{2}\right) ,\quad \forall i=\overline{1,n-1}.  \]
</div>
<p>In the case of the Akima quartic spline with natural type endpoint conditions we obtain the estimate</p>
<div class="equation" id="eq7">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq7} \left\vert m_{0}\right\vert \leq \tfrac {\left\vert m_{1}\right\vert }{4}+\tfrac {1}{4h_{1}}\left( 11\vert y_{1-1/2}-y_{0}\vert \right) +5\vert y_{1}-y_{1-1/2}\vert )\leq \tfrac {19}{4\underline{h}} \omega \left( y,\tfrac {h}{2}\right) \end{equation}
  </div>
  <span class="equation_label">7</span>
</p>
</div>
<p>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</p>
<div class="equation" id="eq8">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq8} \left\vert m_{n}\right\vert \leq \tfrac {\left\vert m_{n-1}\right\vert }{6}+\tfrac {17}{18h_{n}}\vert y_{n-1}-y_{n-1/2}\vert +\tfrac {47}{18h_{n}}\vert y_{n}-y_{n-1/2}\vert \leq \tfrac {73}{18\underline{h}} \omega \left( y,\tfrac {h}{2}\right) \end{equation}
  </div>
  <span class="equation_label">8</span>
</p>
</div>
<p>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,</p>
<div class="equation" id="eq9">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq9} \left\vert m_{0}\right\vert \leq \tfrac {5\left\vert m_{1}\right\vert }{16}+\tfrac {29}{16h_{1}}\vert y_{1-1/2}-y_{0}\vert +\tfrac {13}{16h_{1}}\vert y_{1}-y_{1-1/2}\vert \leq \tfrac {57}{16\underline{h}} \omega \left( y,\tfrac {h}{2}\right) \end{equation}
  </div>
  <span class="equation_label">9</span>
</p>
</div>
<p>and \(\left\vert m_{n}\right\vert \leq \tfrac {57}{16\underline{h}} \omega \left( y,\tfrac {h}{2}\right) .\) </p>
<p>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 <span class="cite">
	[
	<a href="#icata2019" >6</a>
	]
</span>, obtaining</p>
<div class="displaymath" id="a0000000023">
  \begin{align*}  \left\vert S\left( x\right) -y\left( x\right) \right\vert & \leq \max \limits _{x\in \lbrack x_{i-1},x_{i-1/2}]}\left\vert \! A_{i}\! \left( x\right) \! +\! B_{i}\! \left( x\right) \right\vert \max \big\{ \left\vert y_{i-1}-y\left( x\right) \right\vert ,\vert y_{i/2}-y(x) \vert \big\}  \\ & \quad +\max \limits _{x\in [ x_{i-1},x_{i-1/2}]}\left\vert C_{i}( x) \right\vert \cdot \left\vert y_{i}-y\left( x\right) \right\vert + \\ & \quad +\max \limits _{x\in \lbrack x_{i-1},x_{i-1/2}]}\left\vert D_{i}\left( x\right) +E_{i}\left( x\right) \right\vert \max \big\{ \left\vert m_{i}\right\vert :i=\overline{0,n}\big\}  \end{align*}
</div>
<p> 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</p>
<div class="displaymath" id="a0000000024">
  \[  \left\vert S\left( x\right) -y\left( x\right) \right\vert \leq \tfrac {9317}{8192} \omega \left( y,\tfrac {h}{2}\right) +\tfrac {1125}{8192} \omega \left( y,h\right) +\tfrac {\sqrt{3}h_{i}}{18} \max \Big\{ \vert m_{i}\vert :i=\overline{0,n}\Big\}   \]
</div>
<p>for \(x\in [ x_{i-1},x_{i}],\) \(i=\overline{1,n}.\) Thus, concerning the interpolation error estimate, we obtain the following result. </p>
<p><div class="theorem_thmwrapper " id="th1">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">1</span>
  </div>
  <div class="theorem_thmcontent">
  <p>On the interval \([x_{1},x_{n-1}]\) the error estimate for the Akima’s interpolating quartic spline, in terms of the modulus of continuity, is</p>
<div class="equation" id="eq10">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq10} \left\vert S\left( x\right) -y\left( x\right) \right\vert \leq \left( \tfrac {9317}{8192}+\tfrac {\sqrt{3}h^{2}}{6\underline{h}^{2}}\right) \omega \left( y,\tfrac {h}{2}\right) +\tfrac {1125}{8192}\omega \left( y,h\right) . \end{equation}
  </div>
  <span class="equation_label">10</span>
</p>
</div>
<p>On the intervals \([x_{0},x_{1}]\) and \([x_{n-1},x_{n}]\) the error estimates are:</p>
<div class="equation" id="eq11">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq11} \left\vert S\left( x\right) -y\left( x\right) \right\vert \leq \left( \tfrac {9317}{8192}+\tfrac {\sqrt{3}}{18}\left( \tfrac {3h^{2}}{\underline{h}^{2}}+\tfrac {19h}{4\underline{h}}\right) \right) \omega \left( y,\tfrac {h}{2}\right) +\tfrac {1125}{8192}\omega \left( y,h\right) \end{equation}
  </div>
  <span class="equation_label">11</span>
</p>
</div>
<p>for taking the endpoint conditions \(S^{\prime \prime }(a)=S^{\prime \prime }(b)=0,\)</p>
<div class="equation" id="eq12">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq12} \left\vert S\left( x\right) -y\left( x\right) \right\vert \leq \left( \tfrac {9317}{8192}+\tfrac {\sqrt{3}}{18}\left( \tfrac {3h^{2}}{\underline{h}^{2}}+\tfrac {73h}{18\underline{h}}\right) \right) \omega \left( y,\tfrac {h}{2}\right) +\tfrac {1125}{8192}\omega \left( y,h\right) \end{equation}
  </div>
  <span class="equation_label">12</span>
</p>
</div>
<p>when have minimal curvature on the endpoint intervals, and</p>
<div class="equation" id="eq13">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq13} \left\vert S\left( x\right) -y\left( x\right) \right\vert \leq \left( \tfrac {9317}{8192}+\tfrac {\sqrt{3}}{18}\left( \tfrac {3h^{2}}{\underline{h}^{2}}+\tfrac {57h}{16\underline{h}}\right) \right) \omega \left( y,\tfrac {h}{2}\right) +\tfrac {1125}{8192}\omega \left( y,h\right) \end{equation}
  </div>
  <span class="equation_label">13</span>
</p>
</div>
<p>in the case of minimal derivative oscillation near endpoints. </p>

  </div>
</div> </p>
<p>In contrast with the case of cubic splines, by comparing (<a href="#eq7">7</a>)–(<a href="#eq8">8</a>) and (<a href="#eq12">12</a>)–(<a href="#eq13">13</a>), we see that for the quartic splines (<a href="#eq3">3</a>) 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. </p>
<h1 id="a0000000025">5 Error estimates for smooth functions</h1>
<p>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. </p>
<p><div class="lemma_thmwrapper " id="le1">
  <div class="lemma_thmheading">
    <span class="lemma_thmcaption">
    Lemma
    </span>
    <span class="lemma_thmlabel">2</span>
  </div>
  <div class="lemma_thmcontent">
  <p>If \(f\in C^{4}[a,b]\) with Lipschitzian fourth order derivative and if \(H_{4}\left( f\right) \) is the Hermite interpolation polynomial generated by the interpolation conditions given for \(f\left( a\right) ,\) \(f\left( \tfrac {a+b}{2}\right) \), \(f\left( b\right) \), \(f^{\prime }\left( a\right) \), \(f^{\prime }\left( b\right) \), then the error estimate is</p>
<div class="equation" id="eq14">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq14} \left\vert H_{4}\left( f\right) \left( x\right) -f\left( x\right) \right\vert \leq \tfrac {\sqrt{5}\left( b-a\right) ^{5}L}{30000},\quad \forall x\in \lbrack a,b] \end{equation}
  </div>
  <span class="equation_label">14</span>
</p>
</div>
<p>where \(L\) is the Lipschitz constant of \(f^{\left( 4\right) }.\) </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000026">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Consider the fundamental polynomial </p>
<div class="displaymath" id="a0000000027">
  \[  u\left( x\right) =\left( x-a\right)^{2} \left( x-\tfrac {a+b}{2}\right) \left( x-b\right)^{2}  \]
</div>
<p> and for arbitrary fixed \(x\in \lbrack a,b]\) we define \(\varphi _{x}:[a,b]\rightarrow \mathbb {R},\) by </p>
<div class="displaymath" id="a0000000028">
  \[  \varphi _{x}\left( t\right) =\left\vert \begin{array}{cc} u\left( t\right) &  R\left( t\right) \\ u\left( x\right) &  R\left( x\right) \end{array}\right\vert  \]
</div>
<p>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 </p>
<div class="displaymath" id="a0000000029">
  \[  5!\left( v-w\right) R\left( x\right) -u\left( x\right) \left( f^{\left( 4\right) }\left( v\right) -f^{\left( 4\right) }\left( w\right) \right) =0  \]
</div>
<p> and thus,</p>
<div class="displaymath" id="a0000000030">
  \[  \left\vert R\left( x\right) \right\vert =\tfrac {\left\vert u\left( x\right) \right\vert \cdot \left\vert f^{\left( 4\right) }\left( v\right) -f^{\left( 4\right) }\left( w\right) \right\vert }{5!\left\vert v-w\right\vert }\leq \tfrac {L}{5!}\max \limits _{x\in \lbrack a,b]}\left\vert u\left( x\right) \right\vert  \]
</div>
<p>obtaining (<a href="#eq14">14</a>). <div class="proof_wrapper" id="a0000000031">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>Of course, if \(f\in C^{5}[a,b]\), then \(L=\Vert f^{\left( 5\right) }\Vert _{\infty }\) in (<a href="#eq14">14</a>). 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 </p>
<div class="displaymath" id="a0000000032">
  \[  f^{\prime }\left( x\right) -H_{4}\left( f\right) ^{\prime }\left( x\right) =R^{\prime }\left( x\right) =\tfrac {u^{\prime }\left( x\right) \left( f^{\left( 4\right) }\left( v\right) -f^{\left( 4\right) }\left( w\right) \right) }{5!\left( v-w\right) }.  \]
</div>
<p>Consequently, it obtains</p>
<div class="equation" id="eq15">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq15} \left\vert H_{4}\left( f\right) ^{\prime }\left( x\right) -f^{\prime }\left( x\right) \right\vert \leq \tfrac {L\left( b-a\right) ^{4}}{1920},\quad \forall x\in \lbrack a,b]. \end{equation}
  </div>
  <span class="equation_label">15</span>
</p>
</div>
<p>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</p>
<div class="displaymath" id="a0000000033">
  \[  m_{0}=\tfrac {-3y_{0}+4y_{1-1/2}-y_{1}}{h_{1}},\quad m_{n}=\tfrac {y_{n-1}-4y_{n-1/2}+3y_{n}}{h_{n}}  \]
</div>
<p>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 <span class="cite">
	[
	<a href="#beh" >2</a>
	]
</span>. 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. </p>
<p><div class="theorem_thmwrapper " id="th2">
  <div class="theorem_thmheading">
    <span class="theorem_thmcaption">
    Theorem
    </span>
    <span class="theorem_thmlabel">3</span>
  </div>
  <div class="theorem_thmcontent">
  <p>If \(f\in C^{4}[a,b]\) with Lipschitzian fourth order derivative and \(S\in C^{1}[a,b]\) is the Akima’s variant quartic spline interpolating \(f\), then the error estimates are</p>
<div class="equation" id="eq16">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq16} \left\vert S\left( x\right) -f\left( x\right) \right\vert \leq \left\{  \begin{array}{c} \frac{h^{3}\sqrt{3}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }}{288}+\frac{Lh^{5}\sqrt{5}}{30000},\text{ }x\in \lbrack x_{1},x_{n-1}] \\ \frac{h^{3}\sqrt{3}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }}{54}+\frac{Lh^{5}\sqrt{5}}{30000},\text{ }x\in \lbrack x_{0},x_{1}]\cup \lbrack x_{n-1},x_{n}]\end{array}\right. \end{equation}
  </div>
  <span class="equation_label">16</span>
</p>
</div>
<p>and</p>
<div class="equation" id="eq17">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq17} \left\vert S^{\prime }\left( x\right) -f^{\prime }\left( x\right) \right\vert \leq \left\{  \begin{array}{c} \frac{h^{2}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }}{8}+\frac{Lh^{4}}{1920},\text{ }x\in \lbrack x_{1},x_{n-1}] \\ \frac{2h^{2}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }}{3}+\frac{Lh^{4}}{1920},\text{ }x\in \lbrack x_{0},x_{1}]\cup \lbrack x_{n-1},x_{n}]\end{array}\right. \end{equation}
  </div>
  <span class="equation_label">17</span>
</p>
</div>
<p>where \(L\) is the Lipschitz constant of \(f^{\left( 4\right) }\) and \(\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }=\max \limits _{x\in \lbrack a,b]}\left\vert f^{\prime \prime \prime }\left( x\right) \right\vert \). </p>

  </div>
</div> </p>
<p><div class="proof_wrapper" id="a0000000034">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> Consider \(H\left( f\right) \) be the Hermite type piecewise quartic polynomial interpolating on each interval \([x_{i-1},x_{i}]\) the values \(y_{i-1},\) \(y_{i-1/2},\) \(y_{i},\) \(f^{\prime }\left( x_{i-1}\right) ,\) \(f^{\prime }\left( x_{i}\right) ,\) \(i=\overline{1,n}\), and by (<a href="#eq14">14</a>) we get</p>
<div class="displaymath" id="a0000000035">
  \[  \left\vert H\left( f\right) \left( x\right) -f\left( x\right) \right\vert \leq \tfrac {L\sqrt{5}h_{i}^{5}}{30000}\leq \tfrac {L\sqrt{5}h^{5}}{30000},\quad x\in \lbrack x_{i-1},x_{i}].  \]
</div>
<p>Now, by (<a href="#eq4">4</a>) 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</p>
<div class="displaymath" id="a0000000036">
  \begin{align*}  \left\vert f^{\prime }\left( x_{i}\right) -\widetilde{y_{i}^{\prime }}\right\vert = &  \tfrac {h_{i}h_{i+1}}{4}\cdot \tfrac {\left\vert f^{\prime \prime \prime }\left( c_{i}\right) \right\vert }{3!} \\ \left\vert f^{\prime }\left( x_{i}\right) -p_{i}^{\prime }\left( x_{i}\right) \right\vert = &  \tfrac {h_{i}^{2}}{2}\cdot \tfrac {\left\vert f^{\prime \prime \prime }\left( \eta _{i}\right) \right\vert }{3!} ,\quad i=\overline{1,n-1}. \\ \left\vert f^{\prime }\left( x_{i}\right) -p_{i+1}^{\prime }\left( x_{i}\right) \right\vert = &  \tfrac {h_{i+1}^{2}}{2}\cdot \tfrac {\left\vert f^{\prime \prime \prime }\left( \theta _{i}\right) \right\vert }{3!}\end{align*}
</div>
<p> Consequently,</p>
<div class="displaymath" id="a0000000037">
  \begin{align*}  \left\vert T_{-}\left( x_{i}\right) -f^{\prime }\left( x_{i}\right) \right\vert & =\tfrac {1}{2}\left\vert p_{i}^{\prime }\left( x_{i}\right) -f^{\prime }\left( x_{i}\right) +\left( \widetilde{y_{i}^{\prime }}-f^{\prime }\left( x_{i}\right) \right) \right\vert \\ & \leq \tfrac {1}{2}\left( \tfrac {h_{i}^{2}\left\vert f^{\prime \prime \prime }\left( \eta _{i}\right) \right\vert }{12}+\tfrac {h_{i}h_{i+1}\left\vert f^{\prime \prime \prime }\left( c_{i}\right) \right\vert }{24}\right) \leq \tfrac {h^{2}}{16}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }, \end{align*}
</div>
<div class="displaymath" id="a0000000038">
  \begin{align*}  \left\vert T_{+}\left( x_{i}\right) -f^{\prime }\left( x_{i}\right) \right\vert & =\tfrac {1}{2}\left\vert p_{i+1}^{\prime }\left( x_{i}\right) -f^{\prime }\left( x_{i}\right) +\left( \widetilde{y_{i}^{\prime }}-f^{\prime }\left( x_{i}\right) \right) \right\vert \\ & \leq \tfrac {1}{2}\left( \tfrac {h_{i+1}^{2}\left\vert f^{\prime \prime \prime }\left( \theta _{i}\right) \right\vert }{12}+\tfrac {h_{i}h_{i+1}\left\vert f^{\prime \prime \prime }\left( c_{i}\right) \right\vert }{24}\right) \leq \tfrac {h^{2}}{16}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }, \end{align*}
</div>
<p> for all \(i=\overline{1,n-1}\), and by (<a href="#eq5">5</a>) we obtain</p>
<div class="equation" id="eq18">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq18} \left\vert m_{i}-f^{\prime }\left( x_{i}\right) \right\vert \leq \tfrac {h^{2}}{16}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty },\quad i=\overline{1,n-1}. \end{equation}
  </div>
  <span class="equation_label">18</span>
</p>
</div>
<p>On the other hand, at endpoints by the same Lagrange differentiation formula it obtains</p>
<div class="equation" id="eq19">
<p>
  <div class="equation_content">
    \begin{equation}  \label{eq19} \left\vert f^{\prime }\left( x_{0}\right) -m_{0}\right\vert \leq \tfrac {h^{2}}{3}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty },\quad \left\vert f^{\prime }\left( x_{n}\right) -m_{n}\right\vert \leq \tfrac {h^{2}}{3}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }. \end{equation}
  </div>
  <span class="equation_label">19</span>
</p>
</div>
<p>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 (<a href="#eq18">18</a>) and (<a href="#eq19">19</a>) we get</p>
<div class="displaymath" id="a0000000039">
  \begin{align*}  \left\vert S\left( x\right) -H\left( f\right) \left( x\right) \right\vert \leq & \max \limits _{x\in \lbrack x_{i-1},x_{i-1/2}]\cup \lbrack x_{i-1/2},x_{i}]}\left\vert D_{i}\left( x\right) +E_{i}\left( x\right) \right\vert \cdot \left\vert m_{i}-f^{\prime }\left( x_{i}\right) \right\vert \\ \leq &  \tfrac {h_{i}\sqrt{3}}{18}\cdot \tfrac {h^{2}}{16}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }\leq \tfrac {h^{3}\sqrt{3}}{288}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty },\quad x\in \lbrack x_{1},x_{n-1}] \end{align*}
</div>
<p> and</p>
<div class="displaymath" id="a0000000040">
  \[  \left\vert S\left( x\right) -H\left( f\right) \left( x\right) \right\vert \leq \tfrac {h_{i}\sqrt{3}}{18}\cdot \tfrac {h^{2}}{3}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }\leq \tfrac {h^{3}\sqrt{3}}{54}\cdot \left\Vert f^{\prime \prime \prime }\right\Vert _{\infty }  \]
</div>
<p>for \(x\in \lbrack x_{0},x_{1}]\cup \lbrack x_{n-1},x_{n}]\), obtaining (<a href="#eq16">16</a>). Finally, by (<a href="#eq15">15</a>) we get </p>
<div class="displaymath" id="a0000000041">
  \[  \left\vert H\left( f\right) ^{\prime }\left( x\right) -f^{\prime }\left( x\right) \right\vert \leq \tfrac {Lh^{4}}{1920} \quad \textrm{ for all}\  \  x\in \lbrack a,b]  \]
</div>
<p> and since </p>
<div class="displaymath" id="a0000000042">
  \[  \max \limits _{x\in \lbrack x_{i-1},x_{i}]}\left\vert D_{i}^{\prime }\left( x\right) \right\vert =\max \limits _{x\in \lbrack x_{i-1},x_{i}]}\left\vert E_{i}^{\prime }\left( x\right) \right\vert =1 \quad \textrm{for}\   i=\overline{1,n},  \]
</div>
<p> by (<a href="#eq18">18</a>) and (<a href="#eq19">19</a>) we obtain</p>
<div class="displaymath" id="a0000000043">
  \begin{align*}  \left\vert S^{\prime }\left( x\right) -H^{\prime }\left( f\right) \left( x\right) \right\vert \leq &  \tfrac {h^{2}}{8}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty },\quad x\in \lbrack x_{1},x_{n-1}] \\ \left\vert S^{\prime }\left( x\right) -H^{\prime }\left( f\right) \left( x\right) \right\vert \leq &  \tfrac {2h^{2}}{3}\left\Vert f^{\prime \prime \prime }\right\Vert _{\infty },\quad x\in \lbrack x_{0},x_{1}]\cup \lbrack x_{n-1},x_{n}], \end{align*}
</div>
<p> and the estimate (<a href="#eq17">17</a>) follows. <div class="proof_wrapper" id="a0000000044">
  <div class="proof_heading">
    <span class="proof_caption">
    Proof
    </span>
    <span class="expand-proof">â–¼</span>
  </div>
  <div class="proof_content">
  
  </div>
</div> </p>
<p>In (<a href="#eq16">16</a>) we see that the order of approximation by the Akima’s variant quartic spline is \(O\left( h^{3}\right) \) and, as in <span class="cite">
	[
	<a href="#japprox1989" >12</a>
	]
</span> and <span class="cite">
	[
	<a href="#volkov" >16</a>
	]
</span>, we have obtained the corresponding estimate of \(\left\Vert S^{\prime }-f^{\prime }\right\Vert _{\infty }\) for the derivative \(S^{\prime }\), too. </p>
<h1 id="a0000000045">6 Numerical experiment</h1>
<p>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 (<a href="#eq5">5</a>), 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 ??. </p>
<div class="table"  id="tab:my_label">
   <div class="centered"><small class="footnotesize"><table class="tabular">
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(m_{1}=6.583\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(m_{2}=9.95\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(m_{3}=-12.286\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(m_{4}=16.235\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>end-cond.: </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(s^{\prime \prime }\left( a\right) =s^{\prime \prime }\left( b\right) =0\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> min \(J_{2}\left( m_{0},m_{n}\right) \) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> min \(J_{1}\left( m_{0},m_{n}\right) \) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black" 
        rowspan=""
        colspan="">
      <p>\(m_{0}:\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(-8.854\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(-7.9\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black" 
        rowspan=""
        colspan="">
      <p> \(-2.43\) </p>

    </td>
  </tr>
  <tr>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-left:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p>\(m_{5}:\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(-8.94125\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(-8.183\) </p>

    </td>
    <td  style="border-top-style:solid; border-top-color:black; border-top-width:1px; text-align:left; border-right:1px solid black; border-bottom-style:solid; border-bottom-color:black; border-bottom-width:1px" 
        rowspan=""
        colspan="">
      <p> \(-3.8234\) </p>

    </td>
  </tr>
</table> </small> <figcaption>
  <span class="caption_title">Table</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Numerical results</span> 
</figcaption>  </div>
</div>
<p>In <a href="#fig:1">figure 1</a> 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 <a href="#fig:2">figure 2</a> 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 \(\left( x_{0},y_{0}\right) ,\ldots ,\left( x_{5},y_{5}\right) \). </p>
<figure id="fig:1">
  <div class="centered"> <img src="img-0001.png" alt="\includegraphics[height=5.7cm, width=9cm]{Figure1.png}" style="height:5.7cm; width:9cm" />
 </div>
<figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">1</span> 
  <span class="caption_text">Quartic splines, AD ( ... ), EN ( – )</span> 
</figcaption>


</figure>
<figure id="fig:2">
  <div class="centered"> <img src="img-0002.png" alt="\includegraphics[height=5.7cm, width=9cm]{Figure2.png}" style="height:5.7cm; width:9cm" />
 </div>
<figcaption>
  <span class="caption_title">Figure</span> 
  <span class="caption_ref">2</span> 
  <span class="caption_text">Quartic and Akima splines, CM ( ... ), AK ( – )</span> 
</figcaption>


</figure>
<p><br /></p>
<p><small class="footnotesize">  </small></p>
<div class="bibliography">
<h1>Bibliography</h1>
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</dd>
  <dt><a name="behalfa">3</a></dt>
  <dd><p><a href ="https://doi.org/10.1093/imamat/23.3.355"> <i class="sc">G.H. Behforooz</i>, <i class="sc">N. Papamichel</i>, <i class="it">End conditions for cubic spline interpolation</i>, J. Inst. Math. Appl., <b class="bf">23</b> (1979), pp.&#160;355–366. <a href="https://doi.org/10.1093/imamat/23.3.355">https://doi.org/10.1093/imamat/23.3.355</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dd><p><a href ="https://doi.org/10.1016/j.aml.2012.04.016"> <i class="sc">A.M. Bica</i>, <i class="it">Fitting data using optimal Hermite type cubic interpolating splines</i>, Appl. Math. Lett., <b class="bf">25</b> (2012), pp.&#160;2047–2051. <a href="https://doi.org/10.1016/j.aml.2012.04.016">https://doi.org/10.1016/j.aml.2012.04.016</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
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  <dd><p><a href ="https://doi.org/10.1016/j.cagd.2014.03.001"> <i class="sc">A.M. Bica</i>, <i class="it">Optimizing at the end-points the Akima’s interpolation method of smooth curve fitting</i>, Comput. Aided Geom. Design, <b class="bf">31</b> (2014), pp.&#160;245–257. <a href="https://doi.org/10.1016/j.cagd.2014.03.001">https://doi.org/10.1016/j.cagd.2014.03.001</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="icata2019">6</a></dt>
  <dd><p><a href ="https://doi.org/10.33993/jnaat492-1228"> <i class="sc">A.M. Bica</i>, <i class="sc">D. Curilă-Popescu</i>, <i class="sc">M. Curilă</i>, <i class="it">Optimal properties for deficient quartic splines of Marsden type</i>, J. Numer. Anal. Approx. Theory, <b class="bf">49</b> (2020) no. 2, pp.&#160;113–130. <a href="https://doi.org/10.33993/jnaat492-1228">https://doi.org/10.33993/jnaat492-1228</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
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  <dd><p><a href ="https://doi.org/10.1016/S0021-9045(03)00117-5"> <i class="sc">C. de Boor</i>, <i class="sc">A. Pinkus</i>, <i class="it">The B-spline recurrence relations of Chakalov and of Popoviciu</i>, J. Approx. Theory, <b class="bf">124</b> (2003), pp.&#160;115–123. <a href="https://doi.org/10.1016/S0021-9045(03)00117-5">https://doi.org/10.1016/S0021-9045(03)00117-5</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="catmul">8</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/B978-0-12-079050-0.50020-5"> <i class="sc">E. Catmul</i>, <i class="sc">R. Rom</i>, <i class="it">A class of local interpolating splines</i>, Computer Aided Geometric Design, pp.&#160; 317–326, Barnhill, R.E., Reisenfeld, R.F. (Eds.), Academic Press, New York, 1974. <a href="https://doi.org/10.1016/B978-0-12-079050-0.50020-5">https://doi.org/10.1016/B978-0-12-079050-0.50020-5</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="chakalov">9</a></dt>
  <dd><p><i class="sc">L. Chakalov</i>, <i class="it">On a certain presentation of the Newton divided differences in interpolation theory and its applications</i>, Annuaire Univ. Sofia, Fiz. Mat. Fakultet, <b class="bf">34</b> (1938), pp.&#160;353–394 (in Bulgarian). </p>
</dd>
  <dt><a name="dubey2017">10</a></dt>
  <dd><p><i class="sc">S. Dubey</i>, <i class="sc">Y.P. Dubey</i>, <i class="it">Convergence of C2 deficient quartic spline interpolation</i>, Adv. Comput. Sci. Technol., <b class="bf">10</b> (2017) no. 4, pp.&#160;519–527. </p>
</dd>
  <dt><a name="han2018">11</a></dt>
  <dd><p><a href ="https://doi.org/10.1016/j.cam.2017.09.049"> <i class="sc">X. Han</i>, <i class="sc">X. Guo</i>, <i class="it">Cubic Hermite interpolation with minimal derivative oscillation</i>, J. Comput. Appl. Math., <b class="bf">331</b> (2018), pp.&#160;82–87. <a href="https://doi.org/10.1016/j.cam.2017.09.049">https://doi.org/10.1016/j.cam.2017.09.049</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
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  <dd><p><a href ="https://doi.org/10.1016/0021-9045(89)90008-7"> <i class="sc">G. Howell</i>, <i class="sc">A.K. Varma</i>, <i class="it">Best error bounds for quartic spline interpolation</i>, J. Approx. Theory, <b class="bf">58</b> (1989), pp.&#160;58–67. <a href="https://doi.org/10.1016/0021-9045(89)90008-7">https://doi.org/10.1016/0021-9045(89)90008-7</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
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  <dd><p><i class="sc">G. Micula</i>, <i class="sc">S. Micula</i>, <i class="it">Handbook of splines. Mathematics and its Applications</i>, <b class="bf">462</b>, Kluwer Academic Publishers, Dordrecht, 1999. </p>
</dd>
  <dt><a name="popoviciu">14</a></dt>
  <dd><p><a href ="https://www.jstor.org/stable/43769764"> <i class="sc">T. Popoviciu</i>, <i class="it">Sur le prolongement des fonctions convexes d’ordre superieur</i>, Bull. Math. Soc. Roum. Sci., <b class="bf">36</b> (1934), pp.&#160;75–108. <a href="https://www.jstor.org/stable/43769764">https://www.jstor.org/stable/43769764</a> <img src="img-0003.png" alt="\includegraphics[scale=0.1]{ext-link.png}" style="width:12.0px; height:10.700000000000001px" />
</a> </p>
</dd>
  <dt><a name="sch">15</a></dt>
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</a> </p>
</dd>
  <dt><a name="volkov">16</a></dt>
  <dd><p><i class="sc">Y.S. Volkov</i>, <i class="it">Best error bounds for the derivative of a quartic interpolation spline</i>, Siberian Adv. Math., <b class="bf">9</b> (1999) no. 2, pp.&#160;140–150. </p>
</dd>
</dl>


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