Shape-preserving piecewise rational interpolant with quartic numerator and quadratic denominator

doi:10.1016/j.amc.2014.11.067

Applied Mathematics and Computation

Volume 251, 15 January 2015, Pages 258-274

https://doi.org/10.1016/j.amc.2014.11.067 Get rights and content

Abstract

An explicit representation of a piecewise rational interpolant with quartic numerator and quadratic denominator is presented. For positivity data, monotone data and convex data, the shape-preserving properties of the interpolant are given. The interpolant is $C^{2}$ continuous spline with a shape parameter $w_{i}$ on each subinterval. The values of $w_{i}$ to guarantee shape preservation are estimated. A convergence analysis establishes an error bound in terms of $w_{i}$ and shows that the interpolant is $O (h^{2})$ or $O (h^{3})$ accurate. Several examples are supplied to support the practical value of the given interpolation method.

Introduction

In some practical situations it is often required to generate a smooth function that interpolates a prescribed set of data. An interpolant is often needed to preserve certain geometric shape properties of the data such as monotonicity or convexity. At the same time, the interpolant is wanted to be $C^{1}$ or $C^{2}$ continuous.

There are many effective methods for the construction of shape-preserving interpolants. Some methods are appropriate for monotonicity-preserving, see [10], [16], [17], [26], [32]. Some methods are appropriate for convexity-preserving, see [3], [4], [5], [12], [18], [22], [36], [37]. There are some methods available for the construction of a $C^{1}$ shape-preserving interpolant. On the other hand, it is a more difficult task to construct a $C^{2}$ shape-preserving interpolant, see [11], [13], [14], [18], [34], [37].

Some polynomial spline methods have a common feature in that no additional knots need to be supplied, see [1], [2], [6], [8], [9], [14], [16], [20], [21], [23], [33], [35]. In contrast, the papers [15], [26], [27], [34], [37], [38] discussed the methods by adding one or two additional knots on the subinterval so that the monotonicity or convexity of the data is preserved. In [38], it was mentioned that the user could interactively adjust the slopes and knot locations in order to alter the shape of the interpolating curves as desired.

Recently, in application to computer-aided design, some methods for constructing shape-preserving interpolation have been presented, see [7], [19], [24], [25], [30], [31]. In [28], [29], Subdivision methods are discussed.

For $C^{2}$ continuity, almost all the methods are concerned to solve the consistency equations. Thus, shape-preserving properties will lead to some constraint conditions on the interpolation data and changes to any data will require solving again all the equations. Therefore, it is difficult to achieve monotonicity-preserving and convexity-preserving at the same time. In [10], for strict monotone data, the existence and uniqueness of a positive solution of the non-linear equations were proved. In [12], for strict convex data, the existence and uniqueness of a solution of the non-linear equations satisfying the convexity constraint were shown. However, in some other papers, for the derivative parameters, the solution of the non-linear equations satisfying the shape constrains was not discussed. In [13], it was mentioned that the proper choice of the parameter to guarantee shape preservation is still an unsettled question.

There are few $C^{2}$ continuous interpolation methods which are appropriate not only for monotonicity-preserving but also for convexity-preserving. In [10], [26], the convexity-preserving property of the interpolant was not discussed. In [3], [5], [12], [18], [37], the monotonicity-preserving property of the interpolant was not discussed. In [18], the explicit representation of the shape parameters was given.

As mentioned as above, the problem to construct a monotonicity-preserving, convexity-preserving and $C^{2}$ continuous local interpolation method has not been solved well. The motivation of this paper is to present a $C^{2}$ piecewise rational interpolant which is appropriate not only for monotonicity-preserving but also for convexity-preserving. We will construct $C^{2}$ continuous interpolant without solving a global system of consistency equations or adding additional knots on the subinterval. Thus, we can give explicit representations of the shape parameters which do not need any constraint conditions on the positive, strict monotone or strict convex data.

The remainder of this paper is organized as follows. In next section, the piecewise expression of the rational quartic interpolant is described. Some derivative expressions and $C^{2}$ continuity of the given interpolant are shown in Section 3. The shape-preserving properties are discussed in Section 4. In Section 5, we show the convergence order of the interpolant. Some numerical examples and conclusions are given in Section 6 and Section 7 respectively.

Section snippets

Piecewise rational interpolant

Given data $(x_{i}, f_{i}), i = 1, 2, \dots, n$ , with $x_{1} < x_{2} < \dots < x_{n}$ , we put $h_{i} = x_{i + 1} - x_{i}, Δ_{i} = (f_{i + 1} - f_{i}) / h_{i}$ and present the following piecewise rational interpolant $s (x) = s_{i} (x) ≔ \frac{1}{Q_{i} (t)} [{(1 - t)}^{4} f_{i} + {(1 - t)}^{3} {ta}_{i} + {(1 - t)}^{2} t^{2} b_{i} + (1 - t) t^{3} c_{i} + t^{4} f_{i + 1}]$ for $x \in [x_{i}, x_{i + 1}], i = 1, 2, \dots, n - 1$ , where $t = (x - x_{i}) / h_{i} \in [0, 1]$ , $Q_{i} (t) ≔ {(1 - t)}^{2} + (w_{i} - 1) (1 - t) t + t^{2} = 1 + (w_{i} - 3) (1 - t) t .$ Obviously, when $w_{i} > - 1, Q_{i} (t) > 0$ for $t \in [0, 1]$ . When $w_{i} = 3$ , the expression (1) is a piecewise quartic polynomial interpolant. Therefore, in the subsequent work, we would like to choose $w_{i} ⩾ 3$ .

Let $s^{'} (x_{i}) = d_{i}, i = 1, 2$

Derivative expressions and continuity

For the convenience of differentiation, we use the form of Bernstein polynomial. For Bernstein polynomial $B (t) = \sum_{i = 0}^{n} B_{i}^{n} (t) p_{i},$ where $t \in [0, 1], B_{i}^{n} (t) = (\begin{matrix} n \\ i \end{matrix}) {(1 - t)}^{n - i} t^{i}$ , we know that $B^{'} (t) = n \sum_{i = 0}^{n - 1} B_{i}^{n - 1} (t) (p_{i + 1} - p_{i}) .$ According to this, direct differentiate on (1), after much manipulation and by (3), (4), (6), we obtain $s_{i}^{'} (x) = \frac{1}{Q_{i}^{2} (t)} \sum_{j = 0}^{5} {(1 - t)}^{5 - j} t^{j} A_{ij},$ where $\begin{matrix} A_{i 0} = [a_{i} - (w_{i} + 1) f_{i}] / h_{i} = d_{i}, \\ A_{i 1} = [2 b_{i} - a_{i} - (3 w_{i} - 1) f_{i}] / h_{i} = 2 w_{i} Δ_{i} - d_{i} - 2 Θ_{i}, \\ A_{i 2} = [(w_{i} - 1) b_{i} + 3 c_{i} - (2 w_{i} - 1) a_{i} - 4 f_{i}] / h_{i} = (w_{i}^{2} + 2 w_{i} + 3) Δ_{i} - (2 w_{i} - 1) d_{i} - 3 d_{i + 1} - (w_{i} - 1) Θ_{i}, \\ A_{i 3} = [(2 w_{i} - 1) c_{i} + 4 f_{i} \end{matrix}$

Positivity-preserving property

For given set of data $(x_{i}, f_{i}), i = 1, 2, \dots, n$ , the polygon obtained by connecting the data points can be represented by the piecewise linear interpolant $(1 - t) f_{i} + {tf}_{i + 1}, t \in [0, 1], i = 1, 2, \dots, n - 1$ . The piecewise linear interpolant is nonnegativity-preserving. This implies that if $f_{i} f_{i + 1} ⩾ 0$ , then the sign of the interpolant is the same as the sign of the given data.

Theorem 2

For $x \in [x_{i}, x_{i + 1}], i = 1, 2, \dots, n - 1$ , $\lim_{w_{i} \to \infty} s_{i} (x) = (1 - t) f_{i} + {tf}_{i + 1}$ and the convergence is uniform.

Proof

Obviously, $Q_{i} (t) = {(1 - t)}^{2} + (w_{i} - 1) (1 - t) t + t^{2} ⩾ (w_{i} + 1) (1 - t) t$ and then $(1 - t) t$

Convergence analysis

Let f be a given function with which s is compared and $f_{i} = f (x_{i}), i = 1, 2, \dots, n$ . In this section we show that the interpolant (1) has second order accuracy or third order accuracy.

Lemma 6

Let $f \in C^{3} [x_{1}, x_{n}]$ , $δ_{k} (x) = Δ_{k} - Δ_{k - 1} - \frac{1}{2} (h_{k - 1} + h_{k}) f^{″} (x)$ for $k = 2, 3, \dots, n - 1$ , then $| δ_{k} (x) | ⩽ \{\begin{matrix} h^{2} M_{3}, & x \in [x_{k - 1}, x_{k + 1}], \\ 2 h^{2} M_{3}, & x \in [x_{k - 2}, x_{k - 1}] \cup [x_{k + 1}, x_{k + 2}], \end{matrix}$ where $h = \max_{1 ⩽ k ⩽ n - 1} h_{k}, M_{3} = \max_{x \in [x_{1}, x_{n}]} | f^{‴} (x) |$ .

Proof

By Taylor expansion, we have $\begin{matrix} δ_{k} (x) & = \frac{1}{2 h_{k}} [\int_{x}^{x_{k + 1}} {(x_{k + 1} - u)}^{2} f^{‴} (u) du - \int_{x}^{x_{k}} {(x_{k} - u)}^{2} f^{‴} (u) du] \\ - \frac{1}{2 h_{k - 1}} [\int_{x}^{x_{k}} {(x_{k} - u)}^{2} f^{‴} (u) du - \int_{x}^{x_{k - 1}} {(x_{k - 1} - u)}^{2} f^{‴} (u) du] \\ = & \frac{1}{2 h_{k}} \int_{x_{k}}^{x_{k + 1}} {(x_{k + 1} - u)}^{2} f^{‴} (u) du - \end{matrix}$

Numerical examples

We present in this section the performance of the proposed method on a few classical tests and a data set with local monotonicity and convexity. For accuracy, we give an example to compare the proposed method with cubic spline interpolation method which has $O (h^{4})$ accurate.

We first consider two sets of data to show the monotonicity-preserving property by (11). The first set of data may be found in [16], [26], and the interpolation curve is shown in Fig. 1 with $w = (w_{1}, w_{2}, \dots, w_{n - 1}) = (3, 3, 3, 3, 3, 4.5, 15.6$

Conclusion

The presented interpolation method in this paper provides a simple solution of shape-preserving interpolation problem. The interpolant is an explicit representation of $C^{2}$ continuous and piecewise rational interpolant with quartic numerator and quadratic denominator. The interpolant contains shape parameters $w_{i}$ which serve as tension factors. The explicit values of $w_{i}$ to guarantee shape preservation are given. The expressions (10), (11), (12), (13) can be used conveniently for global positivity

Acknowledgments

The author thanks the referees for their careful review and valuable comments. The author’s research was supported by the National Natural Science Foundation of China (Nos. 10871208, 11271376).

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Shape-preserving piecewise rational interpolant with quartic numerator and quadratic denominator

Abstract

Introduction

Section snippets

Piecewise rational interpolant

Derivative expressions and continuity

Positivity-preserving property

Convergence analysis

Numerical examples

Conclusion

Acknowledgments

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