Some studies on nonpolynomial interpolation and error analysis

doi:10.1016/j.amc.2014.07.049

Applied Mathematics and Computation

Volume 244, 1 October 2014, Pages 809-821

https://doi.org/10.1016/j.amc.2014.07.049 Get rights and content

Abstract

In this paper, we propose nonpolynomial and Hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation. The generalized error analysis results for nonpolynomial and Hermite nonpolynomial interpolations are derived. We establish theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series, and propose solution to Runge’s phenomenon through nonpolynomial interpolation. The Hermite nonpolynomial cubic spline, nonpolynomial cubic spline interpolation methods and their error analysis are presented. Numerical simulations are carried out for the analysis of error in cubic spline interpolations. Proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations, and to Empirical Mode Decomposition (EMD) to illustrate practical usefulness of the results.

Introduction

Interpolation is the process of constructing a continuous curve through the known points of a function. Let $f (t)$ be a continuous function on an interval $[a, b]$ and ${t_{j}, f (t_{j}), j = 0, 1, 2, \dots, n}$ be the known distinct points and values of function in the interval. Generally, the interpolation between points is carried out with polynomial or spline of lower degree of the form $p (t) \in span {1, t, t^{2}, \dots, t^{n}}$ . For nonpolynomial interpolation, Sloan [1] introduced interpolating function of the form $p_{n} (t) = \sum_{i = 0}^{n} a_{i} u_{i} (t),$ where ${u_{0} (t), u_{1} (t), \dots, u_{n} (t)}$ is a set of linearly independent real-valued continuous functions on $[a, b]$ , and coefficients $a_{0}, \dots, a_{n}$ are determined by the interpolation conditions. As a special case of (1.1), we consider nonpolynomial interpolating function of the form $p (t) \in span {1, t, \dots, t^{n - 4}, e^{- ω t}, e^{ω t}, \cos (ω t), \sin (ω t)} .$ This reduces to a polynomial interpolating function in the limit as $ω \to 0$ . Runge’s phenomenon states that Lagrange’s interpolation, where $u_{i} (t)$ is a polynomial of degree i in (1.1), cannot guarantee the uniform convergence as $n \to \infty$ and high order polynomial interpolation is unstable [2]. In order to mitigate the Runge’s phenomenon, to make the interpolation curve stable and smooth, polynomial spline is introduced. The polynomial spline interpolation uses low degree polynomials in each of the intervals and constrains the polynomial pieces such that they fit smoothly together. The convergence of some cubic spline interpolation schemes are discussed in [3], and spline interpolation and smoothing on hyperspheres is presented in [4]. There is considerable evidence [5], [6], [7], [8], [9], [10] that the presence of a parameter $ω$ in nonpolynomial spline gives better results, in terms of maximum absolute error of approximation, than the polynomial spline, to approximate the numerical solution of differential equations of given boundary value problems. At the same time we found that methods based on nonpolynomial splines are more accurate and require less time in comparison to numerical methods based on polynomial spline interpolation. It is also well known that the algorithms developed using higher order splines give better results than the algorithms developed using lower order splines [11]. The main results presented here provide sufficient scope to obtain solutions of interpolation problems with more accuracy and less computational effort. In the literature we find that the solutions of nonpolynomial spline with only uniform step size have been discussed, but here, in this work we generalize it for uniform as well as non uniform step size situations. In this work we explore following matters which are not available so far in literature: (1) There is no discussion to obtain optimal value of a parameter as there is no general error analysis result for nonpolynomial interpolation and hence we propose solution to this problem. (2) We propose nonpolynomial interpolation with multiple parameters, determination of these parameters and general error analysis result for nonpolynomial interpolation. (3) We introduce Hermite nonpolynomial interpolation with multiple parameters, determination of those parameters and general error analysis result for Hermite nonpolynomial interpolation. (4) We establish theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series representation. (5) We propose solution to Runge’s phenomenon through nonpolynomial interpolation.

There are various important applications of interpolation like curve fitting, extrapolation, computer graphics, to approximate solutions of boundary value problems, Empirical Mode Decomposition (EMD), and Wang [12] proposed cubic spline wavelet bases for Sobolev space and multilevel interpolation. In EMD interpolation is used to obtain upper and lower envelopes of underlying time series to derive Intrinsic Mode Functions (IMFs). Properties of the IMFs may change by changing the interpolation methods. The EMD, proposed in seminal paper by Huang et al. [13], is an adaptive data-analysis method to analyze stationary as well as non stationary time series. The EMD decomposes a signal into a set of finite band-limited IMFs and final residue as a trend of signal. An IMF is a function that must satisfy two conditions: (1) in the complete range of time series, the number of extrema (i.e. maxima and minima) and the number of zero crossings are equal or differ at most by one. (2) At any point of time in the complete range of time series, the average of the values of upper and lower envelopes, obtained by the interpolation of local maxima and the local minima, is zero.

The first condition ensure that IMFs are narrow band signals and the second condition is necessary to ensure that the instantaneous frequency does not have redundant fluctuations because of asymmetric waveforms.

An ensemble EMD (EEMD) was proposed by Wu and Huang [14] to solve the irregular distribution of local extrema problem. This approach consists of sifting an ensemble of white noise added signal and considers the mean as the final true result. The reason for addition of finite amplitude white noise is to obtain relatively uniform distribution of local maxima and minima such that the upper and lower envelope can be properly identified. The EEMD is computationally very expensive as compare to EMD.

The Compact EMD (CEMD) was proposed by Chu et al. [15] to reduce mode mixing, end effect, and detrend uncertainty present in EMD, and to reduce computation complexity of EEMD as well. The IMFs generated by EMD are dependent on distribution of local extrema of signal and the type of spline used for upper and lower envelope interpolation, and the traditional EMD uses cubic spline for upper and lower envelope interpolation. Singh et al. [16] proposed nonpolynomial cubic spline to obtain upper and lower envelope interpolation in EMD algorithm to reduce mode mixing and detrend uncertainty.

The first aim of present work is to explore nonpolynomial interpolation with multiple parameters and device a method to determine these parameters. The second aim is to extend the results of error analysis of classical polynomial interpolation to derive results of error analysis for nonpolynomial interpolation, and to show that nonpolynomial interpolation generate less error in approximation than polynomial one. The third aim is to study which type of interpolating spline and boundary condition is better and produces minimum error in same order. This paper is organized as follows: the nonpolynomial cubic spline and the Hermite nonpolynomial cubic spline interpolations are discussed in Section 2. Theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series is discussed in Section 3. Section 4 introduces the error analysis for the case of nonpolynomial and Hermite nonpolynomial interpolations. Simulation results are given in Section 5. Section 6 presents conclusions.

Section snippets

Nonpolynomial spline interpolation

If a time series from a physical object or a function is available, spline interpolation is an approach to generate a spline that approximates that time series. For a given knot vectors on interval $[a, b]$ , the splines of degree n form a vector space. The space of all natural cubic splines, for instance, is a subspace of the space of all cubic $C^{2} [a, b]$ splines.

A cubic spline function $S (t)$ , interpolating a function $x (t)$ defined on $[a, b]$ , is such that: (a) In each subinterval $[t_{j}, t_{j + 1}], S (t)$ is a

Nonpolynomial, polynomial interpolation and the Fourier series representation

In this section, we establish the relationship among nonpolynomial, polynomial interpolation and the Fourier series, and propose the nonpolynomial function with n parameters ${ω_{k}}_{k = 1}^{n}$ of the form: $\begin{matrix} T_{2 n} \in span {1, \sin (ω_{1} t), \cos (ω_{1} t), \sin (ω_{2} t), \cos (ω_{2} t), \dots, \sin (ω_{n} t), \cos (ω_{n} t)} \\ T_{2 n} = b_{0} + \sum_{k = 1}^{n} {a_{k} \sin (ω_{k} t) + b_{k} \cos (ω_{k} t)} \end{matrix}$ and this nonpolynomial interpolation belongs to the class $C^{\infty} [a, b]$ . From (3.1), it can be easily seen that:

$\lim_{ω_{1} \to 0} T_{2 n} \in span {1, t, t^{2}, \sin (ω_{2} t), \cos (ω_{2} t), \dots, \sin (ω_{n} t), \cos (ω_{n} t)},$
$\lim_{\binom{ω_{1} \to 0}{ω_{2} \to 0}} T_{2 n} \in span {1, t, t^{2}, t^{3}, t^{4}, \dots,$

The error analysis in nonpolynomial interpolation

We generalize the existing results of error analysis in polynomial interpolations and propose following results for error analysis in nonpolynomial interpolations.

Theorem 4.1

Let $f (t) \in C^{n + 1} [a, b]$ be any function and $t_{0}, \dots, t_{n}$ be distinct nodes in the interval $[a, b]$ . If nonpolynomial function $p (t) \in C^{\infty} [a, b]$ interpolates $f (t)$ , and at the nodes $f (t_{i}) = p (t_{i})$ , then $\forall \bar{t} \in [a, b], \exists ξ \in (a, b)$ such that $f (\bar{t}) - p (\bar{t}) = \frac{f^{(n + 1)} (ξ) - p^{(n + 1)} (ξ)}{(n + 1)!} \prod_{i = 0}^{n} (\bar{t} - t_{i}),$ where $ξ \in (a, b)$ is independent of $\bar{t}$ .

Proof

If $\bar{t} = t_{i}$ for any $i \in \overline{0, n}$ then theorem is

Example 1

To compare the maximum absolute error (MAE) in interpolation among polynomial cubic spline (PCS), nonpolynomial cubic spline (NPCS) and Hermite NPCS (HNPCS) with various boundary conditions (BCs), 50 time series consisting of sum of constant, linearly increasing and various sinusoids, for example, are used as follows: $x_{k} (t_{i}) = 3 + 5 t_{i} + \sum_{m = 1}^{k} m \sin (ω_{m} t_{i}), for k = 1, 2, \dots, 50; ω_{m} = m, t_{i} = (i - 1) Δ t, t_{1} = 0, t_{n} = 1.0 s, Δ t = 50.0 ms, n = 20 .$

Table 1 shows comparison of MAE among PCS, NPCS and HNPCS for few signals, whereas Fig. 1,

Conclusion

We introduced nonpolynomial and Hermite nonpolynomial interpolation with multiple parameters and developed method to determine optimal value of parameters which generate minimum error in approximation. We established theoretical relationship among nonpolynomial, polynomial interpolation and the Fourier series representation of any periodic signal. We have proposed generalized error analysis results for nonpolynomial and Hermite nonpolynomial interpolations as extension of classical error

Acknowledgments

The authors would like to express their appreciation to the anonymous reviewers for their valuable suggestions. The authors would like to express sincere thanks to Jaypee Institute of Information Technology Noida, for providing all required resources and permitting to carry out research at department of electrical engineering, Indian Institute of Technology Delhi (IITD). The authors also would like to thank IITD for providing resources and technical supports required throughout this study.

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Some studies on nonpolynomial interpolation and error analysis

Abstract

Introduction

Section snippets

Nonpolynomial spline interpolation

Nonpolynomial, polynomial interpolation and the Fourier series representation

The error analysis in nonpolynomial interpolation

Example 1

Conclusion

Acknowledgments

J. Approx. Theory

Comput. Math. Appl.

Appl. Math. Comput.

Appl. Math. Comput.

Quartic Octic Spline Adva. Eng. Softw.

Appl. Math. Comput.

Comput. Math. Appl.

Appl. Math. Comput.

On the Stability of Interpolation

J. Comput. Math.

On the convergence of some cubic spline interpolation schemes

SIAM J. Numer. Anal.