The Determination of Derivative Parameters for a Monotonic Rational Quadratic Interpolant

Delbourgo, Roger; Gregory, John

doi:10.1093/imanum/5.4.397

Cited by 36 publications

(23 citation statements)

References 5 publications

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“…In most applications, d i are not given. Consequently, the derivative values are estimated by some appropriate methods available in the literature (see, for e.g., [9]). In this article, we use the arithmetic mean method (amm) which is based on the three-point difference approximation.…”

Section: Estimation Of Derivative Parametersmentioning

confidence: 99%

“…However, for a fixed set of the scaling factors and the derivative parameters, the shape parameters computed using (7) may not meet the conditions of Theorem 6, and hence, in general, the resulting C 1 -rational quadratic spline FIF may not be positive. To get a positive C 1 -rational quadratic spline FIF, one should concern with the solvability of the system governed by (9), (22), and (23).…”

Section: Positive Rational Quadratic Fifmentioning

confidence: 99%

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A rational iterated function system for resolution of univariate constrained interpolation

Viswanathan

Chand

Navascués

2014

RACSAM

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Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function Φ ∈ C 3 [x 1 , x n ] with h 3 order of convergence, where h denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

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Section: Estimation Of Derivative Parametersmentioning

confidence: 99%

Section: Positive Rational Quadratic Fifmentioning

confidence: 99%

A rational iterated function system for resolution of univariate constrained interpolation

Viswanathan

Chand

Navascués

2014

RACSAM

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“…These derivative parameters are usually note given and can be determined by using the method as discussed in [2]. .…”

Section: The Rational (Cubic/quadratic) Spline Interpolationmentioning

confidence: 99%

Convexity Preserving Interpolation by GC2-Rational Cubic Spline

Dube¹,

Rana²

2013

IJCA

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A weighted rational cubic spline interpolation has been constructed using rational spline with quadratic denominator. GC 1 -piecewise rational cubic spline function involving parameters has been constructed which produces a monotonic interpolant to given monotonic data . The degree of smoothness of this spline is GC 2 in the interpolating interval when the parameters satisfy a continuous system. It is observed that under certain conditions the interpolant preserve the convexity property of the data set. We have discussed the constrains for GC 2 -rational spline interpolant in section. Also the error estimate formula of this interpolation are obtained.

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“…There exist many research papers on the construction of rational cubic spline for monotonicity preserving interpolation. Some literature reviews are as follows: Delbourgo and Gregory [6,7] have discussed the monotonicity and convexity preservation by using rational cubic spline (quadratic denominator) with one parameter. Sarfraz [18], Sarfraz et al [19] and Abbas et al [1] also studied the use of rational cubic interpolant for preserving the monotone data.…”

Section: Introductionmentioning

confidence: 99%

“…(ii) Our scheme not involving any knots insertion as appear in the works of Lahtinen [15], Lam [16], Schumaker [21], Fristch and Carlson [8], Schmidt and Hess [20] and Butt and Brodlie [5]. (iii) Our scheme has two degree freedom meanwhile there are no degree freedom in the works of Sarfraz [18], Delbourgo and Gregory [6,7] and Gregory [9].…”

Section: Introductionmentioning

confidence: 99%

Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C² Continuity

Karim

2017

MATEC Web Conf.

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Abstract. Monotonicity preserving interpolation is very important in many sciences and engineering based problems. This paper discuss the monotonicity preserving interpolation for monotone data set by using 2 C rational cubic spline interpolant (cubic/quadratic) with three parameters. The data dependent sufficient conditions for the monotonicity are derived with two degree freedom. Numerical results suggests that the proposed 2 C rational cubic spline preserves the monotonicity of the data and outperform the performance of the other rational cubic spline schemes in term of visually pleasing.

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The Determination of Derivative Parameters for a Monotonic Rational Quadratic Interpolant

Abstract: Explicit formulae are developed for determining the derivative parameters of a monotonic interpolation method of Gregory and Delbourgo (1982).

Cited by 36 publications

References 5 publications

A rational iterated function system for resolution of univariate constrained interpolation

A rational iterated function system for resolution of univariate constrained interpolation

Convexity Preserving Interpolation by GC2-Rational Cubic Spline

Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C² Continuity

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The Determination of Derivative Parameters for a Monotonic Rational Quadratic Interpolant

Abstract: Explicit formulae are developed for determining the derivative parameters of a monotonic interpolation method of Gregory and Delbourgo (1982).

Cited by 36 publications

References 5 publications

A rational iterated function system for resolution of univariate constrained interpolation

A rational iterated function system for resolution of univariate constrained interpolation

Convexity Preserving Interpolation by GC2-Rational Cubic Spline

Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C2 Continuity

Contact Info

Product

Resources

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Rational Cubic Spline Interpolation for Monotonic Interpolating Curve with C² Continuity