×

Shape-preserving piecewise rational interpolant with quartic numerator and quadratic denominator. (English) Zbl 1328.41002

Summary: 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.

MSC:

41A20 Approximation by rational functions
65D05 Numerical interpolation
41A29 Approximation with constraints

Software:

pchip
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abbas, M.; Majid, A. A.; Ali, J. M., Monotonicity-preserving \(C^2\) rational cubic spline for monotone data, Appl. Math. Comput., 219, 2885-2895 (2012) · Zbl 1309.65017
[2] Abbas, M.; Majid, A. A.; Ali, J. M., Positivity-preserving rational bi-cubic spline interpolation for 3D positive data, Appl. Math. Comput., 234, 460-476 (2014) · Zbl 1298.65021
[3] Carnicer, J. M.; Dahmen, W., Characterization of local strict convexity preserving interpolation methods by \(C^1\) functions, J. Approx. Theory, 77, 2-30 (1994) · Zbl 0817.41009
[4] Clements, J. C., Convexity-preserving piecewise rational cubic interpolation, SIAM J. Numer. Anal., 27, 1016-1023 (1990) · Zbl 0702.65010
[5] Clements, J. C., A convexity-preserving \(C^2\) parametric rational cubic interpolation, Numer. Math., 63, 165-171 (1992) · Zbl 0763.41001
[6] Costantini, P., On monotone and convex spline interpolation, Math. Comput., 46, 203-214 (1986) · Zbl 0617.41015
[7] Costantini, P., Curve and surface construction using variable degree polynomial splines, CAGD, 17, 419-466 (2000) · Zbl 0938.68128
[8] Cravero, I.; Manni, C., shape-preserving interpolants with high smoothness, J. Comput. Appl. Math., 157, 383-405 (2003) · Zbl 1027.65009
[9] Dauner, H.; Reinsch, C. H., An analysis of two algorithms for shape-preserving cubic spline interpolation, IMA J. Numer. Anal., 9, 299-314 (1989) · Zbl 0681.65004
[10] Delbourgo, R.; Gregory, J. A., \(C^2\) rational quadratic spline interpolation to monotonic data, IMA J. Numer. Anal., 3, 141-152 (1983) · Zbl 0523.65005
[11] Delbourgo, R.; Gregory, J. A., Shape preserving piecewise rational interpolation, SIAM J. Sci. Stat. Comput., 6, 967-976 (1985) · Zbl 0586.65006
[12] Delbourgo, R., Shape preserving interpolation to convex data by rational functions with quadratic numerator and linear denominator, IMA J. Numer. Anal., 9, 123-136 (1989) · Zbl 0675.65004
[13] Delbourgo, R., Accurate \(C^2\) rational interpolations in tension, SIAM J. Numer. Anal., 30, 595-607 (1993) · Zbl 0772.65003
[14] Dougherty, R. L.; Edelman, A.; Hyman, J. M., Nonnegativity-, monotonicity-, convexity-preserving cubic and quintic Hermite interpolation, Math. Comput., 52, 471-494 (1989) · Zbl 0693.41004
[15] Fiorot, J. C.; Tabka, J., Shape-preserving \(C^2\) cubic polynomial interpolating splines, Math. Comput., 57, 291-298 (1991) · Zbl 0731.65007
[16] Fritsch, F. N.; Carlson, R. E., Monotone piecewise cubic interpolation, SIAM J. Numer. Anal., 17, 238-246 (1980) · Zbl 0423.65011
[17] Gregory, J. A.; Delbourgo, R., Piecewise rational quadratic interpolation to monotonic data, IMA J. Numer. Anal., 2, 123-130 (1982) · Zbl 0481.65004
[18] Han, X., Convexity-preserving piecewise rational quartic interpolation, SIAM J. Numer. Anal., 46, 920-929 (2008) · Zbl 1165.65005
[19] Hong, V. P.; Ong, B. N., Shape preserving approximation by spatial cubic splines, CAGD, 26, 888-903 (2009) · Zbl 1205.65051
[20] Hussain, M. Z.; Hussain, M., Visualization of data preserving monotonicity, Appl. Math. Comput., 190, 1353-1364 (2007) · Zbl 1227.65016
[21] Hussain, M. Z.; Sarfraz, M., Positive-preserving interpolation of positive data by rational cubic, J. Comput. Appl. Math., 218, 446-458 (2008) · Zbl 1143.65010
[22] Kaklis, P. D.; Pandelis, D. G., Convexity preserving polynomial splines of non-uniform degree, IMA J. Numer. Anal., 10, 223-234 (1990) · Zbl 0699.65007
[23] Kvasov, B. I., Methods of Shape-Preserving Spline Approximation (2000), World Scientific Publishing Co Pte. Ltd.: World Scientific Publishing Co Pte. Ltd. Singapore · Zbl 0960.41001
[24] Lavery, J. E., Shape-preserving, first-derivative-based parametric and nonparametric cubic \(L_1\) spline curves, CAGD, 23, 276-296 (2006) · Zbl 1094.65013
[25] Lavery, J. E., Shape-preserving univariate cubic and higher-degree \(L_1\) splines with function-value-based and multistep minimization principles, CAGD, 26, 1-16 (2009) · Zbl 1205.65040
[26] Manni, C.; Sablonnière, P., Monotone interpolation of order 3 by \(C^2\) cubic splines IMA, J. Numer. Anal., 17, 305-320 (1997) · Zbl 0865.41014
[27] Manni, C., On shape preserving \(C^2\) Hermite interpolation, BIT Numer. Math., 41, 127-148 (2001) · Zbl 0985.65008
[28] Merrien, J. L.; Sablonnière, P., Monotone and convex \(C^1\) Hermite interpolants generated by an adaptive subdivision scheme, C. R. Acad. Sci. Paris Ser. I, 333, 493-497 (2001) · Zbl 1013.65009
[29] Merrien, J. L.; Sablonnière, P., Monotone and convex \(C^1\) Hermite interpolants generated by a subdivision algorithm, Constr. Approx., 19, 279-298 (2003) · Zbl 1018.41012
[30] Merrien, J. L.; Sablonnière, P., Rational splines for Hermite interpolation with shape constraints, Comput. Aided Geom. Des., 30, 296-309 (2013) · Zbl 1276.65009
[31] Messac, A.; Sivanandam, A., A new family of convex splines for data interpolation, CAGD, 15, 39-59 (1997) · Zbl 0894.68149
[32] Oja, P., Comonotone adaptive interpolating splines, BIT Numer. Math., 42, 842-855 (2002) · Zbl 1032.65018
[33] Passow, E.; Roulier, J. A., Monotone and convex spline interpolation, SIAM J. Numer. Anal., 14, 904-907 (1977) · Zbl 0378.41002
[34] Pruess, S., Shape preserving \(C^2\) cubic spline interpolation, IMA J. Numer. Anal., 13, 493-507 (1993) · Zbl 0786.65009
[35] Sarfraz, M.; Hussain, M. Z.; Hussain, M., Shape-preserving curve interpolation, Int. J. Comput. Math., 89, 35-53 (2012) · Zbl 1237.68237
[36] Schaback, R., Adaptive rational splines, Constr. Approx., 6, 167-179 (1990) · Zbl 0685.41011
[37] Schmidt, J. W.; He, W., An always successful method in univariate convex \(C^2\) interpolation, Numer. Math., 71, 237-252 (1995) · Zbl 0860.65004
[38] Schumaker, L. L., On shape preserving quadratic spline interpolation, SIAM J. Numer. Anal., 20, 854-864 (1983) · Zbl 0521.65009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.