zbMATH — the first resource for mathematics

Function parametrization by using 4-point transforms. (English) Zbl 0927.65009
Summary: A continuous parametrization of the smooth curve \(f(x)= f(x;{\mathcal R})\) is suggested on a basis of four-point transformations. Coordinates of three reference points of the curve are chosen as parameters \({\mathcal R}\). This approach allows to derive a number of advantages in function approximation and fitting of empiric data. The transformations have made possible to derive a new class of polynomials (monosplines) having the better approximation quality than monomials \(\{x^n\}\). A behaviour of an error of the approximation has a uniform character. A three-point model of the cubic spline is proposed. The model allows to reduce the number of unknown parameters in half and to obtain an advantage in a computing aspect. The new approach to the function approximation and fitting are shown on a number of examples. The proposed approach gives a new mathematical tool and a new possibility in both practical applications and theoretical research of numerical and computational methods.

65D10 Numerical smoothing, curve fitting
65D07 Numerical computation using splines
Full Text: DOI
[1] Dikoussar, N.D., Math. model., 3, 10, 50, (1991), [in Russian]
[2] Dikoussar, N.D., Comput. phys. commun., 79, 39, (1994)
[3] Duda, R.; Hart, P., Pattern classification and scene analysis, (1973), Wiley New York · Zbl 0277.68056
[4] ()
[5] Klein, F., Vorlesungen über Höhere geometrie, (1926), Springer Berlin · JFM 52.0624.09
[6] Lanczos, C., Applied analysis, (1956), Prentice-Hall Englewood Cliffs, Verlag · Zbl 0111.12403
[7] Forsite, G.E.; Malcom, M.A.; Moler, C.B., Computer methods for mathematical computations, (1977), Prentice-Hall Englewood Cliffs, Verlag, H. Wind, Function Parametrization, CERN, 72-21
[8] Ahlberg, J.H.; Nilson, E.N.; Walsh, J.L., The theory of splines and their applications, (1967), Academic Press New York · Zbl 0158.15901
[9] Zavjalov, Y.S.; Kvasov, B.I.; Miroshnichenko, V.L.; Stechkin, S.B.; Subbotin, Y.N., Splines in computer mathematics, (1976), Moscow Nauka, [in Russian]
[10] Popov, B.A., An uniform approximation by splines, (1984), Kiev Naukova Dumka, [in Russian]
[11] Kalitkin, N.N.; Kusmina, L.V., Math. model., 6, 4, 77, (1994), [in Russian]
[12] Roy, D.; Bhattacharya, R.; Bhowmick, S., Comput. phys. commun., 78, 29, (1993)
[13] Nabhan, Tarek M.; Zamaya, Albert Y., Neural networks, 7, 1, 88, (1994)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.