A simple piecewise cubic spline method for approximation of highly nonlinear data
Author(s)
Mehdi Zamani
ABSTRACT
Approximation methods are used in the analysis and prediction of data, especially laboratory data, in engineering projects. These methods are usually linear and are obtained by least-square-error approaches. There are many problems in which linear models cannot be applied. Because of that there are logarithmic, exponential and polynomial curve-fitting models. These nonlinear models have a limited application in engineering problems. The variation of most data is such that the nonlinearity cannot be approximated by the above approaches. These methods are also not applicable when there is a large amount of data. For these reasons, a method of piecewise cubic spline approximation has been developed. The model presented here is capable of following the local nonuniformity of data in order to obtain a good fit of a curve to the data. There is C1 continuity at the limits of the piecewise elements. The model is tested and examined with four problems here. The results show that the model can approximate highly nonlinear data efficiently.
Approximation methods are used in the analysis and prediction of data, especially laboratory data, in engineering projects. These methods are usually linear and are obtained by least-square-error approaches. There are many problems in which linear models cannot be applied. Because of that there are logarithmic, exponential and polynomial curve-fitting models. These nonlinear models have a limited application in engineering problems. The variation of most data is such that the nonlinearity cannot be approximated by the above approaches. These methods are also not applicable when there is a large amount of data. For these reasons, a method of piecewise cubic spline approximation has been developed. The model presented here is capable of following the local nonuniformity of data in order to obtain a good fit of a curve to the data. There is C1 continuity at the limits of the piecewise elements. The model is tested and examined with four problems here. The results show that the model can approximate highly nonlinear data efficiently.
Cite this paper
Zamani, M. (2012) A simple piecewise cubic spline method for approximation of highly nonlinear data. Natural Science, 4, 79-83. doi: 10.4236/ns.2012.41012.
Zamani, M. (2012) A simple piecewise cubic spline method for approximation of highly nonlinear data. Natural Science, 4, 79-83. doi: 10.4236/ns.2012.41012.
References
[1] De Boor, C. (1978) A practical guide to splines. Springer-Verlag, Berlin, 472. doi:10.1007/978-1-4612-6333-3 [2] Conte, S.D. and De Boor, C. (1980) Elementary numerical analysis: An algorithm approach. 3rd Edition, McGraw-Hill, Auckland, 432. [3] Lobo, N.A. (1995) Curve fitting using spline sections of different orders. Proceedings of the 1st International Mathematica Symposium, Southampton, 267-274. [4] Zamani, M. (2009) Three simple spline methods for approximation and interpolation of data. Contemporary Engineering Sciences Journal, 2, 373-381. [5] Lehmann, T.M., Gonner, C. and Spitzer, K. (2001) B-spline interpolation in medical image processing. IEEE Transactions on Medical Imaging, 20, 660-665. doi:10.1109/42.932749 [6] Salomon, D. (2006) Curves and surfaces for computer graphics. Springer, Northridge, 460. [7] Zamani, M. (2009) An investigation of Bspline and Bezier methods for interpolation of data. Contemporary Engineering Sciences Journal, 2, 361-371. [8] Burden, R.L. and Fairs, J.D. (1989) Numerical analysis. 4th Edition, PWS-Kent, Boston, 730. [9] Todd, D.K. (1980) Groundwater hydrology. 2nd Edition, Wiley, New York, 535.
[1] De Boor, C. (1978) A practical guide to splines. Springer-Verlag, Berlin, 472. doi:10.1007/978-1-4612-6333-3 [2] Conte, S.D. and De Boor, C. (1980) Elementary numerical analysis: An algorithm approach. 3rd Edition, McGraw-Hill, Auckland, 432. [3] Lobo, N.A. (1995) Curve fitting using spline sections of different orders. Proceedings of the 1st International Mathematica Symposium, Southampton, 267-274. [4] Zamani, M. (2009) Three simple spline methods for approximation and interpolation of data. Contemporary Engineering Sciences Journal, 2, 373-381. [5] Lehmann, T.M., Gonner, C. and Spitzer, K. (2001) B-spline interpolation in medical image processing. IEEE Transactions on Medical Imaging, 20, 660-665. doi:10.1109/42.932749 [6] Salomon, D. (2006) Curves and surfaces for computer graphics. Springer, Northridge, 460. [7] Zamani, M. (2009) An investigation of Bspline and Bezier methods for interpolation of data. Contemporary Engineering Sciences Journal, 2, 361-371. [8] Burden, R.L. and Fairs, J.D. (1989) Numerical analysis. 4th Edition, PWS-Kent, Boston, 730. [9] Todd, D.K. (1980) Groundwater hydrology. 2nd Edition, Wiley, New York, 535.