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Journal articles on the topic 'Cubic spline interpolation'

Author: Grafiati

Published: 4 June 2021

Last updated: 27 April 2022

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1

Xu, Weizhi. "Elements of Bi-cubic Polynomial Natural Spline Interpolation for Scattered Data: Boundary Conditions Meet Partition of Unity Technique." Statistics, Optimization & Information Computing 8, no. 4 (December 2, 2020): 994–1010. http://dx.doi.org/10.19139/soic-2310-5070-1083.

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This paper investigates one kind of interpolation for scattered data by bi-cubic polynomial natural spline, in which the integral of square of partial derivative of two orders to x and to y for the interpolating function is minimal (with natural boundary conditions). Firstly, bi-cubic polynomial natural spline interpolations with four kinds of boundary conditions are studied. By the spline function methods of Hilbert space, their solutions are constructed as the sum of bi-linear polynomials and piecewise bi-cubic polynomials. Some properties of the solutions are also studied. In fact, bi-cubic natural spline interpolation on a rectangular domain is a generalization of the cubic natural spline interpolation on an interval. Secondly, based on bi-cubic polynomial natural spline interpolations of four kinds of boundary conditions, and using partition of unity technique, a Partition of Unity Interpolation Element Method (PUIEM) for fitting scattered data is proposed. Numerical experiments show that the PUIEM is adaptive and outperforms state-of-the-art competitions, such as the thin plate spline interpolation and the bi-cubic polynomial natural spline interpolations for scattered data.

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2

Xie, Jin, and Xiaoyan Liu. "The EH Interpolation Spline and Its Approximation." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/745765.

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A new interpolation spline with two parameters, called EH interpolation spline, is presented in this paper, which is the extension of the standard cubic Hermite interpolation spline, and inherits the same properties of the standard cubic Hermite interpolation spline. Given the fixed interpolation conditions, the shape of the proposed splines can be adjusted by changing the values of the parameters. Also, the introduced spline could approximate to the interpolated function better than the standard cubic Hermite interpolation spline and the quartic Hermite interpolation splines with single parameter by a new algorithm.

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3

Abdul Karim, Samsul Ariffin, and Kong Voon Pang. "Shape Preserving Interpolation UsingC2Rational Cubic Spline." Journal of Applied Mathematics 2016 (2016): 1–14. http://dx.doi.org/10.1155/2016/4875358.

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This paper discusses the construction of newC2rational cubic spline interpolant with cubic numerator and quadratic denominator. The idea has been extended to shape preserving interpolation for positive data using the constructed rational cubic spline interpolation. The rational cubic spline has three parametersαi,βi, andγi. The sufficient conditions for the positivity are derived on one parameterγiwhile the other two parametersαiandβiare free parameters that can be used to change the final shape of the resulting interpolating curves. This will enable the user to produce many varieties of the positive interpolating curves. Cubic spline interpolation withC2continuity is not able to preserve the shape of the positive data. Notably our scheme is easy to use and does not require knots insertion andC2continuity can be achieved by solving tridiagonal systems of linear equations for the unknown first derivativesdi,i=1,…,n-1. Comparisons with existing schemes also have been done in detail. From all presented numerical results the newC2rational cubic spline gives very smooth interpolating curves compared to some established rational cubic schemes. An error analysis when the function to be interpolated isft∈C3t0,tnis also investigated in detail.

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4

Kumar, Arun, and L. K. Govil. "Interpolation of natural cubic spline." International Journal of Mathematics and Mathematical Sciences 15, no. 2 (1992): 229–34. http://dx.doi.org/10.1155/s0161171292000292.

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From the result in [1] it follows that there is a unique quadratic spline which bounds the same area as that of the function. The matching of the area for the cubic spline does not follow from the corresponding result proved in [2]. We obtain cubic splines which preserve the area of the function.

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5

Rana, S. S., and M. Purohit. "Deficient cubic spline interpolation." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 4 (1988): 111–14. http://dx.doi.org/10.3792/pjaa.64.111.

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6

Dyer, S. A., and J. S. Dyer. "Cubic-spline interpolation. 1." IEEE Instrumentation & Measurement Magazine 4, no. 1 (March 2001): 44–46. http://dx.doi.org/10.1109/5289.911175.

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7

Kim, Jung-Min, Eun-Kook Jung, and Sun-Shin Kim. "Simplification of Face Image using Cubic Spline Interpolation." Journal of Korean Institute of Intelligent Systems 20, no. 5 (October 25, 2010): 722–27. http://dx.doi.org/10.5391/jkiis.2010.20.5.722.

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8

Azizan, Irham, Samsul Ariffin Bin Abdul Karim, and S. Suresh Kumar Raju. "Fitting Rainfall Data by Using Cubic Spline Interpolation." MATEC Web of Conferences 225 (2018): 05001. http://dx.doi.org/10.1051/matecconf/201822505001.

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This study discusses the application of two cubic spline i.e. natural and not-a-knot end boundary conditions to visualize and predict the rainfall data. The interpolation and the analysis of the rainfall data will be done on a monthly basis by using the MATLAB software. The rainfall data is obtained from Malaysia Meteorology Department for Ipoh and Petaling Jaya in year 2014 and 2015. The interpolating curves are then being compared and if there is any negative value on the interpolating curve on some sub-interval, that part will be replaced by using the Piecewise Cubic Hermite Interpolating Polynomial (PCHIP). We discuss the missing data imputation by using both splines.

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9

Li, Jie, Yaoyao Tu, and Shilong Fei. "C˜2 Continuous Cubic Hermite Interpolation Splines with Second-Order Elliptic Variation." Tobacco Regulatory Science 7, no. 6 (November 3, 2021): 6317–31. http://dx.doi.org/10.18001/trs.7.6.106.

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In order to solve the deficiency of Hermite interpolation spline with second-order elliptic variation in shape control and continuity, c-2 continuous cubic Hermite interpolation spline with second-order elliptic variation was designed. A set of cubic Hermite basis functions with two parameters was constructed. According to this set of basis functions, the three-order Hermite interpolation spline curves were defined in segments 02, and the parameter selection scheme was discussed. The corresponding cubic Hermite interpolation spline function was studied, and the method to determine the residual term and the best interpolation function was given. The results of an example show that when the interpolation conditions remain unchanged, the cubic Hermite interpolation spline curves not only reach 02 continuity, but also can use the parameters to control the shape of the curves locally or globally. By determining the best values of the parameters, the cubic Hermite interpolation spline function can get a better interpolation effect, and the smoothness of the interpolation spline curve is the best.

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10

Karim, Samsul Ariffin Bin Abdul, and S. Suresh Kumar Raju. "Wind Velocity Data Interpolation Using Rational Cubic Spline." MATEC Web of Conferences 225 (2018): 04006. http://dx.doi.org/10.1051/matecconf/201822504006.

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Wind velocity data is always having positive value and the minimum value approximately close to zero. The standard cubic spline interpolation (not-a-knot and natural) as well as cubic Hermite polynomial may be produces interpolating curve with negative values on some subintervals. To cater this problem, a new rational cubic spline with three parameters is constructed. This rational spline will be used to preserve the positivity of the wind velocity data. Numerical results shows that the proposed scheme work very well and give visually pleasing interpolating curve on the given domain.

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11

Peng, Bao Ying, and Qiu Shi Han. "NURBS Curve Design and CNC Machining Principle." Applied Mechanics and Materials 141 (November 2011): 392–96. http://dx.doi.org/10.4028/www.scientific.net/amm.141.392.

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Used fixed equal step cubic spline interpolation in non-circular machining, the influence of acceleration may be too strong to cause burns and vibration. To suppress the acceleration influence of feed axis, take cam grinding as an illustration, first derived acceleration and its change from the cubic spline function, then adopting the method of off-line fuzzy model logical reasoning combined with five points of three power smooth, obtained the modified variable interpolation time steps. The experiment results that the feed axis acceleration stability used fuzzy step cubic spline interpolations is much better than equal step spline in cam grinding under the same processing efficiency.

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12

Papamichael, N., and M. J. Soares. "Cubic and quintic spline-on-spline interpolation." Journal of Computational and Applied Mathematics 20 (November 1987): 359–66. http://dx.doi.org/10.1016/0377-0427(87)90153-1.

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13

Jehan Mohammed Al-Ameri. "Cubic Spline Interpolation for Data Infections of COVID-19 Pandemic in Iraq." Al-Qadisiyah Journal Of Pure Science 26, no. 5 (November 12, 2021): 23–32. http://dx.doi.org/10.29350/qjps.2021.26.5.1443.

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In this paper, we use an empirical equation and cubic spline interpolation to fit Covid-19 data available for accumulated infections and deaths in Iraq. For Scientific visualization of data interpretation, it is useful to use interpolation methods for purposes fitting by data interpolation. The data used is from 3 January 2020 to 21 January 2021 in order to obtain graphs to analysing the rate of increasing the pandemic and then obtain predicted values for the data infections and deaths in that period of time. Stochastic fit to the data of daily infections and deaths of Covid-19 is also discussed and showed in figures. The results of the cubic splines and the empirical equation used will be numerically compared. The principle of least square errors will be used for both these interpolations. The numerical results will be indicated that the cubic spline gives an accurate fitting to data.

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14

Nazren, A. R. A., Shahrul Nizam Yaakob, R. Ngadiran, N. M. Wafi, and M. B. Hisham. "Cubic Polynomial as Alternatives Cubic Spline Interpolation." Advanced Science Letters 23, no. 6 (June 1, 2017): 5069–72. http://dx.doi.org/10.1166/asl.2017.7311.

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15

Ramachandran, Dhanya, and Dr V. Madhukar Mallayya. "Two Variable Cubic Spline Interpolation." IOSR Journal of Mathematics 13, no. 01 (March 2017): 01–05. http://dx.doi.org/10.9790/5728-1301060105.

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16

Dyer, S. A., and Xin He. "Cubic-spline interpolation: part 2." IEEE Instrumentation & Measurement Magazine 4, no. 2 (June 2001): 34–36. http://dx.doi.org/10.1109/5289.930984.

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17

CHAND, A. K. B., P. VISWANATHAN, and K. M. REDDY. "TOWARDS A MORE GENERAL TYPE OF UNIVARIATE CONSTRAINED INTERPOLATION WITH FRACTAL SPLINES." Fractals 23, no. 04 (December 2015): 1550040. http://dx.doi.org/10.1142/s0218348x15500401.

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Recently, in [Electron. Trans. Numer. Anal. 41 (2014) 420–442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current paper is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.

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18

Hou, Xiang Hua, and Hong Hai Liu. "Research on Improved Spline Interpolation Algorithm in Super-Resolution Reconstruction of Video Image." Applied Mechanics and Materials 380-384 (August 2013): 3722–25. http://dx.doi.org/10.4028/www.scientific.net/amm.380-384.3722.

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When low-spline interpolation algorithm is adopted by super-resolution reconstruction for video images, there are some defects, such as saw tooth and blur edge, if the result image is magnified. In this paper, high-order spline interpolation algorithm is introduced and it is optimized. Firstly, the common low-spline interpolation algorithms are analyzed and their shortcomings are pointed out. Then cubic spline interpolation algorithm is discussed. If the image is rotated by cubic spline interpolation algorithm, the magnified image may be not correctly displayed and the image can not be registered in super-resolution reconstruction. Finally, the cubic spline algorithm has been improved. Experimental results show that the improved cubic spline interpolation algorithm can not only eliminate the edge blur and saw tooth, but also do registration in reconstruction when image is rotating.

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19

Jin, Yong Qiao, Yu Han Wang, and Jian Guo Yang. "Real-Time B-Spline Interpolator with Look-Ahead Scheme for High-Speed CNC Machine Tools." Key Engineering Materials 455 (December 2010): 599–605. http://dx.doi.org/10.4028/www.scientific.net/kem.455.599.

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NC tool paths of digital CAD models are currently generated as a set of discrete data points. The CNC interpolator must convert these points into continuous machine tool axis motions. In order to achieve high-speed and high-accuracy machining, the development of a real-time interpolation algorithm is really indispensable, which can deal with a large number of short blocks and still maintain smooth interpolation with an optimal speed. In this paper, a real-time local cubic B-spline interpolator with look-ahead scheme is proposed for consecutive micro-line blocks interpolation. First, the consecutive micro-line blocks that satisfy the bi-chord error constraints are fitted into a C1 continuous cubic B-spline curve. Second, machining dynamics and tool path contour constrains are taken into consideration. Third, local cubic B-spline interpolator with an optimal look-ahead scheme is proposed to generate the optimal speed profile. Simulation and experiment are performed in real-time environment to verify the effectiveness of the proposed method. Compared with the conventional interpolation algorithm, the proposed algorithm reduces the machining time by 70%.

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20

KAPOOR, G. P., and SRIJANANI ANURAG PRASAD. "CUBIC SPLINE SUPER FRACTAL INTERPOLATION FUNCTIONS." Fractals 22, no. 01n02 (March 2014): 1450005. http://dx.doi.org/10.1142/s0218348x14500054.

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In the present work, the notion of Cubic Spline Super Fractal Interpolation Function (SFIF) is introduced to simulate an object that depicts one structure embedded into another and its approximation properties are investigated. It is shown that, for an equidistant partition points of [x0, xN], the interpolating Cubic Spline SFIF[Formula: see text] and their derivatives [Formula: see text] converge respectively to the data generating function y(x) ≡ y(0)(x) and its derivatives y(j)(x) at the rate of h2-j+ϵ(0 < ϵ < 1), j = 0, 1, 2, as the norm h of the partition of [x0, xN] approaches zero. The convergence results for Cubic Spline SFIF found here show that any desired accuracy can be achieved in the approximation of a regular data generating function and its derivatives by a Cubic Spline SFIF and its corresponding derivatives.

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21

Tian, Meng, and Hong Ling Geng. "Constrained Control of a Rational Interpolant." Advanced Materials Research 225-226 (April 2011): 170–73. http://dx.doi.org/10.4028/www.scientific.net/amr.225-226.170.

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In this paper, a rational cubic spline interpolation has been constructed using the rational cubic spline with quadratic denominator and the rational cubic spline based on function values. The spline can preserve monotonicity of the data set. The spline not only belongs to in the interpolating interval, but could also be used to constrain the shape of the interpolant curve such as to force it to be the given region. The explicit representation is easily constructed, and numerical experiments indicate that the method produces visually pleasing curves.

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22

Wu, Xinping, Minjie Xu, Yanqiu Gao, and Xianqing Lv. "A Scheme for Estimating Time-Varying Wind Stress Drag Coefficient in the Ekman Model with Adjoint Assimilation." Journal of Marine Science and Engineering 9, no. 11 (November 4, 2021): 1220. http://dx.doi.org/10.3390/jmse9111220.

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In this study, the time-varying wind stress drag coefficient in the Ekman model was inverted by the cubic spline interpolation scheme based on the adjoint method. Twin experiments were carried out to investigate the influences of several factors on inversion results, and the conclusions were (1) the inverted distributions with the cubic spline interpolation scheme were in good agreement with the prescribed distributions of the wind stress drag coefficients, and the cubic spline interpolation scheme was superior to direct inversion by the model scheme and Cressman interpolation scheme; (2) the cubic spline interpolation scheme was more advantageous than the Cressman interpolation scheme even if there is moderate noise in the observations. The cubic spline interpolation scheme was further validated in practical experiments where Ekman currents and wind speed derived from mooring data of ocean station Papa were assimilated. The results demonstrated that the variation of the time-varying wind stress drag coefficient with time was similar to that of wind speed with time, and a more accurate inversion result could be obtained by the cubic spline interpolation scheme employing appropriate independent points. Overall, this study provides a potential way for efficient estimation of time-varying wind stress drag coefficient.

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23

László, Lajos. "Cubic spline interpolation with quasiminimal B-spline coefficients." Acta Mathematica Hungarica 107, no. 1-2 (February 2005): 77–87. http://dx.doi.org/10.1007/s10474-005-0180-4.

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24

Zhang, Wan Jun, Feng Zhang, and Jun Hai Zhao. "Research on Modification Algorithm of Cubic B-Spline Curve Interpolation Technology." Applied Mechanics and Materials 687-691 (November 2014): 1596–99. http://dx.doi.org/10.4028/www.scientific.net/amm.687-691.1596.

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Based on cubic B-Spline curve mathematical properties, theoretical analysis the cubic B-Spline curve recursive formula of Taylor development of first-order, derivation of two order in the interpolation cycle under the condition of certain interpolation increment only and interpolation speed, change the interpolation increments can be amended cubic times B-Spline curves purpose The simulation results show that meet the high-speed and high-accuracy NC machine tool require-ments of CNC systems.

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25

Wang, Jianmin, Yabo Li, Huizhong Zhu, and Tianming Ma. "Interpolation Method Research and Precision Analysis of GPS Satellite Position." Journal of Systems Science and Information 6, no. 3 (June 29, 2018): 277–88. http://dx.doi.org/10.21078/jssi-2018-277-12.

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Abstract According to the precise ephemeris has only provided satellite position that is discrete not any time, so propose that make use of interpolation method to calculate satellite position at any time. The essay take advantage of IGS precise ephemeris data to calculate satellite position at some time by using Lagrange interpolation, Newton interpolation, Hermite interpolation, Cubic spline interpolation method, Chebyshev fitting method respectively, which has a deeply analysis in the precision of five interpolations. The results show that the precision of Cubic spline interpolation method is the worst, the precision of Chebyshev fitting is better than Hermite interpolation method. Lagrange interpolation and Newton interpolation are better than other methods in precision. Newton interpolation method has the advantages of high speed and high precision. Therefore, Newton interpolation method has a certain scientific significance and practical value to get the position of the satellite quickly and accurately.

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26

Charnsamorn, Chapkit, and Suphongsa Khetkeeree. "Symmetric quadratic tetration interpolation using forward and backward operation combination." International Journal of Electrical and Computer Engineering (IJECE) 12, no. 2 (April 1, 2022): 1893. http://dx.doi.org/10.11591/ijece.v12i2.pp1893-1903.

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The existed interpolation method, based on the second-order tetration polynomial, has the asymmetric property. The interpolation results, for each considering region, give individual characteristics. Although the interpolation performance has been better than the conventional methods, the symmetric property for signal interpolation is also necessary. In this paper, we propose the symmetric interpolation formulas derived from the second-order tetration polynomial. The combination of the forward and backward operations was employed to construct two types of the symmetric interpolation. Several resolutions of the fundamental signals were used to evaluate the signal reconstruction performance. The results show that the proposed interpolations can be used to reconstruct the fundamental signal and its peak signal to noise ratio (PSNR) is superior to the conventional interpolation methods, except the cubic spline interpolation for the sine wave signal. However, the visual results show that it has a small difference. Moreover, our proposed interpolations converge to the steady-state faster than the cubic spline interpolation. In addition, the option number increasing will reinforce their sensitivity.

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27

CHAND, A. K. B., and G. P. KAPOOR. "CUBIC SPLINE COALESCENCE FRACTAL INTERPOLATION THROUGH MOMENTS." Fractals 15, no. 01 (March 2007): 41–53. http://dx.doi.org/10.1142/s0218348x07003381.

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This paper generalizes the classical cubic spline with the construction of the cubic spline coalescence hidden variable fractal interpolation function (CHFIF) through its moments, i.e. its second derivative at the mesh points. The second derivative of a cubic spline CHFIF is a typical fractal function that is self-affine or non-self-affine depending on the parameters of the generalized iterated function system. The convergence results and effects of hidden variables are discussed for cubic spline CHFIFs.

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28

Woodward, Charles D. "B2-splines: a local representation for cubic spline interpolation." Visual Computer 3, no. 3 (October 1987): 152–61. http://dx.doi.org/10.1007/bf01962896.

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29

CHAND, A. K. B. "NATURAL CUBIC SPLINE COALESCENCE HIDDEN VARIABLE FRACTAL INTERPOLATION SURFACES." Fractals 20, no. 02 (June 2012): 117–31. http://dx.doi.org/10.1142/s0218348x12500119.

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Fractal interpolation functions provide a new light to the natural deterministic approximation and modeling of complex phenomena. The present paper proposes construction of natural cubic spline coalescence hidden variable fractal interpolation surfaces (CHFISs) over a rectangular grid [Formula: see text] through the tensor product of univariate bases of cardinal natural cubic spline coalescence hidden variable fractal interpolation functions (CHFIFs). Natural cubic CHFISs are self-affine or non-self-affine in nature depending on the IFS parameters of univariate natural cubic spline CHFIFs. An upper bound of the error between the natural cubic spline blended coalescence fractal interpolant and the original function is deduced. Convergence of the natural cubic CHFIS to the original function [Formula: see text], and their derivatives are deduced. The effects free variables, constrained free variables and hidden variables are discussed on the natural cubic spline CHFIS with suitably chosen examples.

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30

PRUESS, STEVEN. "Shape preserving C2 cubic spline interpolation." IMA Journal of Numerical Analysis 13, no. 4 (1993): 493–507. http://dx.doi.org/10.1093/imanum/13.4.493.

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31

Kang, I. G., and F. C. Park. "Cubic spline algorithms for orientation interpolation." International Journal for Numerical Methods in Engineering 46, no. 1 (September 10, 1999): 45–64. http://dx.doi.org/10.1002/(sici)1097-0207(19990910)46:1<45::aid-nme662>3.0.co;2-k.

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32

Chand, A. K. B., and G. P. Kapoor. "Generalized Cubic Spline Fractal Interpolation Functions." SIAM Journal on Numerical Analysis 44, no. 2 (January 2006): 655–76. http://dx.doi.org/10.1137/040611070.

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33

Li, Juncheng, Laizhong Song, and Chengzhi Liu. "The cubic trigonometric automatic interpolation spline." IEEE/CAA Journal of Automatica Sinica 5, no. 6 (November 2018): 1136–41. http://dx.doi.org/10.1109/jas.2017.7510442.

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34

Ramachandran, M. P. "End conditions for cubic spline interpolation." Applied Mathematics and Computation 40, no. 2 (November 1990): 105–16. http://dx.doi.org/10.1016/0096-3003(90)90125-m.

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35

Salkuyeh, Davod Khojasteh. "Stepsize Control for Cubic Spline Interpolation." International Journal of Applied and Computational Mathematics 3, no. 2 (December 22, 2015): 693–702. http://dx.doi.org/10.1007/s40819-015-0126-7.

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36

Feng, Yu Yu, and Jernej Kozak. "OnG 2 continuous cubic spline interpolation." BIT Numerical Mathematics 37, no. 2 (June 1997): 312–32. http://dx.doi.org/10.1007/bf02510215.

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37

Ha, Yujin, Jung-Ho Park, and Seung-Hyun Yoon. "Geodesic Hermite Spline Curve on Triangular Meshes." Symmetry 13, no. 10 (October 14, 2021): 1936. http://dx.doi.org/10.3390/sym13101936.

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Curves on a polygonal mesh are quite useful for geometric modeling and processing such as mesh-cutting and segmentation. In this paper, an effective method for constructing C1 piecewise cubic curves on a triangular mesh M while interpolating the given mesh points is presented. The conventional Hermite interpolation method is extended such that the generated curve lies on M. For this, a geodesic vector is defined as a straightest geodesic with symmetric property on edge intersections and mesh vertices, and the related geodesic operations between points and vectors on M are defined. By combining cubic Hermite interpolation and newly devised geodesic operations, a geodesic Hermite spline curve is constructed on a triangular mesh. The method follows the basic steps of the conventional Hermite interpolation process, except that the operations between the points and vectors are replaced with the geodesic. The effectiveness of the method is demonstrated by designing several sophisticated curves on triangular meshes and applying them to various applications, such as mesh-cutting, segmentation, and simulation.

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38

Vijayakumar, Hannah. "Two-piece Cubic Spline Functions." Mapana - Journal of Sciences 2, no. 1 (October 2, 2003): 25–33. http://dx.doi.org/10.12723/mjs.3.2.

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.M Prenter defines a cubic Spline function in an interval [a, b] as a piecewise cubic polynomial which is twice continuously differentiable in the entire interval [a, b]. The smooth cubic spline functions fitting the given data are the most popular spline functions and when used for interpolation, they do not have the oscillatory behavior which characterized high-degree polynomials. The natural spline has been shown to be unique function possessing the minimum curvature property of all functions interpolating the data and having square integrable second derivative. In this sense, the natural cubic spline is the smoothest function which interpolates the data. Here Two-piece Natural Cubic Spline functions have been defined. An approximation with no indication of its accuracy is utterly valueless. Where an approximation is intended for the general use, one must , of course, go for the trouble of estimating the error as precisely as possible. In this section, an attempt has been made to derive closed form expressions for the error-functions in the case of Two-piece Spline Functions.

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39

Ding, Ke. "Interpolation Technology and its Application in Damage Analysis of Bridge Structure." Advanced Materials Research 446-449 (January 2012): 1261–65. http://dx.doi.org/10.4028/www.scientific.net/amr.446-449.1261.

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The damage analysis based on the modal parameters usually need to compare the test data with the computed data by theoretical analysis. In bridge structure damage analysis, it is impossible that the sensors are set at every node. So the test data need to be extended to every node by using interpolation technology. In the paper, three interpolation methods, cubic interpolation method, cubic spline interpolation method and sinc interpolation method, are introduced. Their applications in bridge structure damage analysis are discussed. The experiment results show that the best interpolation method is cubic spline interpolation method. Using the data processed by interpolation technology can well obtain the damage location of bridge.

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40

Lang, Feng-Gong, and Xiao-Ping Xu. "Error Analysis for a Noisy Lacunary Cubic Spline Interpolation and a Simple Noisy Cubic Spline Quasi Interpolation." Advances in Numerical Analysis 2014 (September 10, 2014): 1–8. http://dx.doi.org/10.1155/2014/353194.

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We mainly present the error analysis for two new cubic spline based methods; one is a lacunary interpolation method and the other is a very simple quasi interpolation method. The new methods are able to reconstruct a function and its first two derivatives from noisy function data. The explicit error bounds for the methods are given and proved. Numerical tests and comparisons are performed. Numerical results verify the efficiency of our methods.

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41

Chen, Pei, Huanguo Chen, Wenhua Chen, Jun Pan, Jianmin Li, and Xihui Liang. "Improved ensemble local mean decomposition based on cubic trigonometric cardinal spline interpolation and its application for rotating machinery fault diagnosis." Advances in Mechanical Engineering 12, no. 7 (July 2020): 168781402094195. http://dx.doi.org/10.1177/1687814020941953.

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Ensemble local mean decomposition has been gradually introduced into mechanical vibration signal processing due to its excellent performance in electroencephalogram signal analysis. However, an unsatisfactory problem is that ensemble local mean decomposition cannot effectively process vibration signals of complex mechanical system due to the constraints of moving average. The process of moving average is time-consuming and inaccurate in complex signal analysis. Therefore, an improved ensemble local mean decomposition method called C-ELMD with modified envelope algorithm based on cubic trigonometric cardinal spline interpolation is proposed in this article. First, the shortcomings in sifting process of ensemble local mean decomposition is discussed and, furthermore, advantages and disadvantages of the common interpolation methods adopted to improve ensemble local mean decomposition are compared. Then, cubic trigonometric cardinal spline interpolation is employed to construct the local mean and envelope curves in a more precise way. In addition, the influence of shape-controlling parameter on envelope estimation accuracy in cubic trigonometric cardinal spline interpolation is also discussed in detail to select an optimal shape-controlling parameter. The effectiveness of cubic trigonometric cardinal spline interpolation for improving the accuracy of ensemble local mean decomposition is demonstrated using a synthetic signal. Finally, the proposed cubic trigonometric cardinal spline interpolation is tested to be effective in gear and bearing fault detection and diagnosis.

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Xie, Jin, and Xiaoyan Liu. "Adjustable Piecewise Quartic Hermite Spline Curve with Parameters." Mathematical Problems in Engineering 2021 (November 28, 2021): 1–6. http://dx.doi.org/10.1155/2021/2264871.

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In this paper, the quartic Hermite parametric interpolating spline curves are formed with the quartic Hermite basis functions with parameters, the parameter selections of the spline curves are investigated, and the criteria for the curve with the shortest arc length and the smoothest curve are given. When the interpolation conditions are set, the proposed spline curves not only achieve C1-continuity but also can realize shape control by choosing suitable parameters, which addressed the weakness of the classical cubic Hermite interpolating spline curves.

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Bogdanov, V. V., and Yu S. Volkov. "Shape-Preservation Conditions for Cubic Spline Interpolation." Siberian Advances in Mathematics 29, no. 4 (October 2019): 231–62. http://dx.doi.org/10.3103/s1055134419040011.

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Abbas, Muhammad, Ahmad Abd Majid, Mohd Nain Hj Awang, and Jamaludin Md Ali. "Positivity-preserving C2 rational cubic spline interpolation." ScienceAsia 39, no. 2 (2013): 208. http://dx.doi.org/10.2306/scienceasia1513-1874.2013.39.208.

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Bergstrom, Theodore, and David Lam. "Recovering event histories by cubic spline interpolation." Mathematical Population Studies 1, no. 4 (January 1989): 327–55. http://dx.doi.org/10.1080/08898488909525283.

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Schohl, G. A., and F. M. Holly. "Cubic‐Spline Interpolation in Lagrangian Advection Computation." Journal of Hydraulic Engineering 117, no. 2 (February 1991): 248–53. http://dx.doi.org/10.1061/(asce)0733-9429(1991)117:2(248).

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刘, 永春. "New Solution of Cubic Spline Interpolation Function." Pure Mathematics 03, no. 06 (2013): 362–67. http://dx.doi.org/10.12677/pm.2013.36055.

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BADEA, E. A., and S. PISSANETZKY. "ACCURATE CUBIC SPLINE INTERPOLATION OF MAGNETIZATION TABLES." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 12, no. 1 (January 1993): 49–58. http://dx.doi.org/10.1108/eb010334.

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Karim, Samsul Ariffin Abdul, Muhammad Aizuddin Mohd Rosli, and Muhammad Izzatullah Mohd Mustafa. "Cubic spline interpolation for petroleum engineering data." Applied Mathematical Sciences 8 (2014): 5083–98. http://dx.doi.org/10.12988/ams.2014.44284.

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Ariffin Abdul Karim, Samsul, and Kong Voon Pang. "Shape Preserving Interpolation using Rational Cubic Spline." Research Journal of Applied Sciences, Engineering and Technology 8, no. 2 (July 10, 2014): 167–78. http://dx.doi.org/10.19026/rjaset.8.956.

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