# cubic spline

## cubic spline

[′kyü·bik ′splīn]
(mathematics)
One of a collection of cubic polynomials used in interpolating a function whose value is specified at each of a collection of distinct ordered values, Xi (i =1, …, n), and whose slope is specified at X1 and Xn ; one cubic polynomial is found for each interval, such that the interpolating system has the prescribed values at each of the Xi, the prescribed slope at X1 and Xn, and a continuous slope at each of the Xi.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Cubic spline histopolation on a general grid is treated in  from several aspects, including methods of the practical construction of the histopolant.
Jung and Taciroglu  have used the natural cubic spline function in the X-FEM formulation to evaluate a semielliptic crack propagation on a two-dimensional plate.
In [11, 22, 23] the construction was significantly improved also for cubic spline wavelet bases.
The airloads in spanwise direction are mostly monotonic, and the classical interpolation schemes such as the cubic spline are suitable to fit the data.
For cubic spline (l = 3), the splines [s.sup.(l)](x) and their first and second derivatives are continuous at each node x; (i = 2, 3,..., m), which give 3(m - 1) conditions.
A function [S.sup.(k+1)](x, t) of class [C.sup.2][a, b] which interpolates [y.sup.(k+1)](x) at the mesh point [x.sub.i] depends on a parameter [tau], reduces to cubic spline in [a, b] as [tau] [right arrow] 0 is termed as parametric cubic spline function.
With regard to the analyses, we used natural cubic spline regressions, which has the advantage of maintaining the continuous nature of exposure variables instead of artificially categorizing it in case of a nonlinear relationship.
Several covariates were incorporated in the main GLM: a) a natural cubic spline smooth function of calendar day with 7 degrees of freedom (df) per year to exclude seasonality in mortality; b) a factor variable for "day of week" to exclude possible variations of mortality within a week; c) a cross-basis function of temperature built by the distributed lag nonlinear model (DLNM) to control for its potentially nonlinear and lagged confounding effects; and d) a natural smooth function with 3 df for the present-day relative humidity.
The exceptions from this category are the Cubic Spline filter with parameter [alpha]=-1 and the Box filter, which sometimes showed average performance, comparable to that of Flat-Top windowed-sinc, or worst performance, depending on the content of the image.
Important special cases were l = 2, the piecewise linear interpolant, and l = 4, a fuzzy cubic spline. Moreover, Kaleva obtained an interpolating fuzzy cubic spline with the not-[alpha]-knot condition.

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