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Discrete spline interpolation : ウィキペディア英語版
Discrete spline interpolation
In the mathematical field of numerical analysis, discrete spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a discrete spline. A discrete spline is a piecewise polynomial such that its central differences are continuous at the knots whereas a spline is a piecewise polynomial such that its derivatives are continuous at the knots. Discrete cubic splines are discrete splines where the central differences of orders 0, 1, and 2 are required to be continuous.
Discrete splines were introduced by Mangasarin and Schumaker in 1971 as solutions of certain minimization problems involving differences.
==Discrete cubic splines==
Let ''x''1, ''x''2, . . ., ''x''''n''-1 be an increasing set of real numbers. Let ''g''(''x'') be a piecewise polynomial defined by
:
g(x)=
\begin
g_1(x) & xg_i(x) & x_\le x < x_i \text i = 2,3, \ldots, n-1\\
g_n(x) & x\ge x_
\end

where ''g''1(''x''), . . ., ''g''''n''(''x'') are polynomials of degree 3. Let ''h'' > 0. If
:
(g_-g_i)(x_i +jh)=0 \text j=-1,0,1 \text i=1,2,\ldots, n-1

then ''g''(''x'') is called a discrete cubic spline.〔

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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