Important: The spline function has been deprecated. Use the superseding function CurveFitting[Spline] instead.  A call to spline automatically generates a call to CurveFitting[Spline].

The spline function computes a piecewise polynomial approximation to the X Y data values of degree d (default 3) in the variable z. The X values must be distinct and in ascending order. There are no conditions on the Y values. The result is returned in the form

which is equivalent to

where the $N=\mathrm{nops}\left(X\right)-1$ segment polynomials $P_i$ are of degree $\le d$.

The segment polynomials $P_1,P_2,\mathrm{...},P_N$ are uniquely given by the following $\left(d+1\right)N$ conditions

$2N$ interpolating conditions$P_i|z=X_i=Y_i$ and$P_i|z=X_{i+1}=Y_{i+1}$, $i=1..N$

$\left(d-1\right)\left(N-1\right)$ continuity conditions$\frac{{\ⅆ}^k}{\ⅆx^k}{P}_i|z=X_{i+1}=\frac{{\ⅆ}^k}{\ⅆx^k}{P}_i|z=X_{i+1}$, $i=1..N-1,k=1..d-1$

$d-1$ additional conditions$\frac{{\ⅆ}^k}{\ⅆx^k}{P}_1|z=X_1=0$, $k=d-\mathrm{iquo}\left(d\,2\right),\mathrm{...},d-1$, and$\frac{{\ⅆ}^k}{\ⅆx^k}{P}_N|z=X_{N+1}=0$, $k=d-\mathrm{iquo}\left(d-1\,2\right),\mathrm{...},d-1$.

The last two conditions give rise to what are called the ``natural splines''. For example, for cubic splines, with $k=2$, we obtain the two conditions

The fourth optional argument must be a positive integer (default 3) or one of the keywords linear, quadratic, cubic, or quartic. It specifies the degree of the segment polynomials.

$\mathrm{spline}\left(\left[0\,1\,2\,3\right]\,\left[0\,1\,4\,3\right]\,x\,\mathrm{linear}\right)$

$\left\{\begin{array}{cc}x& x<1\\ -2+3x& x<2\\ 6-x& \mathrm{otherwise}\end{array}\right.$

$\mathrm{spline}\left(\left[0\&comma;1\&comma;2\&comma;3\right]\&comma;\left[0\&comma;1\&comma;4\&comma;3\right]\&comma;x\&comma;\mathrm{cubic}\right)$

$\left\{\begin{array}{cc}\frac{4}{5}{x}^3+\frac{1}{5}x& x<1\\ -2x^3+\frac{42}{5}{x}^2-\frac{41}{5}x+\frac{14}{5}& x<2\\ \frac{6}{5}{x}^3-\frac{54}{5}{x}^2+\frac{151}{5}x-\frac{114}{5}& \mathrm{otherwise}\end{array}\right.$