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  segment polynomials  are of degree .

The segment polynomials  are uniquely given by the following  conditions

 interpolating conditions and,

 continuity conditions,

 additional conditions, , and, .

The last two conditions give rise to what are called the ``natural splines''. For example, for cubic splines, with , 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.