# Spline Function

An interpolation method in which cell values are estimated using a mathematical function that minimizes overall surface curvature, resulting in a smooth surface that passes exactly through the input points.

A spline function consists of polynomial pieces on subintervals joined together with certain continuity conditions. Formally, suppose that n 1 points

t_{0}, t_{1}, · · · , t_{n} have been specified and satisfy t_{0} < t_{1} < · · · < t_{n}

These points are called knots.

A spline of degree 0 can be given explicitly in the form

The intervals [t_{i-1}, t_{i}) do not intersect each other, and so no ambiguity arises in defining such a function at the knots.

A spline function of degree n with nodes x_{0}, x_{1}, ..., x_{n}, is a function F(x) satisfying the following properties.

(i) F(xi) = f(x_{i}), i = 0, 1, ...., n. (Interpolation conditions).

(ii) On each subinterval [x_{i-1}, x_{i}], 1 ≤ i ≤ n, F(x) is a polynomial of degree n.

(iii) F(x) and its first (n – 1) derivatives are continuous on (a, b).

On each interval, we have four unknowns a_{i}, b_{i}, c_{i} and d_{i}, i = 1, 2, ..., n. Therefore, the total number of unknowns is 4n.

Continuity of F(x), F′(x) and F″(x) on (a, b) implies the following.

(i) **Continuity of F(x) :**

On

On

(ii)** Continuity of F′(x) :
**

**(iii) Continuity of F″ (x) :**

At the end points x_{0} and x_{n}, we have the interpolation conditions

and