# Interpolation

**Interpolation:**

Interpolation is the process of finding a function whose graph passes through a set of given points. It is a method of constructing new data points within the range of a discrete set of known data points. In engineering and science one often has a number of data points, as obtained by sampling or experimentation, and tries to construct a function which closely fits those data points. This is called curve fitting or regression analysis. Interpolation is a specific case of curve fitting, in which the function must go exactly through the data points.

**Finite Difference:
**

Let y = f (x) be a discrete function. Consider the given data set of n 1 values (x_{0}, y_{0}), (x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{n}, y_{n}), where x differ by a quantity h, i.e. values of x are equidistant with interval distance h. The value of x is usually called argument and the corresponding function value y is called entry. In following subsection, we discuss three types of finite differences:

**1**. Forward Differences

**2**. Backward Differences

**3**. Central Differences.

**Forward Differences:**

For the given data set of n 1 values (x_{0}, y_{0}), (x_{1}, y_{1}), (x_{2}, y_{2}), ..., (x_{n}, y_{n}), the quantities y_{1 }− y_{0}, y_{2} − y_{1}, ..., y_{n }− y_{n−1} are called differences, particularly first differences, and are denoted by Δy_{0}, Δy_{1}, ..., Δy_{n−1} respectively. In general, first forward differences are given by;

The symbol Δ is called forward difference operator. Further second forward differences are defined as the differences of the first differences. i.e.,

Here, Δ^{2} is called second forward difference operator. Similarly, other higher order forward difference may be computed. In general,

**Forward Difference Table:**