Introduction:

One of the common tasks faced by scientists and engineers is the problem of estimating the value of dependent variable $y$ for an intermediate value of the independent variable $x$, given a table of discrete data points $(x_{i},y_{i}), \quad i=0,1,2,...n.$ If We had a function $y(x)$ that passes through the given data points then we can use it for evaluating $y(x)$ for the required value of $x$.

Can we construct such a function $f(x)?$
Yes we can and the process of constructing $y(x)$ to fit the given table of data points is called curve fitting.

Depending upon the source from where the data is drawn, the value of the data points may or may not be accurate. For instance, if the data points are drawn from say, logarithmic or trigonometric or interest or steam tables they are treated as accurate as these tables are generated by using well behaved functions. On the contrary the data drawn from experimental measurement tabulations are prone to error's and are treated as not exact.

When the given data is exact then it is meaningful to construct the function passing through the data points. The method of constructing through the data points. The method of constructing a function and estimating value at intermediate points is called interpolation. The functions, thus constructed, are called as interpolating polynomials.

Some of the common methods of interpolation are

1. Lagrange Interpolation
2. Newton's Interpolation
3. Divided Difference Interpolation Formula
4. Spline Interpolation, etc.

The Common strategy to deal with the inexact data is find an approximate function that would represent the general trend of the data, without necessarily passing through the individual points. Linear and non-linear Least Square Regression has been a standard approach in finding such approximating functions.

So, now we discuss about interpolation and later about Least Square Regression process.