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# Interpolation and Regression

Given data points interpolation is the process of finding an equation that passes through above data points and using this equation to find value of .

## Errors in Polynomials

(eq 1)

Let us consider => degree polynomial through the points

Since both and have the same value at

using eq 1

(eq 2)

represting error at the non-tabulated points

using eq 1

(eq 3)

assigning eq 2 in eq 3

let us define a function , which is the function of and

at where

is zero and also at ,
If we impose the law of mean value on , then must be continuous and differentiable. Also there exists a root of between each of the zeros of that is there and a total of zeros of . If there exists then there are zeros and similarly if the derivative exists then there exist at least a zero in the interval of
let us call this as