least squres vs. lagrange interpolation

Question

can some one tell me the differences between these two approximation techniques, what are the strengths and weaknesses of each, and which better one to use. Thanks

Answer 1

Look here for some nice answers: https://stats.stackexchange.com/questions/33659/how-is-interpolation-related-to-the-concept-of-regression.

Also, the following picture shows you a curve, which is an interpolation of the three points, and a line, which is the least square fit of the points.

The error of the least square fit of course comes from its own formulation. The error analysis of interpolation is as follows:

Assume that $P(x)$ is the (degree $n-1$ or less) interpolation polynomial fitting the $n$ points $(x_1,y_1),..., (x_n,y_n)$ on a function $f(x)$. The interpolation error is

$$f(x)-P(x)=\frac{(x-x_1)...(x-x_n)}{n!}f^{(n)}(c)$$,

where $c$ lies between the smallest and largest of the numbers $x,x_1,..., x_n$.

There is no general rule as to which one to use. It largely depends on what kind of problem you are dealing with, how much data you have, and how much you know your model.