Let f:

[a,b] ->

**R** a

continuous function, [a,b] an

interval in

**R** and P

_{n} the

set of

polynomials of

degree lesser or equal n.

Let {x

_{i}| i in {1,...,m} } a set of

points in [a,b].

A polynomial p of P

_{n} is said to

**interpolate** f

iff f(x

_{i})=p(x

_{i}) for all i of {1,...,m}. This is the most common

approximation by polynomials. Other approximations are rarely used.

If m <= n+1 then a p, which interpolates f, exists and if m >= n+1 then p is unique (but doesn't have to exist !), so you would always choose m = n+1.

Note that for increasing n and m=n+1 the solutions don't have to converge to f !

They converge only for special x_{i} and surprisingly the point mustn't have all the same distance. They must be clustered around the ends of the interval, x_{i} must be equal to a + (b-a) arccos(i/m)/pi.

A simple way to calculate the interpolating p is (for m=n+1) using the Lagrange's formula:

__m (x-x1) ... (x-x<i-1>)(x-x<i+1>) ... (x-xm)
p(x) = || ------------------------------------------ f(xi)
i=1 (xi-x0)...(xi-x<i-1>)(xi-x<i+1>)...(xi-xm)

But there is a number of more clever formulas to calculate p.