Formula for coefficients of interpolation polynomial

Question

(Question) What is the (conventional) formula for coefficients of interpolation polynomial?

Consider the interpolation problem: find the polynomial through a given set of points $(x_0,y_0),...,(x_n,y_n)$. Suppose we want the polynomial in canonical form (monomial basis): coefficients times powers of $x$.

Is there a standard solution to this problem? What would be the formula for the coefficients and how is it obtained?

Note: I found my own solutions for this problem, but I would like to know the ‘standard’ solution. Maybe it involves some Linear Algebra that I am not aware of yet.

gt6989b · Accepted Answer · 2019-02-11T16:33:04.673

2

Here is a general way to obtain it. Your unique polynomial will have to be of degree at most $n$, so assume we have $$p(x) = a_0 + a_1x + +\ldots + a_nx^n = \sum_{k=1}^n a_k x^k.$$

Then, plugging in your points, you get the equations $$ y_i = \sum_{k=1}^n a_k x_i^k, \quad \forall i \in [n]. $$ This is a linear system of $n+1$ equations in $n+1$ unknowns, guaranteed to be invertible due to the nature of the system. Hence, there is a unique solution, which you can find by Gaussian Elimination, for example.

UPDATE - EXAMPLE

Perhaps an example will help. Consider finding the unique linear polynomial through the points $(0,0)$ and $(1,1)$. We assume the form $p(x) = a_0 + a_1x$ which leads to the equations $$ \begin{split} 0 = a_0 + a_1 \cdot 0\\ 1 = a_0 + a_1 \cdot 1 \end{split} $$ and the first equation simplifies to $a_0=0$, plugging into the second yields $a_1=1$, so we have $$p(x) = 0 + 1 \cdot x = x,$$ which is, of course, the unique line through $(0,0)$ and $(1,1)$, as expected.

edited Feb 11 '19 at 16:33

answered Feb 11 '19 at 16:24

gt6989b

54,422

How would you define $a_k$ specifically? – Max Feb 11 '19 at 16:26
@Max you are not defining $a_k$, they are variables for which you are solving – gt6989b Feb 11 '19 at 16:29
Indeed, instead of define, I meant how do you get $a_k$ directly (so a formula for $a_k$). – Max Feb 11 '19 at 16:32
@Max see update for an example. One can use something like Cramer's Rule if you insist on calculating coefficients directly via a formula instead of a general algorithmic procedure. – gt6989b Feb 11 '19 at 16:34
I would like the formula for $a_k$ in closed form. As used in the link. – Max Feb 11 '19 at 16:38
@Max Please read up on Cramer's Rule I cited in the above comment. It is a closed form as you ask. – gt6989b Feb 11 '19 at 16:39
See also Lagrange interpolation which is a mechanical way of generating the polynomial through a set of given points (although it would need simplification to get to the form you want). – Paul Aljabar Feb 11 '19 at 16:49
@gt6989b Thanks, I understand Cramer's rule but this would probably lead to exponential computing time. – Max Feb 11 '19 at 17:39
@Max take a pick: elegant closed form with slow computation (Cramer's Rule) or cubic computation without closed form (Gaussian Elimination) – gt6989b Feb 12 '19 at 04:33
@Max +1, funny :) – gt6989b Feb 12 '19 at 16:42
If you don’t mind answering, how do we know that the system is invertible? – Ovi Apr 06 '19 at 18:56
@Ovi you can reduce it to the Vandermonde matrix, which is known to always have non-zero determinant – gt6989b Apr 07 '19 at 01:34

Formula for coefficients of interpolation polynomial

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