Foundations — Error in polynomial interpolation

This page assumes you know nothing yet. Before you can read the error formula $E(x)=\frac{f^{(n+1)}(\xi )}{(n+1)!}{\prod }_{i=0}^n(x-x_i),$ you must own every single symbol in it. We build them one at a time, each one earning its place before the next appears. When you finish, go read Error in polynomial interpolation.

1. $f(x)$ — the true function we don't fully know

The picture. Draw a smooth curve on graph paper. For every horizontal position $x$, the curve has one height — that height is $f(x)$.

Why the topic needs it. Interpolation is about guessing this curve when we only know a few of its heights. Without a "true curve" there is nothing to be wrong about.

2. Nodes $x_0,x_1,\dots ,x_n$ — the dots we actually know

The picture. On the same curve, stab a few dots. Each dot sits at a known horizontal position $x_i$ and a known height $f(x_i)$. Everything between the dots is unknown territory.

Why the topic needs it. These dots are the only hard facts we have. The whole error story is about the gaps between them.

3. Distinct — the dots must not stack

The picture. Two dots at the exact same horizontal spot would be one dot pretending to be two — you can't fit a single unique curve through that. Spread them out sideways.

Why the topic needs it. Distinctness is what guarantees there is exactly one polynomial of degree $\le n$ through the dots. See Lagrange Interpolation and Newton Divided Differences for the two standard ways to build that polynomial.

4. Polynomial $p_n(x)$ and "degree"

The picture. Degree $0$ is a flat horizontal line. Degree $1$ is a straight slanted line. Degree $2$ is a single U-shaped bend (a parabola). Higher degree $=$ more allowed wiggles.

Why the topic needs it. $p_n$ is our guess: the unique polynomial of degree $\le n$ that passes through all $n+1$ dots. The error compares this guess to the truth.

5. $E(x)$ — the error, "truth minus guess"

The picture. At each $x$, draw a vertical stick from the guess curve up (or down) to the true curve. The length-with-sign of that stick is $E(x)$. At every node the two curves touch, so the stick has zero length: $E(x_i)=0$.

Why the topic needs it. This is the quantity the whole parent page measures. Everything else exists to describe the shape of these vertical sticks.

6. The product ${\prod }_{i=0}^n(x-x_i)$ — distance from the dots

Read $(x-x_i)$ as a signed distance. If $x$ is far from node $x_i$, then $(x-x_i)$ is big in size; if $x$ is the node, $(x-x_i)=0$.

The picture. $\omega$ is itself a polynomial that dips to zero at each dot and bulges in the gaps. In each gap it makes a hump; the biggest humps live near the ends of the interval.

Why the topic needs it. $\omega (x)$ is the "how far from the dots" knob you control by choosing where to place nodes — the reason Chebyshev Nodes beat evenly spaced nodes and dodge the Runge Phenomenon.

7. Derivatives $f^{\prime }$, $f^{\prime \prime }$, and $f^{(n+1)}$ — measuring twistiness

Why we suddenly need derivatives here. The straighter the true curve, the easier it is to guess between dots; the more it bends, the more it can sneak away from our polynomial between the dots. "Bending" is exactly what higher derivatives measure — that is why a derivative, and not some other tool, shows up in the error.

Why the $(n+1)$-th derivative specifically? With $n+1$ dots, our guess $p_n$ already matches the truth in "$n+1$ pieces of information." The first thing it cannot control is the $(n+1)$-th derivative — so that is precisely the leftover twistiness that leaks into the error. This mirrors Taylor's Theorem with Remainder, where the leftover term also uses the next derivative.

8. $\xi$ — the mystery sample point

The picture. Somewhere between the leftmost and rightmost dots there is a hidden spot $\xi$ whose bending value $f^{(n+1)}(\xi )$ makes the formula come out exactly right. It hides at a different place for each test point $x$, so really $\xi =\xi (x)$.

Why the topic needs it — and where it comes from. This existence guarantee is handed to us by Rolle's Theorem (the special case of the Mean Value Theorem where the endpoints have equal height). Rolle says: between two points of equal height, the slope is zero somewhere — but never says exactly where. That "somewhere" is our $\xi$.

9. Factorial $(n+1)!$ — the softening denominator

The picture. A staircase that gets steep fast: $1,2,6,24,120,720,\dots$ It grows quicker than any fixed power.

Why the topic needs it. In the error formula $(n+1)!$ sits in the denominator, shrinking the error. It appears naturally in the derivation: differentiating the top power of $\omega$ exactly $n+1$ times spits out $(n+1)!$. Miss it and your bound is wildly wrong.

10. $C^{n+1}[a,b]$ — the smoothness ticket

The picture. A curve you could draw without lifting the pen, with no sharp corners even after bending it many times over.

Why the topic needs it. The error formula secretly differentiates $n+1$ times. If those derivatives didn't exist or jumped around, the whole argument collapses. $C^{n+1}[a,b]$ is the fine print that lets the machinery run.

How these feed the topic

Read it top to bottom: the true function and its known dots build the guess and the error; distances give $\omega$; derivatives give twistiness; Rolle gives $\xi$; the factorial and smoothness ticket finish the formula.

Equipment checklist

Cover the right side and answer each before revealing.

When every reveal is instant, go read Error in polynomial interpolation.