The experiment that breaks your intuition
You have just learned to build the unique interpolating polynomial through a set of points — in Lagrange form or via Newton's divided differences — and you have seen the error formula that says how far off it can be. So here is the natural next move: to nail a function down really well, sprinkle in more points. More points, higher degree, smaller error. That is the whole optimistic story of polynomial interpolation. The trouble is that on equally spaced points it is, for some perfectly innocent functions, flatly false.
The classic test case is Carl Runge's 1901 function f(x) = 1 / (1 + 25 x^2) on the interval [-1, 1]. It is smooth, gentle, a single tidy bell sitting in the middle — nothing dramatic. Interpolate it at equally spaced nodes and watch what happens as you add points. At degree 5 the fit is rough but reasonable. At degree 10 the polynomial starts to wobble near x = -1 and x = +1. At degree 20 those end wobbles have grown into towering spikes: the polynomial overshoots the true value by a factor of dozens, then plunges below zero, all in the regions between the outermost nodes. Adding points made it worse. Much worse.
Why the error formula allows it
Recall the error formula from the previous guide. At a point x, the interpolant of degree n misses the true f by f(x) - p_n(x) = f^{(n+1)}(c) / (n+1)! times the node polynomial omega(x), where omega(x) = (x - x_0)(x - x_1)...(x - x_n) is the product over all nodes and c is some hidden point in the interval. To make the error vanish as n grows, that product of three pieces must shrink. The factorial (n+1)! in the denominator helps a lot. But the other two factors can fight back hard.
First problem: the derivatives. For Runge's function the high derivatives f^{(n+1)} grow explosively — like 25 raised to roughly the order of the derivative, courtesy of the 25 in the denominator pushing the poles of the function (at x = +/- i/5 in the complex plane) close to the real interval. That growth can outrun the factorial. Second problem: the node polynomial omega(x). On equally spaced nodes, omega(x) is small near the center but balloons near the ends — its peaks between the outer nodes grow exponentially with n. Multiply a blowing-up derivative by a blowing-up omega and divide by a factorial that loses the race, and the end-region error grows without bound.
So Runge's phenomenon is not a numerical-arithmetic bug — it is a genuine fact about the exact interpolating polynomial. It would happen with perfect real arithmetic. (On a real machine it gets a second, separate wound: the equispaced Vandermonde system behind naive interpolation is wildly ill-conditioned, so even forming the coefficients loses many digits — but fix the conditioning and the divergence still stands.) The lesson generalizes: equally spaced high-degree polynomial interpolation is fragile, and you should reach for it only at low degree.
f(x) - p_n(x) = [ f^{(n+1)}(c) / (n+1)! ] * omega(x)
omega(x) = (x - x_0)(x - x_1) ... (x - x_n)
equispaced nodes : omega peaks BLOW UP near the ends (bad)
Chebyshev nodes : omega is EQUI-OSCILLATING, tiny everywhere (good)Cure one: stop using equally spaced nodes
The end-region peaks of omega(x) are the villain, and they come straight from the choice of equally spaced nodes. So change the nodes. The brilliant idea — the subject of guide 5 in this rung — is to bunch the nodes more densely toward the two ends, exactly where the trouble lives. The optimal bunching is given by the Chebyshev nodes: project equally spaced points off a semicircle down onto the interval, so they cluster near +/- 1 like x_k = cos(k pi / n). With these nodes, omega(x) no longer blows up at the ends; instead it equi-oscillates, staying small and even across the whole interval.
The payoff is dramatic. Interpolate the very same Runge function at Chebyshev nodes and the error now shrinks smoothly as you add points, converging fast all the way to the ends. The same degree-20 polynomial that spiked to disaster on equispaced nodes hugs the bell beautifully on Chebyshev nodes. The function did not change; only where you sampled it did. This is one of the most quietly profound facts in the whole subject: WHERE you look matters as much as HOW MUCH you look.
Cure two: keep the degree low and stitch pieces together
The other escape route refuses to play the high-degree game at all. Instead of one towering polynomial across the whole interval, lay down many low-degree pieces — a different cubic on each small subinterval — and stitch them together smoothly. That is a cubic spline, the topic of the next guide. Because no single piece ever has high degree, the omega-blowup that drives Runge's phenomenon never gets a chance to start. Splines converge reliably on equally spaced data, which is exactly why they, not high-degree polynomials, are what your plotting software actually draws through your points.
There is a real trade-off to be honest about. A global Chebyshev interpolant of a smooth function converges spectrally — faster than any fixed power of 1/n, error like c^n for some c below 1 — which is breathtaking when the function is genuinely smooth. A cubic spline converges only like O(h^4) where h is the spacing — polynomially, not spectrally. But the spline asks nothing of you: equally spaced data is fine, the matrix you solve is well-conditioned and tridiagonal, and it never blows up. Smooth function plus freedom to choose nodes, reach for Chebyshev; messy or fixed-grid data, reach for splines.
The deeper moral: interpolation is not the same as best approximation
Runge's phenomenon teaches a humbling distinction. Interpolation only insists that the curve hit your chosen points; it says nothing about behavior in between. Best approximation asks the truly useful question instead: among all polynomials of degree n, which one minimizes the worst error over the WHOLE interval? That is minimax approximation, and its answer never diverges the way equispaced interpolation does. By the Weierstrass theorem, some sequence of polynomials always converges uniformly to any continuous function — the failure was never about polynomials being weak, only about a bad way of choosing them.
The beautiful punchline, which the last guide of this rung makes precise, is that Chebyshev-node interpolation is almost as good as the true minimax best approximation — within a slowly growing factor (the Lebesgue constant, which grows only like log n for Chebyshev nodes, versus exponentially for equispaced ones). So the practical recipe falls out cleanly: if you can choose where to sample, choose Chebyshev nodes and you get near-best accuracy almost for free; if you are handed fixed equally spaced data, use a spline and keep the degree low.
Guard against one common misreading: 'high-degree polynomials are bad.' Not so. High-degree polynomials on the RIGHT nodes are superb — the degree-20 Chebyshev fit above is proof. The villain is the equispaced grid at high degree, not the degree itself. Same tool, different hands. And keep the universal honesty of this whole subject in view: every fit is an approximation, on a real machine it carries rounding error too, and a curve that nails your samples can still be wildly wrong between them — so always test a candidate interpolant at points you did NOT feed in.