The nodes were evenly spaced

A condition imposed at points is not a condition measured how much a collocated fit hides. A boundary condition satisfied exactly at a finite set of samples departs from it between them by a factor of 8.3, the solver’s own residual cannot see any of that, and the rung named three repairs.

Two were made. More samples close the gap in the report; more free coefficients on fixed samples open it. The third was named and not tried:

The collocation nodes are evenly spaced, which the rung shows to be the worst case, and clustering them where the map works hardest is the standard remedy. Every construction in the condition anchor uses even spacing.

Trying it takes one line in the boundary generator and produces a result the previous rung’s arithmetic did not predict.

Where the nodes go #

A collocation is a set of places where a condition is required to hold exactly. This anchor’s constructions all impose the boundary scale is constant along a stated region’s edge, and they all place the nodes at equal steps along it — which is the obvious thing to do and is what a loop over an edge produces.

The Chebyshev positions are $t_k = \tfrac12\left(1 - \cos\frac{\pi k}{m}\right)$ along each edge, which puts about $\sqrt{m}$ of the $m$ nodes into the last $1/m$ of the interval at each end. They are the standard answer to a standard problem: a fit whose error concentrates at the ends of an interval should have its samples concentrated there too, and evenly spaced nodes are the textbook worst case for exactly that reason.

The reason a conformal map of a polygon has its error at the corners is not a numerical accident. The map is singular there — the Schwarz–Christoffel construction has a branch point at each corner, and the series being fitted is smooth — so the boundary condition is hardest to hold in the neighbourhood a uniform sampler treats like any other.

What changes #

Read the second column and there is no convergence in it. The value at 160 nodes is larger than the value at 80, and larger than the value at 48; it goes down and up and down again with no trend, across a five-fold increase in the number of equations. The whole range is 43 per cent wide.

Read the third and the fit reaches a floor at forty-eight nodes and stays on it to four significant figures for the rest of the ladder.

That is a different statement from “clustering is better”. At any single node count the improvement is modest — between 1.06 and 1.75 times — and a comparison at one count would report a mild gain and move on. What the ladder shows is that the two behave differently in kind: one converges and one does not.

Why the even fit cannot converge #

The previous rung has the mechanism, and this is that mechanism read at the other end.

With twelve terms the fit has twenty-five unknowns. At forty-eight nodes it has ninety-six equations, so it is a least-squares problem rather than an interpolation and there is a genuine residual. The residual is spread over the nodes according to where they are, and evenly spaced nodes ask the series to hold the condition at the corners no harder than anywhere else — so the fitted coefficients trade accuracy at the corners against accuracy along the flat parts, and the trade depends on how many nodes happen to fall near each corner.

Adding nodes changes that count. It does not change it monotonically, because how close the outermost node of an edge sits to the corner depends on the arithmetic of $m$, and a quantity that depends on where the last sample happens to land is a quantity with no limit. That is the same structure as the doubling ladder that cannot see a comb: a feature the sampler’s arithmetic decides whether to resolve, producing a sequence that looks like noise and is not.

Clustering removes it by construction. The Chebyshev nodes approach each corner at a rate set by the cosine rather than by $1/m$, so the ratio of node spacing to the distance from the corner is asymptotically constant, and the corner is always resolved to the same relative depth however many nodes there are.

The same ladder at a shorter series #

The floor a clustered fit settles on is the truncation’s, so shortening the series should raise it without changing the shape of either curve. It does.

That separation matters for how the result should be used. Somebody who wants a better map needs more terms; somebody who wants a trustworthy one needs the nodes in the right places, and the two are independent decisions that this anchor’s constructions have been making as one.

The control #

An ellipse has no corners, its conformal map is analytic on the boundary, and the series converges to $4.5\times10^{-12}$ at either node placement. Moving the nodes changes that by three per cent, which is nothing.

So the improvement on the square belongs to its corners rather than to the arithmetic of moving nodes about. That is the control the rung needs: a version of this essay in which clustering helped everywhere would be reporting a property of the solver, and one in which it helped nowhere would have nothing to report.

The assertion states both halves. On the square the clustered fit’s departure must be constant to within two per cent across the refined counts while the even one spreads by more than twenty; on the ellipse the two must agree to within twenty per cent and the residual must be below $10^{-9}$, or the control is not converging and is not a control.

What this says about the anchor’s other constructions #

Every solved map in this anchor collocates, and every one of them uses even spacing. It is worth being precise about which of their published results this rung disturbs and which it does not.

The nearest map to an impossible request is a least-squares fit against a stated target field, collocated on a boundary. Its residual is a property of the request rather than of the fit — the request is unachievable and the residual measures by how much — so a node placement that fits the achievable part better does not change the finding. It changes the number by something of the order measured here.

A map with no formula is the construction itself, and its accuracy claims are of the kind this rung is about: stated as a boundary residual, on a fit whose refinement does not converge. Those numbers are as good as any single member of a non-convergent sequence, which is to say they are indicative rather than settled.

The Chebyshev cap, where the answer is known in closed form — the stereographic projection centred on the cap — is unaffected, and it is the reason this rung’s finding did not surface earlier. A cap is smooth and its collocation converges at any placement, so the one case with a known answer is the one case where the defect cannot appear. The verification that licensed the whole machinery was run on the shape immune to its worst failure, which is not a mistake anybody made deliberately and is exactly how a blind spot survives.

What was computed, and how #

The fit is chebyshevMap’s, unchanged: a truncated series solved by least squares against a stated number of boundary samples, wrapped in the ordinary projection interface so that every measurement this site owns applies to it. The only change is one function that decides where along each edge the $k$-th of $m$ samples sits.

The dense boundary that measures the residual is always evenly spaced, whatever the fit’s nodes were. That matters: measuring a clustered fit on a clustered sample would commit the rung’s own defect twice, and would report a residual that is small because the instrument shares the fit’s blind spots.

What a solver could report instead #

The previous rung’s complaint was that the solver’s residual is not the map’s, and its repair was to measure the map on a refined boundary. This rung adds a second thing a solver could hand back for nothing.

The residual at a stated refinement, at two node placements. Running the fit twice — once evenly, once clustered — and reporting both costs one extra solve, which for a twenty-five-unknown least-squares problem is microseconds. If the two agree, the fit has converged in the sense that matters. If they differ, the number being reported is a member of a sequence with no limit and should be labelled as one.

That is a cheaper and more informative diagnostic than refining, because refining a non-convergent sequence produces more members of it and says nothing about whether it has a limit. Two placements at one count is a better test than one placement at two counts, and the arithmetic above is the reason: the failure lives in the placement axis and the standard test explores the other one.

Where the model stops #

One region and one series. A spherical square of 25° circumradius, fitted with twelve terms. The floor’s value — $7.03\times10^{-5}$ — is a property of the truncation rather than of the nodes, and a longer series reaches a lower one. What the rung claims is about the shape of the two ladders rather than about that number.

Clustering per edge, not globally. The nodes here are Chebyshev along each edge separately, so they crowd at every corner equally. A boundary whose corners differ in interior angle has singularities of differing strength and would want the crowding weighted, which is standard in the Schwarz–Christoffel literature and is not done here.

And this does not repair the previous rung. A clustered fit still satisfies the condition exactly at its own nodes and approximately between them, and the solver still reports the first. The hidden ratio is smaller and it is not zero, and nothing about node placement changes the fact that a condition imposed at points is a condition imposed at points.

The generalisation #

The rule this pays out is one about how to read a refinement.

The previous rung established that adding samples improves what the solver says and not what the map does. The obvious inference — that the collocation is already as good as it gets and the remaining error is the truncation’s — is wrong, and this rung is why. The remaining error was the nodes’ placement, and it was invisible because refining is the only knob anybody turns.

A refinement sequence explores one direction through a space of methods. Node count is one axis and node placement is another, and a method that will not converge along the first can converge immediately along the second. The threshold is not a percolation and the first break is mostly its denominator are the same lesson from the neighbouring anchor: a quantity fitted along one axis produced an exponent that turned out to belong to a different variable entirely.

The practical form: when a refinement stops paying, check whether the thing being refined is the thing that is wrong. Here the answer was no, the fix cost one line, and it had been named in a shortfall note before it was tried. Not every distortion can be asked for is the constraint that made this anchor need a solver at all, and it is unchanged by any of this.

The number that is left over #

The clustered fit settles on 7.030 × 10⁻⁵ and nothing here says what that number is. It is the truncation’s, not the collocation’s — a shorter series settles higher and a longer one lower — so it is the residual a map with no formula would report if its nodes were in the right places.

Quantifying how it falls with the term count is a ladder along the third axis of the same method and it is not run here.

Who found it, and when #

Everything about the numerics is old. Runge’s example of 1901 is the canonical demonstration that evenly spaced interpolation nodes fail on a function that clustered ones handle, and the Chebyshev points are the standard remedy in every approximation-theory text since. The concentration of a conformal map’s error at a polygon’s corners is Schwarz’s and Christoffel’s from the 1860s, and the practice of crowding collocation points there is standard in the numerical conformal mapping literature — Trefethen’s treatment of the Schwarz–Christoffel transformation is explicit about it.

What is new here is only the join, and the join is the same shape as this collection’s other findings. The cartographic literature on solving for a projection under stated conditions — Chebyshev’s criterion and everything descended from it — is written in the language of variational calculus and states the conditions as conditions rather than as a numerical scheme. The numerical analysis literature knows exactly where to put the nodes and has no reason to be talking about maps. So a construction that solves a two-hundred-year-old cartographic problem with a twentieth-century numerical method inherited the wrong default, and the default survived because refining it made the report look better.

Why refining it made the report look better #

The half-sentence at the end of the account above is the part with the most reach, and it deserves stating as a rule rather than as a remark.

The residual was evaluated where the condition was imposed. A collocation scheme forces the condition to hold at its nodes, so the residual at those nodes is zero up to the solver’s own floor — by construction, whatever the nodes are and however badly they are placed. Adding nodes adds places where the residual is zero.

So the diagnostic improved as the fit got worse in the gaps. With evenly spaced nodes the error between them grows as the count rises, which is exactly Runge’s phenomenon, and none of that growth is visible to a check that only looks at the nodes. The reported number falls, the map deteriorates, and the two are not in contradiction because they are measurements of different things.

The rule is that a residual must be evaluated somewhere the fit was not imposed. A separate, denser, differently placed set of points — offset from the nodes, ideally between them where the error is largest. That is one extra evaluation pass of a function the solver already has, and it converts a quantity that cannot fail into one that can.

And the failure mode is general. Any scheme that enforces a condition at a finite set and then reports its residual at that same set is reporting its own arithmetic. It applies to collocation, to interpolation, to a least-squares fit checked on its training points, and to any gate that tests a construction at the places the construction was built to satisfy.

It cost nothing to fix and it had been wrong for eleven rungs.

Which is this collection’s own habit stated from the inside. A check that cannot come out the other way proves nothing, and the residual is zero at the nodes is a statement that could never have come out the other way. The repair was not a better solver; it was measuring somewhere the answer was not guaranteed.

Where the ladder goes next #

Eleven rungs have taken a condition from a statement about a map to a system of equations and back. Every one of them has imposed the condition on a boundary, because that is where Chebyshev’s criterion says it belongs, and the interior has been left to look after itself. Whether it does — whether a map whose boundary scale is constant to five decimal places has an interior worth having — is a question this anchor has assumed the answer to since its first rung.