Abstract
Take one function, sample it at the same points, and add points. This paper asks what decides whether the approximation improves or worsens——the answer is that there are two things one can move, independently. No new mathematical theorem and no new law is claimed.
First, from the same $29$ points the polynomial gives $\mathbf{1131.420474}$ and the cubic spline $\mathbf{0.001039}$ (Section 2).
Second, this is the core of the paper. There are two cures, and either one alone suffices (Section 3).
Third, moving the nodes gives $\mathbf{0.00365254}$ and moving the family gives $\mathbf{0.00103905}$. Improvements of $309762.9$ and $1088894.0$ (Section 3).
Fourth, what breaks lies in the placement of the nodes, not in the function. The Lebesgue constant at $n=29$ is $1803783.3902$ against $3.0829$ (Section 4).
Fifth, the equispaced polynomial passes an error of $1$ at $n=9$. That is larger than the maximum of the function itself (Section 4).
Sixth, the separator is whether the nodes were moved or the family was (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——Runge’s example of $1901$, Chebyshev nodes, cubic splines, and the Lebesgue constant are all standard. We do not build approximation theory——all we use is one function and three approximations of it. We prove no convergence theorem——the general question of which family converges under which hypotheses is not entered. Only numbers actually measured for one function are put down. We do not seek optimal nodes——that Chebyshev nodes are near-optimal is used, but no optimality is proved. We do not treat finite elements——using splines to solve differential equations is not entered. It is mentioned and no more. Rounding error is not the subject——the numbers here come out to the same digits in extended precision. Loss of significance belongs to Paper 221 and Paper 330. Relation to earlier papers: Paper 221 showed that changing the last bit moves an eigenvalue by $0.16$——there the input is perturbed; here the input is exact and the approximating family collapses anyway. Both are “the numbers cannot be trusted”, from different causes. Paper 300 showed whether two things share a root is decidable——by that criterion the two cures here have distinct roots. Moving one leaves the other where it was. Paper 182 showed “the curse of dimensionality” is not one curse——the same form, with “the cure for the Runge phenomenon” not being one cure here. Paper 324 showed that no search is universal because of how the problems are counted——that every approximating family has functions it handles badly comes from the same root. Paper 316 showed the same integer returning from three different counts——there different routes reach the same answer; here different routes reach answers equally good. What is added is putting all three on one function under one way of measuring, placing the Lebesgue constant in the same table so that the “why” is a number, identifying the $n$ at which the error passes the maximum of the function, and putting the separator on nodes against family.
1. Notation and the Question
Setting: On the interval $[-1,1]$ take
$$f(x)=\frac{1}{1+25x^2}.$$The function is smooth on $[-1,1]$ and its values run only from $1/26$ to $1$——as a target for approximation it is not a particularly nasty shape.
Observation 1.1 More points ought to mean a better fit, one naively thinks. There is no intuitive reason why more samples should hurt——and yet the interpolating polynomial on equispaced points gets worse the more points one adds.
Observation 1.2 The way of measuring is fixed first. Error means the maximum absolute error over $[-1,1]$ sampled at $20001$ points——all three are measured with the same ruler.
Question: What is it that breaks, and what must be moved to repair it?
2. Same Points, Same Function, Two Outcomes
Calculation 2.1: Take $n$ equispaced points and compare (a) the polynomial through them with (b) the cubic spline through them (computed in this paper).
| Points $n$ | Equispaced polynomial | Equispaced cubic spline |
|---|---|---|
| $5$ | $0.438357$ | $0.317082$ |
| $13$ | $3.663393$ | $0.006909$ |
| $21$ | $59.822309$ | $0.003183$ |
| $29$ | $\mathbf{1131.420474}$ | $\mathbf{0.001039}$ |
Observation 2.2 The left column rises and the right one falls. The same points, the same function——and opposite directions.
Observation 2.3 At $n=5$ there is almost nothing between them. $0.438357$ against $0.317082$, a factor of $1.38$——tried at a small number of points, the difference is invisible.
Observation 2.4 At $n=29$ they are apart by a factor of $1088894$. $1131.420474$ against $0.001039$——from one and the same sample, answers six orders of magnitude apart.
Observation 2.5 What the left column means, said exactly. The values of $f$ never exceed $1$. An error of $1131$ therefore means——that the approximation swings to a thousand times the height of what it approximates.
3. The Core: There Are Two Cures and They Are Independent
Setting: To repair what is broken there are two things one may move.
| What is moved | What it changes |
|---|---|
| Placement of the nodes | where the samples are taken |
| The approximating family | which set of functions is fitted |
Calculation 3.1: Fix $n=29$ and move one at a time (computed in this paper).
| Nodes | Family | Maximum error | |
|---|---|---|---|
| Baseline | equispaced | polynomial | $\mathbf{1131.420474}$ |
| Move the nodes only | Chebyshev | polynomial | $\mathbf{0.00365254}$ |
| Move the family only | equispaced | cubic spline | $\mathbf{0.00103905}$ |
Observation 3.2 This is the core of the paper. Moving either one alone repairs it——leave the nodes equispaced and the spline gives $0.00103905$; leave the family polynomial and Chebyshev nodes give $0.00365254$.
Observation 3.3 The improvements are $309762.9$ and $1088894.0$. Both are five to six orders of magnitude——this is not a difference between working and not working; both work decisively.
Observation 3.4 The two answers are not the same. Chebyshev over spline is $3.5152$——equally good, but not equal. Two separate mechanisms happen to land in neighbouring decades.
Observation 3.5 Apply the criterion of Paper 300.
| Criterion | Result |
|---|---|
| Is there a correspondence preserving an identity | No. One is about nodes, one about a function space |
| Does the same quantity appear | No. A Lebesgue constant against a piecewise degree |
| Does moving one move the other | No. Changing nodes barely moves the spline error |
Observation 3.6 All three rows point to distinct roots. One phrase, “the cure for the Runge phenomenon”, names two things that do not meet——the same shape Paper 182 found in “the curse of dimensionality”.
4. Why: What Breaks Does Not Lie in the Function
Setting: Does interpolation error come from a bad function or from a bad placement of points? These can be separated. There is a quantity fixed by the nodes alone, the Lebesgue constant $\Lambda_n$——an amplification factor for error computable from the positions of the points without ever looking at the function.
Calculation 4.1: Compute $\Lambda_n$ for the same node sets (computed in this paper).
| Points $n$ | Equispaced | Chebyshev |
|---|---|---|
| $5$ | $2.2078$ | $1.7988$ |
| $13$ | $89.3249$ | $2.5393$ |
| $21$ | $10986.6965$ | $2.8678$ |
| $29$ | $\mathbf{1803783.3902}$ | $\mathbf{3.0829}$ |
Observation 4.2 $f$ does not appear once in this table. Only the positions of the points went into it——so the cause of the collapse does not lie on the side of the function being approximated.
Observation 4.3 Equispaced grows by a factor of $816995.9$ from $n=5$ to $n=29$, while Chebyshev goes only from $1.7988$ to $3.0829$. Same interval, same count——placement alone moves the amplification by six orders of magnitude.
Observation 4.4 The equispaced polynomial passes an error of $1$ at $n=9$. And $1$ is the maximum of $f$ itself——past nine points, answering $0$ and doing nothing is the better policy.
Observation 4.5 To be stated honestly——the spline wins here for this function. Cubic spline convergence saturates at a fixed order, so for an analytic function wanted to high accuracy Chebyshev is the faster of the two——what this paper claims is that two cures act independently, not that splines are always better.
Observation 4.6 Here is the parting from Paper 221. There, moving the last bit of the input moved the answer——here the input is exact and extended precision returns the same numbers. What is broken is not the arithmetic but the construction of the approximation.
5. The Separator: Were the Nodes Moved, or the Family?
Observation 5.1 The separator——whether the remedy changed where the samples are taken, or which set of functions is fitted.
| Changed | Freedom in nodes | How error is held down | |
|---|---|---|---|
| Chebyshev nodes | position | given up | keeps $\Lambda_n$ small |
| Cubic spline | family | kept | has no high-order terms |
Observation 5.2 The third column is what divides practice. If you choose your own sample points Chebyshev is available; if you do not, only the spline remains——against data already taken, one of the two hands is missing from the start.
Observation 5.3 Both may be moved at once. And doing both merely doubles what one of them already achieved in five orders of magnitude——they do not add, because the two stop the same illness from opposite sides.
Observation 5.4 To be stated honestly——“equispaced is bad” is too strong. Equispaced is harmless while $n$ is small ($0.438357$ at $n=5$)——what is bad is not the placement but the placement together with the degree.
Observation 5.5 Where this stands relative to Paper 324. Paper 324 wrote that no search is universal because of how the problems are counted——this is the approximation-theoretic version, and that every family has functions it handles badly comes from the same root. What is shown here is that there are two independent routes around the bad cases.
Observation 5.6 One last thing——taking more samples is not the cause of this. More samples are welcome. The cause is having demanded that one single polynomial pass through all of them——the data were not bad; the demand was too strong.
6. The Census and the Close
- $1131.420474$ for the equispaced polynomial at $n=29$ (Calculation 2.1): the function’s maximum is $1$.
- $0.001039$ for the spline on the same points (Calculation 2.1): the opposite direction.
- Only a factor of $1.38$ at $n=5$ (Observation 2.3): invisible at few points.
- $0.00365254$ from moving the nodes (Calculation 3.1): family still polynomial.
- $0.00103905$ from moving the family (Calculation 3.1): nodes still equispaced.
- Chebyshev over spline is $3.5152$ (Observation 3.4): equally good, not equal.
- Lebesgue constants $1803783.3902$ against $3.0829$ (Calculation 4.1): $f$ never appears.
- The error passes $1$ at $n=9$ (Observation 4.4): answering $0$ becomes better.
- Doing both only doubles it (Observation 5.3): they do not add.
- Without a choice of points, one hand is missing (Observation 5.2): this divides practice.
Closing: Approximate $f(x)=1/(1+25x^2)$ on equispaced points of $[-1,1]$ and the polynomial gets worse as points are added while the spline on the very same points gets better——at $n=29$, $\mathbf{1131.420474}$ against $\mathbf{0.001039}$, six orders of magnitude out of one sample. Since $f$ never exceeds $1$, the polynomial is swinging to a thousand times the height of what it approximates. At $n=5$ the two are only $1.38$ apart——tried at few points, the difference is invisible. And there are two cures, either of which alone repairs it——move only the nodes to Chebyshev and the error is $\mathbf{0.00365254}$ (an improvement of $309762.9$); move only the family to splines and it is $\mathbf{0.00103905}$ (an improvement of $1088894.0$). The ratio between the two answers is $3.5152$: equally good, not equal——two separate mechanisms landing in neighbouring decades. The cause of the collapse does not lie on the side of the function——the Lebesgue constant, computed from the node positions alone, is $1803783.3902$ for equispaced against $3.0829$ for Chebyshev at $n=29$, and $f$ does not enter that calculation at all. The equispaced polynomial passes an error of $1$ at $n=9$, beyond which answering $0$ and doing nothing is the better policy. One thing separates them——whether the remedy changed where the samples are taken or which set of functions is fitted. If you choose your own sample points Chebyshev is available; if you do not, only the spline remains——against data already taken, one of the two hands is missing from the start. Under the criterion of Paper 300 all three rows point to distinct roots——one phrase names two things that do not meet. One last thing——taking more samples is not the cause of this. More samples are welcome. The cause is having demanded that one single polynomial pass through all of them. The data were not bad; the demand was too strong.
References
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- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. Zenodo. 10.5281/zenodo.22104170
- Yamagishi, Y. (2026). Change the Last Bit and an Eigenvalue Moves by 0.16 [Paper 221]. Zenodo. 10.5281/zenodo.22120012
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Same Integer Comes Back from Three Different Counts [Paper 316]. Zenodo. 10.5281/zenodo.22208229
- Yamagishi, Y. (2026). That No Search Is Universal Follows from How the Problems Are Counted [Paper 324]. Zenodo. 10.5281/zenodo.22209395
- Yamagishi, Y. (2026). That the Condition Number Squares Is an Identity, Not an Approximation [Paper 330]. Zenodo. 10.5281/zenodo.22681423
Reference
Yamagishi, Y. (2026). From the Same Points, One Method Diverges by a Thousand and the Other Converges to a Thousandth [Paper 329]. YOI: https://www.yamagishiyuuki.com/yoi/329
DOI: 10.5281/zenodo.22681343
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 329. This paper is a preprint and has not been peer reviewed.