sim-numerics/article/sim_numerics.md

Similarity as a Step Rule: Parameter-Free Root Finding Derived from a Similarity Function

V. Sitnikov

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Abstract

Bisection places the next iterate at the midpoint because it uses no information about the function's value. We replace that rule with a similarity rule: the step is proportional to how dissimilar the current outcome f(x) is to the expectation 0, measured by a similarity function S:\mathbb{R}\to[0,1] with S(0)=1. We derive the contraction factor of the resulting iteration in closed form; it predicts a linear rate to four digits when 1-S has a non-zero slope at the origin, and a sublinear e_n\sim C/n with a predicted C when it does not, both confirmed. Three consequences follow. First, with the ratio similarity scaled by the far end — and with that combination only, of the five tested — the rule reproduces the false-position formula identically; so does the truncated linear similarity scaled by a divided difference over the bracket ends, which shows that the boundary of novelty runs through which points the difference is taken over. Second, any similarity fixed in advance with the scale also fixed yields at best linear convergence at a rate that is a property of the problem rather than of the method; on the eight standard problems of §2.2, five to eight of them fail. Third, the rate vanishes exactly when the scale equals the local scale of the value axis; reading that scale off the data — a divided difference, not a fitted coefficient — turns the step into |f|/|f'| and the convergence superlinear.

The derivation fixes the method completely, before any benchmark is consulted: a similarity whose approach to 1 is linear, the finest available data-derived scale, and a safeguard that projects the step onto the admissible interval rather than replacing it. What results is, explicitly, a Newton step with a three-point divided-difference derivative, clipped to the bracket and projected onto the bisection budget; the similarity function is scaffolding of the derivation and does not appear in the final rule. It carries a proven worst case of twice bisection plus one step and contains no constant calibrated on a problem or on the benchmark. On the 154 problems of Alefeld, Potra and Shi it needs 2806 evaluations of f against 2820 for TOMS 748 — a margin of 0.5%, which is a tie — and 2890 for Brent, whom it beats on 102 problems to 40 head to head. On the 45 problems of Chandrupatla it beats Brent 30 to 4 and TOMS 748 31 to 8, and loses to ITP. The result is parity with methods built on tuned constants, obtained without any. For every data-derived scale, the ranking of the nine similarity functions matches the derived law without exception.

Reading the same similarity in one more place — which three points the local model is built on, selected by the shape of their configuration rather than by recency — lowers the Alefeld–Potra–Shi count to 2527, below every method measured here including Chandrupatla's 2785, with the guarantee intact and still no tuning constant; reading it instead on the choice of far end gives 5511 over both sets, and reading it in the decision to trust the model or halve lowers the canonical count further, to 2523. Where the similarity is read turns out to matter more than which similarity it is. The gap that remains on the wide-bracket set is taken apart on the same evaluations: a third of it is the order of the model — a convex two-point step cannot carry the negative weight of quadratic interpolation — and two thirds is not a model at all but the decision when to give the model up, which Chandrupatla takes by bisecting on up to all of his steps and which no reading measured here takes correctly for ours.

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1. Introduction

A bracketing root finder must decide, at every step, where inside the bracket to evaluate next. Bisection answers "in the middle", which is the correct answer when nothing is known. Every improvement since has answered "where a local model of f says the root is": a secant in false position, an inverse quadratic in Brent [2], a cubic in TOMS 748 [1], an interpolate–truncate–project cycle in ITP [4]. Each of those answers carries constants: Illinois halves the stale ordinate, Brent accepts an interpolation subject to three empirical inequalities, ITP is governed by \kappa_1,\kappa_2,n_0.

This paper asks a different question. Suppose the only thing available is a similarity function — a map S from an outcome to [0,1], where 1 means "identical to what was wanted" and 0 means "not similar at all", in the sense used for parameter-free control in [6]. Can a root finder be built out of S alone, and how does it compare with methods built out of polynomial models and tuned constants?

The answer is sharp in both directions, and the negative half is as informative as the positive half.

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2. The similarity step rule

Definition 1 (Similarity step). Let [a,b] bracket a root, x be the current iterate and x_\mathrm{far} the opposite end. Let S:\mathbb{R}\to[0,1] with S(0)=1 be a similarity function and \sigma>0 a scale. The next iterate is

x' \;=\; x \;+\; \bigl(1 - S(f(x)/\sigma)\bigr)\,(x_\mathrm{far} - x). \tag{1}

The rule is exactly the informal statement "step further when the outcome is dissimilar to the expectation, and less far when it is similar". Bisection is the degenerate member S\equiv\tfrac12: a similarity function that carries no information about f.

2.1 The two-point normalisation is false position

The most obvious way to make (1) scale-free is to normalise the similarity by the two ends of the bracket, S(a)=|f(b)|/(|f(a)|+|f(b)|) and S(b)=1-S(a), and to place the next iterate at the similarity-weighted average of the ends.

Proposition 1. For the ratio similarity S(z)=1/(1+|z|) with \sigma=|f(x_\mathrm{far})|, equivalently for the endpoint weights above,

S(a)\,a + S(b)\,b \;=\; \frac{a f(b) - b f(a)}{f(b)-f(a)}.

That is the false-position formula. Two things must be said about how far it reaches, because the obvious generalisation is false.

It is specific to that similarity. On f=x^2-2 over [1,4], where false position gives 1.200000, the same construction gives 1.206812 for e^{-|z|}, 1.015228 for 1/(1+z^2), 1.007624 for 1/\sqrt{1+z^2} and 1.214286 for the truncated linear form: one of five reproduces it. So "a similarity rule scaled by the bracket is not a new method" would be too broad a conclusion, and this paper's own Table 3 refutes it — bracket-scaled configurations take first and third place on Chandrupatla and one of them solves 41 of 45, where false position solves 6.

Proposition 1'. It is, however, specific to the bracket ends rather than to the ratio form. Take the truncated linear similarity S(z)=\max(0,1-|z|) and the divided difference over the two ends, \sigma=|f[a,b]|\cdot|b-a|=|f(b)-f(a)|=|f(a)|+|f(b)|. Then 1-S=|f(a)|/(|f(a)|+|f(b)|), which is false position again, and the agreement is exact to the last bit.

Together the two say where the boundary of novelty runs, and it is not where one would guess: it runs through the words most recent in the definition of \hat f' in §3. A divided difference over the bracket ends is three centuries old; the same difference over the most recent iterates is the secant. In our measurements plain false position solves 127 of the 154 Alefeld–Potra–Shi problems, against 153 for bisection.

2.2 The contraction law

Write e_n = x_n - r for the error, D = x_\mathrm{far} - r, and let c be the slope of 1-S at the origin, 1-S(z) = c\,z + O(z^2).

Proposition 2 (Contraction law). For a simple root and e_n small, rule (1) satisfies

e_{n+1} \;=\; e_n\left(1 - \frac{c\,|f'(r)|\,D}{\sigma}\right) + O(e_n^2), \qquad \text{rate} = \left|1 - \frac{c\,|f'(r)|\,D}{\sigma}\right|. \tag{2}

Derivation. |f(x_n)| = |f'(r)|\,|e_n| + O(e_n^2), hence 1-S(f(x_n)/\sigma) = c\,|f'(r)|\,|e_n|/\sigma + O(e_n^2), and x_\mathrm{far}-x_n = D - e_n. Substituting into (1) and discarding second-order terms gives (2). \square

Table 1 checks (2) against measurement on f(x)=x^2-2 over [0,2] with \sigma=|f(0)|, for which |f'(r)|=2\sqrt2, D=2-\sqrt2 and \sigma=2.

Table 1. The contraction law against measurement, all rows at the chosen scale \sigma=|f(x_0)|. c is the slope of 1-S at the origin and c_2 the coefficient of z^2 when c=0.

A similarity with c=0 does not converge at a rate at all. Expanding 1-S=c_2z^2+O(z^4) in the same way as (2) gives e_{n+1}=e_n-c_2|f'(r)|^2|D|e_n^2/\sigma^2+O(e_n^3), hence

e_n \;\simeq\; \frac{\sigma^2}{c_2\,|f'(r)|^2\,|D|}\cdot\frac1n, \tag{2a}

sublinear rather than stagnant. Reporting a "measured rate" for these rows would report the value of e_{n+1}/e_n at whatever n the window happened to end — it creeps to 1 monotonically (0.925 at n=10, 0.991 at n=100, 0.9998 at n=4000) — so the prediction tested is the coefficient in (2a) instead.

S(z)	c	predicted	measured
1/(1+\lvert z\rvert)	1	rate 0.1716	0.1716
e^{-\lvert z\rvert}	1	rate 0.1716	0.1716
1/(1+z^2)	0, c_2=1	n\,e_n\to0.8536	0.8503
e^{-z^2}	0, c_2=1	n\,e_n\to0.8536	0.8498
1/\sqrt{1+z^2}	0, c_2=\tfrac12	n\,e_n\to1.7071	1.6988
\max(0,1-\lvert z\rvert)	1	rate 0.1716	captured, see below

The last row is not a slow rate but an immediate failure, and it matters because that similarity is the one this paper goes on to propose. At x_0=0 the residual is z=-1, so S=\max(0,1-1)=0, the step is the whole bracket, the iterate lands exactly on x_\mathrm{far} and never leaves: the error is 0.5858 after one step and after two hundred. The failure is a property of the chosen scale, not of the similarity. Under the derived scale of §3 the residual |f|/\sigma stays below 1 — on this problem the clip does not engage on a single one of the seven steps the method takes — the truncation never fires, and the same function is the best of the nine (Table 4). This is the paper's own thesis, that the content sits in the scale and not in the shape of S, showing up as a discontinuity in the behaviour of one function.

Corollary 1 (A chosen similarity is not enough). For fixed S and fixed \sigma the rate in (2) depends on |f'(r)|, D and \sigma, none of which is under the method's control. Convergence is at best linear and occurs only when the product lands in (0,2).

Empirically, with \sigma taken as |f| at the starting point, five to eight of the eight standard problems fail outright for the six similarity functions tested. Those eight, used throughout as a quick set and never as evidence on their own, are: x^2-2 on [0,2]; \cos x-x on [0,1]; e^x-2 on [0,2]; x^3-x-2 on [1,2]; \sin x-\tfrac12 on [0,1.5]; x^{10}-1 on [0,1.3], which stalls false position; e^{-x}(x-1) on [-1,40], whose right end is flat to 10^{-16}; and \arctan x on [-1,3].

Corollary 2 (Missing the calibration in either direction costs). The rate degrades as the product c\,|f'(r)|\,|D|/\sigma departs from 1, upwards or downwards. We claim no symmetry between the two directions, and the measurements do not support one: similarities with unbounded slope — 1-\sqrt{|z|} has c\approx3.2\cdot10^{3} at the resolution used, 1-|z|^{1/3} has c\approx4.6\cdot10^{4} — fail on five and six of the eight problems with a chosen scale, yet over the full benchmark they finish at 10497 and 11941, ahead of all three c=0 similarities at 14314, 15102 and 16394, and one of them heads Table 3. Overshooting is the milder failure.

Corollary 3 (The scale must come from the data). The rate is zero exactly when c\,|f'(r)|\,D/\sigma = 1, i.e. \sigma = c\,|f'(r)|\,D. Substituting this \sigma back into (1) with c=1 gives x' - x = |f(x)|/|f'|, the Newton step. The similarity rule is therefore superlinear precisely when its scale is the local scale of the value axis, and that scale is not available a priori: it must be read off the evaluations.

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3. The method

Corollaries 1–3 fix every choice in the method before any benchmark is consulted. We record them here as a prediction, and Section 5.2 tests it against the full grid.

Similarity: any with c\neq0. The law makes the shape of S almost irrelevant and its slope at the origin decisive. We take the plainest member, the truncated linear or tent form S(z)=\max(0,1-|z|). It is not the ratio form 1/(1+|z|), which is a separate row of every table here; the two are distinguished throughout because Proposition 1 applies to the ratio form and Table 1's last row to the tent.

Scale: the finest data-derived one. We set \sigma = |\hat f'|\cdot|x_\mathrm{far}-x| with \hat f' a divided difference of the most recent evaluations — not of the bracket ends, which by Proposition 1' would give false position. A divided difference is a computation, not a fitted coefficient: nothing is chosen, regressed or tuned. The two-point difference has slope error O(h) and gives the secant order \varphi\approx1.618; the three-point difference — the derivative at x_n of the parabola through the last three evaluations, \hat f' = f[x_{n-1},x_n] + f[x_{n-2},x_{n-1},x_n](x_n-x_{n-1}) — has error O(h^2). A Newton step whose derivative carries an O(h^2) error satisfies e' = e-f/\hat f' = e-(e-ce^2+\dots)(1+O(e^2)) = ce^2+O(e^3), so the predicted order is 2 — not Muller's 1.84, which belongs to a method using the parabola's root rather than its derivative. Measured medians are 1.584 for the two-point scale and 2.000 for the three-point, against 1.809 for Brent and 2.654 for TOMS 748.

What the rule reduces to. With S(z)=\max(0,1-|z|) one has 1-S=\min(1,|z|), and (1) becomes

x' \;=\; x + \operatorname{sign}(x_\mathrm{far}-x)\cdot\min\!\Bigl(\frac{|f(x)|}{|\hat f'|},\;|x_\mathrm{far}-x|\Bigr), \tag{4}

a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. The similarity function does not appear in (4): it is scaffolding of the derivation, not machinery of the algorithm.

We state this ourselves rather than leave it to be discovered, because it locates the contribution precisely. It is not a new family of methods. It is (i) a derivation that fixes every choice — which similarity, which scale, which safeguard — before any benchmark is consulted, and then predicts the ranking of all nine similarity functions correctly; (ii) a safeguard of ITP's projection type with ITP's truncation stage, and hence all three of its constants, removed; (iii) the negative results of §8; (iv) the measurements.

Algorithm. Given [a,b] with f(a)f(b)<0 and tolerance \varepsilon_x:

evaluate fa = f(a), fb = f(b)
A, B <- a, b                          the proven bracket
x  <- the endpoint with the smaller |f|;  xp <- the other;  xp2 <- none
n_max <- 2 * ceil(log2(|b-a| / xtol));   allow <- 2^n_max * xtol/2
while |B - A| > xtol:
    far   <- the end of [A,B] further from x
    f'    <- derivative at x of the parabola through (x,xp,xp2)
             (two-point difference while xp2 does not yet exist)
    sigma <- |f'| * |far - x|
    if sigma is 0 or not finite:  xn <- midpoint(A,B)
    else:
        xn <- x + min(1, |f(x)|/sigma) * (far - x)         # the step, eq (4)
        xn <- clamp(xn, hi - allow, lo + allow)            # projection, eq (3)
    if xn outside (A,B) or already evaluated:  xn <- midpoint(A,B)
    evaluate fn = f(xn);   allow <- allow / 2
    xp2, xp, x <- xp, x, xn
    update the bracket: whichever of A,B keeps the sign change
return midpoint(A,B)

Every quantity in it is either an evaluation of f, a divided difference of evaluations, or the halving that defines bisection. There is no line in which a value was chosen.

3.1 The safeguard: project, do not replace

An interpolation-only rule has no worst-case bound. The obvious safeguards replace the computed step with a midpoint whenever some progress test fails, and all of them are ruinous here, for a reason specific to a fast method: a step that collapses the error need not halve the bracket, so a bracket-based test fires on exactly the steps that were best. With the three-point scale, replacing on that test drops the observed order from 1.94 to 1.00 and the Alefeld–Potra–Shi total from 2806 to 7260 — worse than bisection.

The remedy is to keep as much of the computed step as the budget allows. Let n_\max=2\lceil\log_2(w_0/\varepsilon_x)\rceil be twice the bisection step count for the initial width w_0 and $x$-tolerance \varepsilon_x, and let \epsilon=\varepsilon_x/2. At step j, with the bracket [\ell,h], define the allowance and the projection

\alpha_j = 2^{\,n_\max-j}\epsilon, \qquad x \;=\; \operatorname{clamp}\bigl(x_\mathrm{sim},\; h-\alpha_j,\; \ell+\alpha_j\bigr). \tag{3}

The next bracket is at most \max(x-\ell,\,h-x), so (3) gives w_{j+1}\le\alpha_j directly, and induction bounds the run by n_\max steps in exact arithmetic. In floating point the bracket can land exactly on the boundary, and the termination test then asks for one more; the guarantee stated and verified is therefore 2\lceil\log_2(w_0/\varepsilon_x)\rceil+3 evaluations.

Three remarks on why this introduces nothing.

• The doubling is not a tuning constant. It is the factor already claimed in the bound. Zero slack — n_\max=\lceil\log_2(w_0/\varepsilon_x)\rceil, a worst case of exactly bisection — is degenerate and was measured as such: a worst case of exactly bisection demands a halving at every step, so the projection has no freedom, the radius on a unit bracket is 0.05 from the first step, and the guard behaves as bisection.
• This is ITP's projection stage without ITP's truncation stage. All three of ITP's constants live in the truncation, whose purpose is to repair regula falsi; the similarity step needs no repair, so the truncation and every constant with it are dropped.
• Writing the clamp in the form (3) rather than as a radius \alpha_j-w_j/2 about the midpoint is not cosmetic. That radius is a difference of two nearly equal numbers; the cancellation made the computed radius too large by a relative 8.9\cdot10^{-5}, which broke the invariant on eleven Chandrupatla problems.

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4. Experimental protocol

Problem sets. The 154 problems of Alefeld, Potra and Shi [1] and the 45 of Chandrupatla [3], taken programmatically from SciPy's scipy.optimize._tstutils so that no bracket or root is transcribed by hand.

Cost model. Evaluations of f. A run counts as solved when it reports convergence and lands within 10^{-6} of the published root; $x$-tolerance 10^{-12}; budget 200 evaluations. Failures are charged the full budget, so no method can buy a low total by giving up early, and totals are reported over all problems rather than over the subset a method happened to solve. Rankings are by failures first, then by total.

Fairness to the competitors. Three deliberate concessions.

1. ITP is natively written against a half-width \epsilon and terminates on b-a\le2\epsilon; feeding the common tolerance in as \epsilon would let it stop at twice its competitors' bracket width, so \epsilon=\varepsilon_x/2 is used.

2. ITP is reported both at the authors' recommended constants and as a per-problem oracle, the best of a (\kappa_1,n_0) grid chosen separately for every problem — not an achievable method, included so that the state of the art cannot be said to have been handicapped.

3. Our implementation of ITP is validated against the method's own theorem, that it never needs more than n_{1/2}+n_0 iterations, on all 154 problems for n_0\in\{0,1,2,5\}.

4. Where SciPy ships the method, the competitor is run at the better of SciPy's implementation and ours, problem by problem. This is not a formality. Our implementations turned out to be slightly cheaper than the library's — 6991 evaluations against brentq's 7004, and 5209 against _chandrupatla's 5228 — so quoting the library figure alone would have flattered us by a fifth of a percent, which on a 0.5% margin is not nothing. The two agree on 191 of 199 problems for Brent and 179 of 199 for Chandrupatla, the differences being one or two evaluations from the relative-tolerance term in the stopping rules. ITP has no library implementation, which is why it is checked against its own theorem instead; bisection and the false-position family are not in SciPy either and are ours.

Reproducibility. All numbers come from a single Linux run. The test functions call libm, whose last digit differs between platforms; the same SciPy version scores 2819 for TOMS 748 on Windows and 2820 on Linux for the same set. Every number stated in this paper is checked against the run data by a script that is part of the test suite.

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5. Results

The grid is nine similarity functions × four scales × six safeguards = 216 configurations, plus ten baselines. The configuration proposed in Section 3 — ratio similarity, three-point scale, projection safeguard — is one entry of that grid, fixed by the theory rather than chosen from the ranking.

Table 2. Alefeld–Potra–Shi, 154 problems, out of 226 configurations.

#	configuration	tuned	solved	total evals	>2\times
1	proposed: tent, 3-point, projection	no	153/154	2806	0
2	the same rule with no safeguard at all	no	153/154	2806	0
3	TOMS 748	yes	153/154	2820	0
4	Brent	yes	153/154	2890	0
11	ITP, per-problem oracle	yes	153/154	3260	0
18	Ridders	no	153/154	3313	0
31	ITP, recommended constants	yes	153/154	3957	0
61	Illinois	yes	153/154	7045	0
65	bisection	no	153/154	7530	0
206	false position	no	127/154	10560	—

On the problems both solve, the proposed configuration beats Brent 102/11/40 (wins/ties/losses), TOMS 748 86/16/51, ITP at its recommended constants 132/3/18, the per-problem ITP oracle 121/4/28, and bisection 152/1/0.

Table 3. Chandrupatla, 45 problems, brackets up to 10^{12} wide.

#	configuration	tuned	solved	total evals
1	similarity, 1-\sqrt{\lvert z\rvert}, bracket scale, projection	no	41/45	4516
2	ITP, per-problem oracle	yes	40/45	2611
3	similarity, e^{-\lvert z\rvert}, bracket scale, both guards	no	40/45	2733
5	ITP, recommended constants	yes	40/45	2857
6	bisection	no	40/45	3090
10	proposed: tent, 3-point, projection	no	40/45	3220
127	TOMS 748	yes	40/45	4000
189	Brent	yes	38/45	4101

The set's nominal winner is a similarity configuration that solves one problem more than anything else, 41 of 45, at a worse total; ranking by failures first puts it on top, and we report it as such rather than by the total that would flatter our proposal. The proposed configuration is tenth by total and beats Brent 30/4/4 and TOMS 748 31/1/8, while losing to ITP 17/0/23 and to the ITP oracle 16/0/24.

5.1 What the guarantee costs

Nothing on one set, and less than nothing on the other. Against the identical rule with no safeguard at all:

	solved	total evals	runs over the 2\times bound
APS, no safeguard	153/154	2806	0
APS, projection	153/154	2806	0
Chandrupatla, no safeguard	35/45	4104	10
Chandrupatla, projection	40/45	3220	0

On the canonical set the projection never binds and the totals are identical to the last evaluation. On the wide-bracket set it solves five more problems for 21% fewer evaluations. This is the opposite of the usual trade, and it is the point of projecting rather than replacing: the two safeguards that replace cost 7260 and 7239 on Alefeld–Potra–Shi, against 2806.

5.2 Wall-clock time, and why it is not the cost model

Evaluations of f are the cost model because in any real problem f is the expensive part. Wall-clock time measures something narrower — the method's own arithmetic per step — and it is implementation- and language-bound. We report it because a reader will ask, and because what it shows is instructive.

On the 199 problems as they stand, every f being a formula of a few operations, the proposed rule is the slowest of the pure-Python methods compared, at 32.8 ms against 18.4 for Brent, 21.1 for ITP, 14.2 for Ridders and 22.9 for bisection — while using the fewest evaluations of all of them, 4991 against 5932, 5675, 6421 and 9474. Bisection, needing nearly twice the evaluations, finishes in two thirds of our time.

Those evaluation counts are raw: they are what the timing loop actually spent, and a failed run contributes only the evaluations it made before the budget ran out. They are therefore not the totals of §5, where a failure is charged the full 200-evaluation budget so that no method can buy a low score by giving up — 4991 raw against 6026 charged for the proposed rule, 5932 against 6991 for Brent, 9474 against 10620 for bisection. Wall-clock time can only be compared against what was really executed, so the raw counts are the right companions to the milliseconds; every comparison in §5 uses the charged ones. Both are stated to avoid the appearance of two incompatible sets of numbers for the same methods.

That inversion is the point. Making f progressively more expensive by a controlled amount of arithmetic and timing the sweep directly — nothing subtracted, nothing modelled — the proposed rule overtakes ITP once one call to f costs about 9 µs and Brent at about 20 µs, which on this machine is of the order of a hundred arithmetic operations. Below that the ranking is decided by per-step overhead; above it, by evaluations.

Three caveats, all of them limitations of the measurement rather than of the methods. Timings vary by tens of percent between runs even on an idle machine. An attempt to separate our per-step cost into "algorithm" and "Python call overhead", by timing a flattened rewrite of the same rule, produced a variant that diverged from the original on 42 of the 199 problems and was therefore discarded rather than quoted; the question stays open. And an earlier version of this measurement estimated a per-evaluation overhead by subtracting the measured cost of f, which returned negative overheads for four of six methods — an impossibility that condemned the estimator, and the reason nothing is subtracted here.

5.3 The law across the whole grid

Corollaries 1–3 predict an ordering of the nine similarity functions and of the four scales. Table 4 reports it. The prediction holds without exception for every data-derived scale: the three similarities with c=1 occupy the top three places and the three with c=0 the bottom three. For the chosen scale the two groups are indistinguishable (16102 against 15345), which is consistent — with a chosen scale nothing converges well and the ordering carries no information.

Table 4. Mean total evaluations over both sets, step guard, grouped by the slope c of 1-S at the origin.

scale	c=1 (linear approach)	c=0 (quadratic approach)
3-point derived	6809	22294
2-point derived	6906	22167
bracket	10263	19137
start (chosen)	16102	15345

Ranked individually and best over scales, the similarities come out 1-|z| (6496), e^{-|z|} (6822), 1/(1+|z|) (7038), (1+|z|)^{-2} (9502), 1-\sqrt{|z|} (10497), 1-|z|^{1/3} (11941), e^{-z^2} (14314), 1/(1+z^2) (15102), 1/\sqrt{1+z^2} (16394); the scales come out 3-point (6496), 2-point (6628), bracket (9081), start (14540). Both orderings are those the law predicts, and both were predicted before the grid was run.

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6. Where the evaluations go

Splitting each run at the geometric mean of the initial width and the attainable tolerance — the log-midpoint of the journey, at which bisection by construction has spent half its evaluations — separates the approach from the endgame. On Alefeld–Potra–Shi the two-point rule spends 2235 on the approach and 668 on the endgame against Brent's 2175 and 515: its deficit is almost entirely the endgame, i.e. the order, which is what motivated the three-point scale. On Chandrupatla it spends 1068 and 1161 against Brent's 1343 and 1358, winning both phases.

Two natural explanations of the Chandrupatla advantage were tested and refuted. Brent discards 53.3% of its interpolations to its safeguard on the set it wins and only 34.0% on the set it loses, so safeguard rate is not the mechanism; and on Chandrupatla Brent is the bolder of the two — 56.3% of its probes land in the outer tenths of the bracket against our 41.8% — so step length is not the mechanism either.

The mechanism is step quality. Reconstructing each method's bracket from the signs it observed, the median probe removes 0.500 of the bracket for every method on Alefeld–Potra–Shi, but on Chandrupatla it removes 0.275 for the similarity rule against 0.058 for Brent and 0.068 for TOMS 748 — and that comparison already flatters them, since 69 and 141 of their probes respectively fall outside the proven bracket and are excluded, against none of ours. High-order interpolation is a wager that f is locally polynomial. When the wager pays, order 1.84 or 2.65 wins the endgame; when it fails, the probes are wasted. The similarity rule makes the weakest available assumption — a single local slope — and degrades gracefully.

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7. Where the similarity is read

Everything above reads the similarity in one place: the length of the step, (1-S(z)) times the distance to the far end. But a bracketing iteration takes several decisions per step, and the similarity was not consulted in any of the others: which end counts as far (decided by distance), which points the local slope is estimated from (decided by recency), and whether the step is worth taking at all (decided after the fact, by the safeguard). Each of those is a place where a similarity can be read, and the placement turns out to matter more than the form of S.

Three readings were added to the proposed rule of §3 (tent similarity, three-point scale), each parameter-free and each measured on the protocol of §4; every number below is re-checked against its own sweep file by verify_article.py, and every configuration in Table 5 keeps the 2\lceil\log_2(w_0/\varepsilon_x)\rceil+3 guarantee (0 violations) and solves 193 of the 199 problems.

Where a safeguard other than the projection of §3.1 appears, it is one of the grid's. safe is the projection preceded by a step-length test — a step not at most half the step taken two evaluations earlier is replaced by a halving — with the projection still carrying the bound. credit replaces the scheduled allowance of (3) by one earned step by step: a surplus is banked whenever a step shrinks the bracket by more than half and drawn on when it shrinks it by less, which bounds the run by bisection plus one step. bracket and both are the replacing safeguards of §3.1, which keep a bound at the price of the order; step alone and none carry no bound.

1. Trust before the step. The step trusts a local slope, so the question of trust is whether that slope has settled: if two consecutive estimates have opposite signs the local model is not established, and the evaluation is spent halving instead. A sign is compared, not a magnitude.
2. Which points the model uses. For a linear f the ratio of positions \xi=(x-u)/(v-u) and the ratio of values \varphi=(f(x)-f(u))/(f(v)-f(u)) coincide, so |\varphi-\xi| measures how unlike a linear configuration a triple is. Among the declared candidate triples the one minimising it is selected — an argmin over what is already in hand, with no threshold.
3. Which end is far. Read by dissimilarity instead of distance: the end whose |f| is larger is the one less similar to the target.

Table 5. Reading the similarity in one more place is worth more than changing S. Evaluations of f; failures charged the full budget.

configuration (all with the guarantee, 193/199)	APS-154	CHA-45	both
proposed: tent, 3-point, projection (§3)	2806	3220	6026
(1) trust before the step, projection	2848	3074	5922
(1)+(2) points chosen by shape, safe	2527	3108	5635
(1)+(3) far end by dissimilarity, projection	2726	2785	5511
(1)+(2)+(3), safe	2884	2859	5743
Chandrupatla	2785	2423	5208
TOMS 748	2820	4000	6820
ITP, per-problem oracle	3260	2611	5871

Two results here are worth stating plainly. First, the canonical set improves from 2806 to 2527, which is below every method measured, including Chandrupatla's 2785, and it is obtained with the worst-case guarantee intact and still without a tuning constant. Second, on the two sets together the 5511 of reading (3) passes ITP's oracle (5871) but not Chandrupatla (5208): the remaining gap is entirely on the wide-bracket families, where Chandrupatla needs 2423 against our 2683 (reading (3) under safe).

The readings do not compose. Taken together, (2) and (3) give 2884 and 2859 — worse than either alone. Two selections that are each an improvement interfere, which is a reason to measure composition rather than assume it.

7.1 The similarity function belongs to the place, not to the method

Everything above varies where the similarity is read while keeping one function S for the whole method. That is itself an assumption, and it is measurable. Proposition 1 says that the ratio similarity scaled by the far end is false position; what it does not say is why false position stalls. Write |f|\sim K e^m near the current point, e=|x-r|. Then the residual ratio against the far end needs no slope at all,

z \;=\; \frac{|f(x)|}{|f(x_\mathrm{far})|} \;=\; \Bigl(\frac{e}{E}\Bigr)^{m}, \qquad\text{so}\qquad z^{1/m} \;=\; \frac{e}{E}, \tag{5}

and under the ratio similarity S=1/(1+|z|) the step fraction is z/(1+z), which is exactly inverse linear interpolation. False position therefore reads a dimensionless ratio at order one on an approach of order m: its three-century-old stall is a mismatch between the similarity function and the place it is read in. The exponent 1/m is not a constant — m=1/(1-ff''/f'^2) comes from three evaluations already paid for, as in item 5 of §8, and is 1 whenever the power model does not apply.

Measured over the 252 cells of the grid, the correction behaves exactly as that reading predicts and not at all as a universal improvement. Paired inside each (function, safeguard) cell of the far-end reading it is cheaper in 29 of 60 cells — indistinguishable from a coin. But in the one pairing where that reading is exact inverse interpolation, the ratio similarity, it is cheaper under all seven safeguards:

Table 6. The measured order in the step place, on the far-end reading with the ratio similarity 1/(1+|z|). Cheaper in 7 of 7 safeguards; in 29 of 60 cells over all nine functions.

safeguard	APS, m=1	APS, measured m	CHA, m=1	CHA, measured m
both	4522	3675	3387	3216
bracket	4606	3542	3571	3303
credit	5190	4686	3013	2884
projection	6932	6364	4794	4559
safe	5987	5229	4108	3923
step	6435	5551	4257	4022
none	10676	10043	8265	7794

Two limits must be stated with it. The estimator is biased low: on a pure cube, with no noise at all, it returns 2.830 rather than 3, because a parabola through three points carries only the second difference. The bias is towards m=1, that is towards the uncorrected reading, which is the harmless direction, but it is a bias. And the correction does not rescue the far-end reading as a method: the best cell it reaches on the wide-bracket set is 2884, against 2683 for reading (3) above and 2423 for Chandrupatla. In the slope reading of §3 the same exponent is harmful, 2806\to2854, which is the same statement read from the other side — a function that does not belong to a place does not become right by being applied there.

7.2 The decision place

Three places have now been measured: the step length, the far end, and the points the model is built on. The fourth is the decision — whether the local model deserves this evaluation or halving is the better use of it. Chandrupatla's method is strong exactly there: it refuses its model on most steps by the acceptance band 1-\sqrt{1-\xi}<\varphi<\sqrt{\xi}, with \xi and \varphi the ratios of positions and of values defined above. The rule of §3 has no similarity in this place at all: a sign gate and a projection that repairs after the step.

Three readings of that place were measured, each on quantities already in hand and each without a constant. Two are ours: refuse when the configuration is less like a linear one than the iterate is off-centre, |\varphi-\xi|>\min(\xi,1-\xi) ("defect"); and refuse when the configuration's similarity to linear falls below the iterate's similarity to the root, S(|\varphi-\xi|)<S(z), with the same S in both places ("simpair"). The third is Chandrupatla's own band, placed inside our step rule. It is not a proposal; it is an instrument, and it answers whether the remaining gap lives in this place.

In total, no. With Chandrupatla's band in our step, the wide-bracket set costs 3066 under the projection and 2899 under credit, against his 2423, while the canonical set rises from 2806 to 5432. The band is not idle — it refuses 63.6% of steps — so this is a decision that was taken and did not pay in total. Per family it did pay, and where: it closes almost the whole gap on the flat families and gives back more than that on the families where our step is good. §7.4 takes this apart; the short form is that the decision place does hold most of the gap, but no reading measured so far makes that decision correctly for our step.

Our own readings do not close it either (best 2868 on the wide-bracket set), but "defect" sets a new best on the canonical set: 2523 at 153/154 under the projection, below the 2527 of reading (2) and below every method measured, with the guarantee intact and no constant. It refuses only 7.0% of steps, the fewest of the three. Over both sets it costs 5733 against the 5511 of reading (3), because it gives back on the wide brackets (3210) what it gains on the canonical problems — so it is a second point on the same trade, not a replacement.

7.3 Six problems of the two sets are scored unsolved for every method

Chasing the remaining gap to Chandrupatla led to a property of the benchmarks themselves. On the five problems of the wide-bracket family fun7, and on the single canonical problem aps.13.00, the declared root is 0 while f vanishes identically on a neighbourhood of it in floating point: f(0\pm t)=0 for every t up to at least 3.2\times10^{-2}, because the exponential factor underflows — at the returned points |f(x)| is below the smallest normal double. Every method therefore stops after 8 to 28 evaluations at a point where f(x)=0 exactly — a genuine root — and every method is then scored as having failed, because the harness asks for |x-\text{root}|\le10^{-6} while the returned point sits up to 2.8\times10^{-2} away.

The consequence is arithmetic. Each such problem charges the full budget of 200 evaluations to every method: about 1150 evaluations of every reported two-set total — ours, Chandrupatla's, TOMS 748's, ITP's, Brent's, bisection's — are this artefact, and the "153 of 154" and "40 of 45" that every method reports are statements about the scoring rather than about the methods. Re-scored with the criterion "a root is a root" (|x-\text{root}|\le10^{-6} or f(x)=0), the same runs solve 199 of 199 for every method except Brent's 197, and the totals fall as follows.

Table 7. The same runs under the harness criterion and under "a root is a root". The ranking is unchanged; the magnitudes are not, and neither is the size of the remaining gap on the wide-bracket set: 253 evaluations, or 17%, not 11%.

method	APS as scored	APS re-scored	CHA as scored	CHA re-scored
ours, (1)+(2) points by shape	2527	2337	3108	2159
ours, proposed (projection)	2806	2634	3220	2357
Chandrupatla	2785	2593	2423	1487
TOMS 748	2820	2636	4000	3088
Brent	2889	2707	4099	3222
ITP	3957	3767	2857	1907
bisection	7530	7338	3090	2136

7.4 Where the wide-bracket limit actually is

The remaining gap looks diffuse — per family +57, +54, +46, +43, +35 and smaller — but part of it has one address. The fun1 family is the same function with brackets of growing width, and our cost against Chandrupatla's across them is 7/8, 12/12, 19/15, 59/24, 63/44: we tie or win while the bracket is narrow and lose progressively as it widens. The endgame is not the cause — our observed order on those very problems is the higher one (3.63 against 2.02 on the $2\times10^4$-wide bracket). The cause is the rate of progress far from the root: 59 evaluations on a $2\times10^4$-wide bracket is what bisection itself needs (\lceil\log_2(2\times10^{16})\rceil\approx54), i.e. there we advance by about one halving per evaluation while Chandrupatla advances by about 2.3.

It is not the guarantee that caps us. With the safeguard removed entirely the same problems cost 46, 57, 61 and 68 against bisection's 57, 77, 45 and 45 — better on the wide brackets, worse than bisection on fun3.1 and fun5.1 — and still far from Chandrupatla's 24 and 44.

Nor is it that the step fails to be relative to the bracket. With the tent similarity 1-S=\min(1,|f|/(|\hat f'|D)), so by (4) the step is exactly \min(D,|f|/|\hat f'|): a fraction of the distance to the far end, and the similarity is that fraction. Nor does the iterate creep along one side without moving the proven bracket: measured probe by probe, 91.5% of our evaluations on the $2\times10^4$-wide bracket do shrink the bracket, slightly more often than Chandrupatla's 87.5%.

What differs on fun1 is how much each probe removes. Ours shrinks the bracket by 0.93 halvings per evaluation there, Chandrupatla's by 2.26. On a wide bracket of a convex function the Newton step systematically undershoots, because a linear model of the value axis cannot see the curvature that separates the iterate from the root; a model that carries the curvature — inverse quadratic interpolation — lands far closer.

That mechanism is right for fun1 and wrong as a statement about the whole gap. Replaying our own trajectories on the 33 problems where we lose, and placing both models on the same evaluations — one extra evaluation each, from the same state — splits the 262 evaluations of the gap in two. On fun1 (58 of them) inverse quadratic interpolation from our own three points removes far more of the bracket than our step did: 10.37 halvings against 0.62 on the $2\times10^4$-wide bracket. There the gap is the order of the model, and the reason is structural: our step x+(1-S)(x_\mathrm{far}-x)=S\,x+(1-S)\,x_\mathrm{far} is a convex combination of two points, while the quadratic interpolant's Lagrange weights at 0 always include exactly one negative weight when the triple brackets the root (verified on all 428 non-degenerate states; in the other five the negative weight underflowed to -0). A similarity in [0,1] cannot produce a negative weight — Proposition 1 extended from two points to three.

But 178 of the 262 — two thirds — sit on fun3–fun6, which are pure powers of order 3, 5, 9 and 19, and there no interpolating model works: our step removes 0.41–0.81 halvings of the bracket per evaluation, inverse quadratic interpolation from the same points only 0.01–0.17. Chandrupatla does not win those families with his model — his traced run bisects 93, 96, 100 and 100% of his steps. We lose them because our step is slower than bisection there and our guarantee, at most twice bisection, licenses exactly that. His band inside our step confirms it family by family: under the projection it takes fun4, fun5, fun6 from 573, 456, 456 to 296, 239, 235 — against his own 293, 228, 228 — and costs +164, +111, +170, +144 on fun1, fun2, fun8, fun9, where it refuses steps that were good.

So most of what remains is not the order of the model but the decision when to give the model up, and it is open because every detector measured so far confounds "the model is useless here" with something else: the band with "the configuration is not shaped for inverse quadratic interpolation", bracket progress with "the step has not moved the bracket yet" (4449 on the canonical set, §8), and the local order with nothing at all — m\ge2 is exactly the condition under which a tangent step loses to a halving, since it lands at e(1-1/m), and it is clean (0.2% of canonical states) but blind: it reports m\ge2 on only 5.8% of states of families whose order is by construction 3 to 19. For the fun1 part, three parameter-free ways round the model order have been measured and all three fail in the same way, buying the wide brackets at the price of the canonical set: a fixed fraction of the bracket (the cube-root similarity), a logarithmic change of variable, and the curvature-corrected scale of §8 item 5. The fourth way is to model the curvature directly, which is inverse quadratic interpolation — a different method rather than another reading of this one.

On the re-scored wide-bracket set of Table 7 our shape reading (2159) is worse than bisection (2136); only the dissimilarity reading (1740) is clearly ahead, which is the same statement seen from the other end: where the step cannot march, the reading that helps least is the one that only improves the local model.

Every statement in §§7.3–7.4 is reproduced by diag_rescore_cha.py, diag_rescore_aps.py, diag_fun7_root.py, diag_cha_gap.py, diag_model_order.py, diag_model_order_checks.py, diag_flat_detectable.py and diag_band_by_family.py, whose outputs are kept beside them.

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8. Negative results

1. The prototype-aggregation form of the method loses. A richer construction — prototypes over pairs of evaluated points, softmax weights over the whole memory, similarity on both the value and the abscissa axis — leads on eight hand-picked problems but loses decisively on the real set: 0 wins, 6 ties and 147 losses against Brent. The simplest form of the idea won.

2. Every safeguard that replaces the step destroys the order. Bracket-based and step-based replacement both drop the three-point rule from order 1.94 to 1.00 and cost 7260 and 7239 evaluations on Alefeld–Potra–Shi. Only projection preserves both. We record this because it is the trap the construction naturally falls into.

3. Zero slack in the projection is degenerate. A worst case of exactly bisection demands a halving at every step and leaves the projection no freedom at all.

4. Regimes exist, at the level of a function family, and cannot be exploited by a switch. The cube-root similarity beats the proposed rule on all five members of Chandrupatla's fun4 — 32/32/40/47/59 against 89/93/113/125/153, a factor of 2.6 — and consistency across a family at different parameters is the signature of structure rather than luck. But the same similarity over the whole Chandrupatla set costs 4099 against 3220, so a switch chosen per benchmark is worse than no switch at all: 6905 against 6026 over both sets. The scale axis offers only 5.3% headroom even to an oracle handed the best scale per problem, two thirds of it inside a single problem family, and the winners are not separated by anything the method can see before it commits — the proposed scale wins on bracket widths from 10^0 to 10^{12}.

5. The phase detector: derivation sound, estimator not. Redoing the contraction law without the assumption |f|\approx|f'(r)|e — taking |f|\approx K e^m instead — does produce the march exponent on its own, p = L/(L-\ln m) with L=\ln(e/D), giving p=1 at a simple root and dropping to 0.50 and 0.23 as the approach flattens, which is the neighbourhood of the cube root that wins those families. Substituted back it collapses to the multiplier m, measurable as m = 1/(1-\theta) with \theta = f f''/(f')^2 from three divided differences and no constant. It preserves the order (2.001) and improves the eight standard problems (82 against 84) — and loses on the full benchmark, 6304 against 6026. The estimator is the failure: on Alefeld–Potra–Shi, where almost every root is simple and the truth is m=1, it reports m>1 on 27.9% of steps with excursions to m=34.8, against 35.8% on the flat families. 27.9 against 35.8 is no separation. A cruder variant, correcting by the fraction of its own promise the previous step kept, fails everything: the order collapses to 0.904 and the standard problems double to 161.

6. One problem is scored unsolved for every method, aps.13.00, f(x)=x e^{-1/x^2}, all of whose derivatives vanish at the root and which underflows to exactly zero throughout |x|<0.086. Every method returns a point where f(x)=0 exactly — a genuine root — which the harness scores as a failure because it lies more than 10^{-6} from the declared root. It is a property of the scoring, not a failure of the methods; §7.3 finds five more such problems in the wide-bracket set.

7. The symmetric ratio similarity degenerates. For S(x,y)=\min(|x|,|y|)/\max(|x|,|y|) with y the target 0, S\equiv0; the only usable reading is 1-|f|/\sigma, which saturates and stagnates (Table 1, last row).

8. Placement beats form, and the wrong placement is expensive. The same "refuse before the step" idea as reading (1) of §7, with refusal decided by promised bracket progress instead of by the shape of the model, costs 4449 on the canonical set against 2527 — a difference of 1922 evaluations for one choice of criterion. A secant step routinely collapses the error while barely moving the proven bracket, so bracket progress rejects steps that were excellent; this is the same mechanism that makes the bracket guard of §3.1 cost 7260.

9. Making the scale independent of the bracket does not help either. The natural objection to the derived scale is that it defeats the very principle of a similarity step: with \sigma=|\hat f'|D the normalised residual is |f|/(|\hat f'|D), which on the $2\times10^4$-wide bracket of x^3-2x-5 at x=10^4 equals 0.167, so S=0.833 — "similar" — and the step is small although the root is some five thousand step-lengths away. The similarity is reading what fraction of the bracket the Newton step is, not how far the iterate is from the root. Selecting the scale between the two quantities already computed, \sigma=\min(|\hat f'|D,\max(|f(A)|,|f(B)|)) — no new constant, a minimum over what is in hand — does give a full-bracket step far away and Newton's step near the root, and on eight hand-picked wide problems it improves 403 evaluations to 390. Over the whole grid it is worse on both sets: 3288 on the canonical set against 2726, and 3111 on the wide-bracket set against 2683. Its apparent extra solves (194 instead of 193) are two fun7 problems where it lands inside the flat zero region close enough to satisfy the harness — the artefact of §7.3, not a capability.

10. Reading the similarity at the measured order does not cure false position. Since false position's stall is a similarity read at order one (§7.1), the obvious hope is that reading it at the measured order is the parameter-free version of the classical fix. It is not. Unguarded, the correction takes the far-end reading from 127/154 and 10676 to 128/154 and 10043 — one problem and six per cent — where Illinois, at the price of one constant (halve the stale ordinate), reaches 153/154 and 7045. A sixth of the gap is not the gap. The correction is real and it is in the right place; it is simply much smaller than a constant that was chosen for exactly this failure. Our 10676 and the 10560 of Table 2 are not the same algorithm — our iterate starts at the midpoint of the bracket, plain false position at an end — and the two must not be differenced.

11. A logarithmic change of variable does not buy the wide brackets. Working in u=\operatorname{sign}(x-c)\log(1+|x-c|/\varepsilon_x) makes the step multiplicative, which looked like the right diagnosis of the flat families — for the same similarity, scale and safeguard ((1+|z|)^{-2}, two-point, safe) the wide-bracket set improves from 3643 to 3474 — but the canonical set gets worse, 5822 to 6940, against the 2806 of the proposed rule. An earlier version compared the 3474 with 4516, which is the total of a different configuration. A march needs a ratio of distances while an identity on narrow brackets needs an absolute unit, and no parameter-free unit provides both. §7.4 shows the diagnosis itself was wrong: on the flat families the right move is to halve, not to march.

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9. Conclusion

A root finder can be derived from a similarity function alone, and the derivation shows where the content of such a method actually sits. It is not in the shape of S — nine shapes were tried and their ranking is dictated by a single number, the slope of 1-S at the origin. It is not in making the similarity change faster — that overshoots exactly as readily as flatness stagnates. It is in the scale against which similarity is judged: chosen, the method is brittle and merely linear; taken from the bracket, it is false position and three centuries old; read off the data, it is superlinear. And the safeguard that makes such a method safe must project the step, not replace it: replacing costs the order, projecting costs nothing on one benchmark and improves the method on the other.

The resulting rule has no tuning constant and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 — 2806 evaluations against 2820 is a margin of 0.5% and should be read as a tie — and ahead of Brent by 2.9%, which it also beats 102 problems to 40 head to head. The claim the measurements support is parity with methods that carry tuned constants, not superiority over them.

This is the same movement as in parameter-free similarity control [6], where membership functions are eliminated by normalising the input space from the data rather than by choosing widths. Here the object normalised is the value axis, and the price of choosing it by hand is measured rather than asserted.

Future work

The diagnosis of §7.4 leaves one question, and it is narrower than the one this section used to pose. An earlier version proposed a logarithmic change of variable to turn an additive step into a geometric march; that was measured and refuted (§8, item 11), and the step turned out to be a fraction of the bracket already.

What is open is a decision, not a model. On two thirds of the remaining gap the right move is to halve, and Chandrupatla halves; our guarantee permits a step that is slower than halving, and nothing we have measured tells our step when it is. The condition itself is known exactly — a tangent step on an approach of local order m lands at e(1-1/m) and loses to a halving when m\ge2 — so the constant is bisection's \tfrac12 and not a choice. What is missing is a reading of m that sees it: the three-point estimator reports m\ge2 on 0.2% of canonical states, which is the right silence, and on 5.8% of states of pure powers of order 3 to 19, which is blindness. Two successive residuals of the run itself carry the order more directly than a parabola through three points — on an order-m approach under tangent steps their ratio settles at (1-1/m)^m, while at a simple root it goes to zero — and that is the reading we would test next, under the same rule as every other here: pre-registered, with the canonical set as the price it must not charge.

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