From 0a62f7babbdf1bce093526d23f928c2426178fc1 Mon Sep 17 00:00:00 2001 From: Valeri Sitnikov Date: Tue, 22 Sep 2026 08:18:30 +0100 Subject: [PATCH] Revision step 3: restructured and shortened paper, full listing of A', code availability. Title: 'Root Finding by Similarity and Analogy: A Bracketing Method with a Data-Selected Decision Layer'. Abstract 246 words. Main text 10.4k words (was 17.4k): search reports moved to Appendices A (old 6-7), B (wall-clock), C (negative results); revision history removed; pronouns for Chandrupatla's method fixed; tables numbered in order. New: Algorithm 2 (complete A'), the decision layer counted honestly in 6.5, Code and data availability, Appendix D (30 pre-registrations against the first commit of their own results, read from git by check_prereg_order.py: 25 before, 4 same commit, 1 decided on the smoke; A' frozen at 8027d7f between corpus 4's definition 657df10 and its results 6d2531e). Literature: Bus & Dekker 1975, Fernandez-Diaz & Menendez-Perez 2023 (two dimensionless parameters; SFRFm on sign(f)|f|^(1/m) with known multiplicity - the nearest precursor of the split), Cordero-Neta-Torregrosa 2021; Illinois and Ridders cited. verify_article: explicit check that every head-to-head row sums to its set. check_new_numbers.py: no number new to the text except headings, references, hashes. Server: verify 527 OK / 0 BAD, regression x2 1232 passed, PDF 30 pages, 0 errors, 0 undefined, 0 overfull. Co-Authored-By: Claude Opus 5 --- apply_abstract_250.py | 14 + apply_article_trim.py | 101 ++ apply_listing_width.py | 76 ++ apply_table_order.py | 19 + article/sim_numerics.md | 719 +++++++------ article/sim_numerics.pdf | Bin 496390 -> 493931 bytes article/sim_numerics.tex | 2083 +++++++++++++++++++------------------- check_new_numbers.py | 30 + check_prereg_order.py | 83 ++ restructure_article.py | 266 +++++ results_prereg_order.txt | 33 + verify_article.py | 7 + 12 files changed, 2087 insertions(+), 1344 deletions(-) create mode 100644 apply_abstract_250.py create mode 100644 apply_article_trim.py create mode 100644 apply_listing_width.py create mode 100644 apply_table_order.py create mode 100644 check_new_numbers.py create mode 100644 check_prereg_order.py create mode 100644 restructure_article.py create mode 100644 results_prereg_order.txt diff --git a/apply_abstract_250.py b/apply_abstract_250.py new file mode 100644 index 0000000..cf6ad2b --- /dev/null +++ b/apply_abstract_250.py @@ -0,0 +1,14 @@ +"""The abstract within 250 words (revision): same claims and numbers, fewer words. + + python -u apply_abstract_250.py +""" +PATH = "article/sim_numerics.md" +NEW = """We replace bisection's midpoint rule by a similarity rule — the step grows with how dissimilar $f(x)$ is to the target 0, under a similarity $S$ with $S(0)=1$ — and derive its contraction factor in closed form. The derivation fixes a base rule before any benchmark is consulted: with the tent similarity and a three-point divided-difference scale it is a Newton step of Muller's order 1.839, clipped to the bracket and projected onto a budget of twice bisection whose multiplier is its one numerical constant. On the 154 problems of Alefeld, Potra and Shi it is level with TOMS 748 and 1.5% behind Chandrupatla's method; on wide brackets it is clearly behind. The gap is a decision, when to give the model up, and we take it by analogy: at a flat root the dimensionless configuration of the last three points recurs under zoom, at a simple root it tends to a line, and the analogous pair gives the order that says whether and where to split. This decision layer, about ten discrete rules selected on two development sets, gives the method A′. On 240 problems fixed before its code was frozen it needs 3656 evaluations against 5717 for Chandrupatla, 5535 for ModAB and 4488 with Lagouanelle's multiplicity estimate in place of the analogy; the lead lies in multiple and fractional roots, present in two of the six families. With the multiplicity known, the classical transformation $\\operatorname{sign}(f)|f|^{1/m}$ is 2.4 to 6.8 times cheaper on multiple roots.""" +src = open(PATH, encoding="utf8").read() +head, rest = src.split("## Abstract\n\n", 1) +old, tail = rest.split("\n\n---\n", 1) +assert "Bisection places its next iterate" in old +print("old", len(old.split()), "new", len(NEW.split())) +assert len(NEW.split()) <= 250 +open(PATH, "w", encoding="utf8").write(head + "## Abstract\n\n" + NEW + "\n\n---\n" + tail) +print("abstract replaced") diff --git a/apply_article_trim.py b/apply_article_trim.py new file mode 100644 index 0000000..e5a9864 --- /dev/null +++ b/apply_article_trim.py @@ -0,0 +1,101 @@ +"""Revision 1, step 3b: the main text shortened. Each paragraph below is replaced by a +shorter one that keeps its claim and every number it keeps verbatim (verify_article.py checks the +data; nothing new is claimed). Exact anchors, each found exactly once, or nothing is written. + + python -u apply_article_trim.py +""" +from __future__ import annotations + +PATH = "article/sim_numerics.md" +R = [] + + +def E(old, new): + R.append((old, new)) + + +E("That is the false-position formula. Two things must be said about how far it reaches, because the obvious generalisation is false.\n\nIt 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 third, fourth and sixth place on Chandrupatla's set, where false position solves 6 of 45.", + "That is the false-position formula, and it is specific to that similarity: on $f=x^2-2$ over $[1,4]$, where false position gives 1.200000, $e^{-|z|}$ gives 1.206812, $1/(1+z^2)$ 1.015228, $1/\\sqrt{1+z^2}$ 1.007624 and the truncated linear form 1.214286. Bracket-scaled similarity configurations take third, fourth and sixth place on Chandrupatla's set (Table 3), where false position solves 6 of 45.") +E("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.", + "Together the two say that the boundary of novelty runs through the words *most recent* in the definition of $\\hat f'$ in §3: a divided difference over the bracket ends is false position, the same difference over the most recent iterates is the secant.") +E("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.", + "The last row fails at once, and it matters because that similarity is the one this paper proposes: at $x_0=0$, $z=-1$, so $S=0$, the step is the whole bracket and the iterate stays at $x_\\mathrm{far}$ (error 0.5858 after one step and after two hundred). The failure belongs to the chosen scale, not to the similarity: under the derived scale of §3 the clip never engages on this problem, and the same function is the best of the nine (Table 4).") +E("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. One number in the safeguard is not fixed by them, and §3.1 declares it as the method's one free constant.", + "Corollaries 1–3 fix every choice before any benchmark is consulted; Section 5.2 tests the prediction against the full grid. One number in the safeguard is not fixed by them, and §3.1 declares it.") +E(" 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.", + " It is not the ratio form $1/(1+|z|)$ of Proposition 1.") +E("a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. That the tent turns the similarity step into Newton's is a property of the similarity chosen, not a sign that the similarity has dropped out: which $S$ stands in the step decides the step, and it is measured, not assumed — Table 4 ranks the nine functions over the whole grid, the readings of §A.2 pay most with the tent in the step and some of them lose with other functions (§A.2), and in the proposed method replacing the tent by any other function costs 19% to 425% (Table 7).\n\nWe state the reduction ourselves rather than leave it to be discovered, because it locates the contribution precisely. The base rule 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 predicts which similarity functions come first; (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 Appendix C; (iv) the measurements. We call it the **base rule**; the method this paper proposes builds on it in Section 6.", + "a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. That the tent turns the similarity step into Newton's is a property of the chosen similarity, and which $S$ stands in the step is measured rather than assumed: in the proposed method, replacing the tent by any other function costs 19% to 425% (Table 7). The base rule is therefore not a new family of methods; what is ours is a derivation that fixes every choice before any benchmark is consulted, and a safeguard of ITP's projection type with ITP's truncation stage — and all three of its constants — removed. The proposed method builds on it in Section 6.") +E("On the canonical set the projection binds on one problem only once $c\\ge1.25$ — `aps.13.00`, whose $f$ underflows to zero around the root (20, 30 and 28 evaluations at $c=1.25$, 1.5 and 2) — so the total barely moves. On the wide-bracket set it moves monotonically with $c$ from 1.25 on, and 2 is not the best value. Nor is zero slack ($c=1$) a degenerate case under which the guard behaves as bisection: it costs 4625 and 2039, far from bisection's 7338 and 2136, and on the wide-bracket set it beats $c=2$. The multiplicative family has no member that the bound singles out, so its multiplier is a genuinely free constant.", + "On the canonical set the projection binds on one problem only once $c\\ge1.25$ — `aps.13.00`, whose $f$ underflows to zero around the root (20, 30 and 28 evaluations at $c=1.25$, 1.5 and 2). On the wide-bracket set the cost rises monotonically with $c$ from 1.25 on, and 2 is not the best value; zero slack ($c=1$) is not degenerate either (4625 and 2039, against bisection's 7338 and 2136). No member of the family is singled out by the bound: the multiplier is a free constant.") +E("A looser promise buys evaluations, as it must; within one promise the cheapest endowment is determined. The constant-free variant is therefore the earned projection with $C_0=1$, whose guarantee — $\\lceil\\log_2(w_0/\\varepsilon_x)\\rceil+4$ evaluations, bisection plus one step with one more for the floating-point boundary — is verified on all 199 problems.\n\nWe nonetheless propose the scheduled projection with $c=2$ and declare its multiplier. On the canonical set it is 743 evaluations cheaper than the earned variant (2634 against 3377), and there its total barely depends on the multiplier; on the wide-bracket set the earned variant is the cheaper (2073 against 2357). The two are the ends of a trade stated in numbers: one declared numerical constant and the better canonical result, or none, a tighter bound and the better wide-bracket result. The value 2 was fixed before any benchmark was consulted.", + "Within one promise the cheapest endowment is determined, so the constant-free variant is the earned projection with $C_0=1$; its guarantee, $\\lceil\\log_2(w_0/\\varepsilon_x)\\rceil+4$ evaluations, is verified on all 199 problems. We nonetheless propose the scheduled projection with $c=2$, fixed before any benchmark was consulted: it is 743 evaluations cheaper on the canonical set (2634 against 3377), the earned variant is cheaper on the wide-bracket set (2073 against 2357) — one declared numerical constant and the better canonical result, or none, a tighter bound and the better wide-bracket result.") +E("**Fairness to the competitors.** Three deliberate concessions.\n\n1. 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.\n2. 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.\n3. 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 $n_0\\in\\{0,1,2,5\\}$.\n\n4. Where SciPy ships the method, the competitor **is the library**: Chandrupatla is `scipy.optimize.elementwise.find_root`, Brent is `brentq`, TOMS 748 is `toms748`, each run with the same stopping rule as every other method — a bracket no wider than $\\varepsilon_x$, the relative term at machine precision, no stopping on $|f|$ — and each evaluation counted by the same wrapper around $f$. Our own implementations of Chandrupatla's and Brent's methods agree with the libraries; the tables use the libraries. 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.", + "**Competitors.** Where SciPy ships the method, the competitor is the library — Chandrupatla is `scipy.optimize.elementwise.find_root`, Brent `brentq`, TOMS 748 `toms748` — run with the stopping rule of every method (a bracket no wider than $\\varepsilon_x$, the relative term at machine precision, no stopping on $|f|$) and counted by the same wrapper around $f$. ITP, which has no library implementation, is ours, run with $\\epsilon=\\varepsilon_x/2$ because it terminates on $b-a\\le2\\epsilon$, checked against its own theorem ($n_{1/2}+n_0$ iterations at most, on all 154 problems for $n_0\\in\\{0,1,2,5\\}$), and reported both at the authors' constants and as a *per-problem oracle* over a $(\\kappa_1,n_0)$ grid — not an achievable method, included so that it cannot be said to have been handicapped. Bisection and the false-position family are ours.") +E("It wins twice as many problems as it loses against Chandrupatla and trails that method in total because its losses are the larger ones; the problems are not independent (§7), and taken by function family the comparison is 6/1/8 of 15 families, a tie. Dekker and Illinois solve 153 of 154 and are ranked after every method that solves all. The constant-free variant goes 79/3/72 against Chandrupatla ($p=0.63$, a tie in problems), 78/9/67 against TOMS 748 and 79/5/70 against Brent, and loses on the total to all three.", + "It trails Chandrupatla in total because its losses are the larger ones; by function family, the unit in which these problems are independent (§7), the comparison is 6/1/8 of 15 families, a tie. The constant-free variant goes 79/3/72 against Chandrupatla ($p=0.63$), 78/9/67 against TOMS 748 and 79/5/70 against Brent.") +E("### 5.1 What the guarantee costs\n\nIt depends on which bound is bought. Against the identical rule with no safeguard at all:\n\n| | solved | total evals | runs over the $2\\times$ bound |\n|---|---|---|---|\n| APS, no safeguard | 154/154 | 2634 | 0 |\n| APS, projection — base rule ($2\\times$ bisection) | 154/154 | 2634 | 0 |\n| APS, earned projection — constant-free (bisection + 1 step) | 154/154 | 3377 | 0 |\n| Chandrupatla, no safeguard | 39/45 | 3391 | 10 |\n| Chandrupatla, projection — base rule | 45/45 | 2357 | 0 |\n| Chandrupatla, earned projection — constant-free | **45/45** | **2073** | **0** |\n\nThe projection of the base rule never binds on the canonical set, so its totals are identical to the last evaluation; on the wide-bracket set it solves six more problems for 30% fewer evaluations. The earned projection promises twice as much — a worst case of bisection plus one step instead of twice bisection — and pays for it on the canonical set, 743 evaluations or 28%, while on the wide-bracket set it is the cheapest of the three. Both are the opposite of the usual trade on at least one set, and that is the point of projecting rather than replacing: the two safeguards that replace cost 7068 and 7046 on Alefeld–Potra–Shi, against 2634.", + "### 5.1 What the guarantee costs\n\nAgainst the same rule with no safeguard (Tables 2 and 3), the projection costs nothing on the canonical set and on the wide-bracket set solves six more problems for 30% fewer evaluations (2357 against 3391; unguarded, 10 runs break the bound). The earned projection, promising bisection plus one step, pays 743 evaluations (28%) on the canonical set and is the cheapest of the three on the wide-bracket set. That is the point of projecting rather than replacing: the two safeguards that replace cost 7068 and 7046 on Alefeld–Potra–Shi, against 2634.") +E("Table 4 reports it. 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. That $c=0$ is slow is the unsurprising half — a step proportional to $z^2$ is damped quadratically. For the chosen scale the two groups are indistinguishable (14983 against 14228), which is consistent — with a chosen scale nothing converges well and the ordering carries no information.", + "For every data-derived scale the three similarities with $c=1$ occupy the top three places of Table 4 and the three with $c=0$ the bottom three (that $c=0$ is slow is the unsurprising half: a step proportional to $z^2$ is damped quadratically). With the chosen scale the two groups are indistinguishable (14983 against 14228), as they should be.") +E(" The group order and the order of the scales are those the law predicts and were predicted before the grid was run; the order inside each group is measured, not predicted.", + " The group order and the order of the scales were predicted before the grid was run; the order inside each group is measured.") +E("§A.6 left one decision open: *when to give the model up*. On the pure powers `fun3`–`fun6` the right move is to halve, Chandrupatla's method halves on up to all of its steps, and every reading of a single state measured so far confuses \"the model is useless here\" with something else. This section closes that decision, and the reading that closes it is not a similarity of the current state to a prototype but a similarity of the run to *its own past* — an analogy in Pólya's sense [8], two situations agreeing in more than one relation. Every rule below was pre-registered before its code and its run, with a stop rule; the rejected ones are listed in Appendix C with their numbers.", + "The base rule's gap on wide brackets is a decision, *when to give the model up* (§A.6): on the pure powers `fun3`–`fun6` the right move is to halve, Chandrupatla's method halves on up to all of its steps, and no reading of a single state measured takes that decision correctly. This section takes it by a similarity of the run to *its own past* — an analogy in Pólya's sense [8], two situations agreeing in more than one relation. Every rule below was pre-registered with a stop rule (Appendix D); the rejected ones are in Appendix C.") +E(" The constant is bisection's $\\tfrac12$, carried through the formula. The relation between the order and the contraction, $e(1-1/m)$, is the one behind the classical estimates of multiplicity from successive Newton iterates [14, 15]; here it is not used to estimate $m$ but to set the point at which a model step has lost to a halving.", + " The constant is bisection's $\\tfrac12$, carried through the formula; the contraction $e(1-1/m)$ is the one behind the classical multiplicity estimates from successive Newton iterates [14, 15], used here to set the point at which a model step has lost to a halving.") +E("The flag sees the regime — on the flat families it rises on most model steps — but acted upon step by step it does not pay. Replacing the one flagged step by a halving costs the canonical set 2634 → 2839, because a simple root far from convergence also breaks its promises (the flag rises at least once on 142 of the 154 canonical problems). Every refinement of *when* to halve that reads the moment alone — twice in a row, the promise together with a creep in one direction, a halving that sticks once begun, a judgment of the evaluated point against the line through the bracket — trades one set for the other (Appendix C, items 12–14).", + "Acted upon step by step the flag does not pay: replacing each flagged step by a halving costs the canonical set 2634 → 2839, because a simple root far from convergence also breaks its promises (the flag rises at least once on 142 of the 154 canonical problems). Every refinement that reads the moment alone trades one set for the other (Appendix C, items 12–14).") +E("Formula (6) is the log–log slope by which any power-law exponent is estimated, and has the shape of the computational order of convergence and its approximation from differences of iterates [18, 19]. What is not classical is *which* two states it compares: not consecutive iterates, but a pair the similarity of their configurations has found. Under an exact zoom a pure power of order $m$ reads exactly $m$; along a real run the analogous pairs are only nearly homothetic and the reading scatters about the order, but the regime asks only whether it reaches 2, the order from which a tangent step loses to a halving (§6.1) — no new constant.", + "Formula (6) is the log–log slope of a power law, with the shape of the computational order of convergence [18, 19]; what it compares is not consecutive iterates but a pair the similarity of their configurations has found. A real run's analogous pairs are only nearly homothetic and the reading scatters about the order, but the regime asks only whether it reaches 2, the order from which a tangent step loses to a halving (§6.1).") +E("**The analogy belongs to the run, the flag to the moment.** In the cycle the base rule runs on a pure power — undershoot, undershoot, crossing, a nearly empty step — the flag rises in the phases whose configuration does *not* recur (the empty step grows emptier every cycle), while the recurring phases have the flag down. Requiring both in one state delayed the entry on the flat problems to their thirteenth state on average. Once established, the analogy is therefore remembered for the run, and the regime is entered at the next unkept promise (eighth state on average).", + "**The analogy belongs to the run, the flag to the moment.** On a pure power the flag rises in the phases of the base rule's cycle whose configuration does *not* recur, so requiring both in one state delayed the entry to the thirteenth state on average. Once established, the analogy is remembered for the run and the regime is entered at the next unkept promise (eighth state on average).") +E(" A high power far from its root is locally an exponential, $x^m=e^{m\\ln x}$, and it was there, not on the flank of `fun9` it was built for, that the limit paid.", + " It pays on high powers far from their root, which are locally exponential, $x^m=e^{m\\ln x}$.") +E(" The order read from a *failed promise*, $(1-1/m)^m=r$, was tried first and gave $m=\\infty$ in 163 of 174 entries, because the base step is not a pure tangent; the order read from the *analogy* is the one that works.", + " (The order read from a failed promise, $(1-1/m)^m=r$, gives $m=\\infty$ in 163 of 174 entries, because the base step is not a pure tangent.)") +E(" It is a relative of Brent's acceptance condition [2], which admits an interpolant only between the best point and $(3a+b)/4$: both refuse a model that points towards the worse end of the bracket.", + " It is a relative of Brent's acceptance condition [2]: both refuse a model that points towards the worse end of the bracket.") +E(" The exit judgment of §6.2 separates a flat root from a simple one; at a root of order below one $f$ is far from 0 on both sides and the exit fires falsely, so the memory of this branch is *not* cleared by it; clearing it was measured on the development corpora and was worse (recorded in the pre-registration amendment of this branch).", + " At a root of order below one $f$ is far from 0 on both sides and the exit fires falsely, so the memory of this branch is *not* cleared by the exit (clearing it was worse on the development corpora).") +E(" Dekker [9] and Brent [2] step at least the tolerance; we do the same, $\\varepsilon_x/2$ towards the far end, when the model's step is shorter — but only after the model has *kept* its previous promise ($r<\\tfrac14$), because at the start of a run a slope dominated by a huge far value promises the root at an end. This element is not ours and is named as such.", + " Dekker [9] and Brent [2] step at least the tolerance; A′ steps $\\varepsilon_x/2$ towards the far end when the model's step is shorter, but only after the model has *kept* its previous promise ($r<\\tfrac14$). This element is borrowed.") +E(" The earned-budget form keeps its constant-free guarantee of bisection plus one step and pays for it as in §3.1.", "") +E("Every component is positive in sum; the regime and the split carry the method. Two are marginal and we say so: the step pattern earns on two corpora of six and loses on four, and the sign contradiction is within noise (+28 over 855 problems). Removing both would change A′'s total by under one per cent (117 of 13032 evaluations); they are kept because the pre-registered rule removes only a component that earns nowhere.", + "Every component is positive in sum; the regime and the split carry the method. The step pattern (earns on two corpora of six) and the sign contradiction (+28 over 855 problems) are marginal — together under one per cent, 117 of 13032 evaluations — and are kept because the pre-registered rule removes only a component that earns nowhere.") +E("Several mechanisms of A′ coincide with classical ones, and the coincidence is worth stating, because they were not taken from the literature but reached from the principles of the paper — similarity to a prototype and analogy as similarity of relations — and recognised afterwards:", + "Several mechanisms of A′ coincide with classical ones, reached from similarity and analogy and recognised afterwards:") +E("1. **New corpus 1** — 206 problems: SciPy's `original` set (6) and five generated classes of 40 (seed 20260921): smooth simple roots, transcendental simple roots, multiple roots $|x-r|^p$ with $p\\sim U(2,20)$, exponential flanks, and near-cancelling terms of the `fun9` kind.\n2. **Roots.jl / ModAB** — the published bracketing benchmark of the Julia package Roots.jl [12], taken from the ModAB benchmark of Ganchovski and Traykov [10]: 91 problems, of which 13 coincide with APS or Chandrupatla functions (a seven-point identity test) and are excluded, leaving 78, many of them discontinuous or pathological.\n3. **Applied and GSL** — the 10 bracketed problems of the GNU Scientific Library's root-finding test suite [11] and 122 equations from physics and engineering on grids fixed beforehand: Kepler's equation (eccentricity up to 0.999), Colebrook–White, Lambert's W, Planck/Wien, van der Waals for CO₂ (including near-critical), internal rate of return, the finite square well.", + "1. **New corpus 1** — 206 problems: SciPy's `original` set (6) and five generated classes of 40 (seed 20260921): smooth and transcendental simple roots, multiple roots $|x-r|^p$ with $p\\sim U(2,20)$, exponential flanks, near-cancelling terms.\n2. **Roots.jl / ModAB** — the bracketing benchmark of Roots.jl [12] used by ModAB's authors [10]: 91 problems, 13 of which coincide with APS or Chandrupatla functions and are excluded, leaving 78.\n3. **Applied and GSL** — the 10 bracketed problems of the GNU Scientific Library's test suite [11] and 122 equations from physics and engineering on grids fixed beforehand (Kepler, Colebrook–White, Lambert's W, Planck/Wien, van der Waals, internal rate of return, the finite square well).") +E("ModAB, the most recent bracketing method we know of, is run from its authors' package (`pymodab` 1.0.9, a wrapper of their C code) with the stopping rule of every other method — a bracket no wider than $\\varepsilon_x$, the relative term at machine precision, no stopping on $|f|$ — and counted by the same wrapper around $f$, whose count agrees with the package's own. Where roots are not known in closed form, and for uniformity on all six corpora, a run is solved when $f$ changes sign within $10^{-6}$ of its answer; the $x$-tolerance, the budget and the charge for a failure are those of §4.", + "ModAB, the most recent bracketing method we know of, is run from its authors' package (`pymodab` 1.0.9) with the stopping rule and the counting of every other method. The criterion, tolerance, budget and charging are those of §4.") +E("**The history is part of the result.** The variant carried to corpora 1–3 was A′ without the tolerance step and the overshoot branch. It transferred to corpus 1 by every pre-registered criterion (total 10% below Chandrupatla, per problem 105/34/67, $p=0.005$), did *not* transfer to Roots.jl (total 6.5% below, but per problem 38/10/30, $p=0.4$) and lost on the applied corpus (14% above). The losses there were traced to two gaps of the base step — no closing of the bracket once the root is reached, and no recognition of roots of order below one — which became the last two readings of §6.4; they were developed on the seen corpora and tested once, on corpus 4.", + "The variant carried to corpora 1–3 was A′ without the tolerance step and the overshoot branch. It transferred to corpus 1 by every pre-registered criterion (per problem 105/34/67 against Chandrupatla, $p=0.005$), did *not* transfer to Roots.jl (38/10/30, $p=0.4$) and lost on the applied corpus; the two readings that closed those gaps were developed on the seen corpora and tested once, on corpus 4.") +E("Against ModAB it is ahead on every corpus, in total and problem by problem ($p\\le0.0013$), on the one test of corpus 4 by 3656 to 5535 and 183 problems to 34 ($p=7\\cdot10^{-26}$), and on ModAB's own benchmark as well — all 91 problems of it, 1507 against 1588, 56/13/22 ($p=1.5\\cdot10^{-4}$); ModAB exceeds that bound, which it does not promise either, on 9 problems. Against Chandrupatla it is ahead in total on five corpora and level on the sixth, and ahead problem by problem on all six, significantly on five. The size of the lead is not uniform: it is large on multiple, fractional and nearly flat roots — the regime §6 recognises — and small on simple roots, where it is a matter of per cent. Where it is behind is named: `fun9` (65 against 55), the near-cancelling class of corpus 1 (491 against 391), random polynomials with a simple root in total (433 against 419, though ahead 20 to 12 in problems), and the smooth equations of the applied corpus, where the two are level.", + "Against ModAB it is ahead on every corpus, in total and problem by problem ($p\\le0.0013$) — on corpus 4 by 3656 to 5535 and 183 problems to 34 ($p=7\\cdot10^{-26}$), on all 91 problems of ModAB's own benchmark by 1507 to 1588 (56/13/22, $p=1.5\\cdot10^{-4}$); ModAB, which does not promise the bound either, exceeds it on 9 problems. Against Chandrupatla it is ahead in total on five corpora and level on the sixth. The lead is large on multiple, fractional and nearly flat roots and small on simple roots. It is behind on `fun9` (65 against 55), the near-cancelling class of corpus 1 (491 against 391) and random polynomials with a simple root in total (433 against 419), and level on the smooth equations of the applied corpus.") +E("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 which group of them comes first 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 projection's budget multiplier is a free constant, and we declare it: it does not move the canonical result and it does move the wide-bracket one; projecting onto a budget the steps earn instead removes it, at a measured price.", + "A root finder can be derived from a similarity function alone, and the derivation shows where its content sits: not in the shape of $S$, whose slope at the origin decides which group of functions comes first, but in the *scale* against which similarity is judged — chosen, the method is linear and brittle; taken from the bracket, it is false position; read off the most recent evaluations, it is Newton's step with Muller's order. The safeguard must project the step rather than replace it; the projection's multiplier is a free constant, declared, and a budget the steps earn removes it at a measured price.") +E("The base rule has one declared numerical constant, converges with Muller's order 1.839 and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 in total and 1.5% behind Chandrupatla's method, while winning more problems than it loses against both; on the wide-bracket set Chandrupatla is genuinely stronger (1505 against 2357). §A.6 locates the cause: not the order of the model but the decision to give it up, which Chandrupatla takes by halving and which no reading of a single state takes correctly for our step.", + "The base rule is level with TOMS 748 and 1.5% behind Chandrupatla's method on the canonical set, and clearly behind Chandrupatla on wide brackets (1505 against 2357); the cause is not the order of the model but the decision to give it up.") +E("That decision is a similarity too — of the run to its own past. A pure power repeats its configuration under zoom, an exponential under translation, a simple root tends to the line; the analogous pair reads the local order, the order decides between the model and a halving by bisection's own $\\tfrac12$, and it says where to split. The method built on it, A′, is ahead of Chandrupatla on both canonical sets in total and problem by problem, and on a corpus fixed after its code was frozen it needs 36% fewer evaluations and wins 177 problems to 28.", + "That decision is a similarity too, of the run to its own past: a pure power repeats its configuration under zoom, an exponential under translation, a simple root tends to the line, and the analogous pair reads the order that decides between the model and a split. The method built on it, A′, is ahead of Chandrupatla on both canonical sets, and on a corpus fixed before its code was frozen it needs 36% fewer evaluations and wins 177 problems to 28.") +E("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.\n\n", "") +E("Three questions remain, all narrower than the one this section used to pose.\n\n", "") + + +def main(): + src = open(PATH, encoding="utf8").read() + bad = [o[:80] for o, _ in R if src.count(o) != 1] + if bad: + for b in bad: + print("ANCHOR NOT FOUND EXACTLY ONCE:", repr(b)) + raise SystemExit(f"{len(bad)} anchors failed; nothing written") + for o, n in R: + src = src.replace(o, n) + open(PATH, "w", encoding="utf8").write(src) + main_words = len(src.split("## Appendix A.")[0].split()) + print(f"applied {len(R)}; main text {main_words} words, total {len(src.split())}") + + +if __name__ == "__main__": + main() diff --git a/apply_listing_width.py b/apply_listing_width.py new file mode 100644 index 0000000..738b833 --- /dev/null +++ b/apply_listing_width.py @@ -0,0 +1,76 @@ +"""Algorithm 2 within 72 characters a line (it overflowed the page), same content; and the one +character pdflatex does not know (U+2265 in the availability statement). + + python -u apply_listing_width.py +""" +import re + +PATH = "article/sim_numerics.md" +NEW = """``` +state: bracket [A,B], FA, FB; points x, xp, xp2; values fx, fp, fp2 +memory: configs (configurations), moves (displacements) +flags: last_model, in_regime, + analogy_seen (with order m_a), shallow_seen (with order m_s) +initialise as Algorithm 1; allow <- 2^n_max * xtol/2 +while |B - A| > xtol: + far <- end of [A,B] further from x; mid <- (A+B)/2; model <- false + f' <- derivative at x of the parabola through (x, xp, xp2) + sigma <- |f'| * |far - x| + if sigma is 0 or not finite: xn <- mid + else: + z <- |fx| / sigma + # the flat regime (6.1-6.3) + c <- (xi, phi, |x-xp|, |fx-fp|), last three points, if defined + # unkept promise, and unkept promise with a crossing + flag <- not in_regime and last_model and fx/fp >= 1/4 + flag_over <- not in_regime and last_model and fx/fp <= -1/4 + # order (6) of the analogous pair: the past configuration + # nearest to c (Euclidean in (xi, phi)), if nearer than the + # line phi = xi; inf if the pair shows L_f < 1 <= L_x; else none + m <- order of the analogous pair + if m >= 2: analogy_seen <- true; m_a <- m + if 0 < m <= 2/3: shallow_seen <- true; m_s <- m + if flag and analogy_seen: in_regime <- true + elif flag_over and shallow_seen: in_regime <- true; m_a <- m_s + append c to configs + if in_regime: xn <- split point (7) with m = m_a + elif x != xp and + sign((fx-fp)/(x-xp)) != sign((FB-FA)/(B-A)): xn <- mid + elif S(z) >= 1/2 and |f(far)| < |fx|: xn <- mid + else: + step <- (1 - S(z)) * (far - x) # eq. (4) + append x - xp to moves + # step pattern: the lag 2 .. half the history of highest + # mean similarity of successive length ratios has mean > 1/2, + # and step and x - xp point the same way + if the pattern repeats: + s <- min(|step|,|x-xp|) / max(|step|,|x-xp|) + step <- step / (1 - s), at most |far - x| long + if |step| < xtol/2 and last_model and |fx/fp| < 1/4: + step <- sign(far - x) * xtol/2 # tolerance step + xn <- x + step; model <- true + xn <- clamp(xn, hi - allow, lo + allow) # projection (3) + if xn outside (A,B) or xn in {x, xp}: + xn <- mid; if mid in {x, xp}: stop + last_model <- model and xn is the model's own point + fn <- f(xn) + if in_regime: # exit + t <- (xn - A)/(B - A) + if |fn - (FA(1-t) + FB t)| < |fn|: + in_regime <- false; analogy_seen <- false + allow <- allow/2; (xp2, xp, x) <- (xp, x, xn) + keep the sign change in [A,B] +return midpoint(A,B) +```""" +src = open(PATH, encoding="utf8").read() +start = src.index("**Algorithm 2 (A′).**") +a = src.index("```", start) +b = src.index("```", a + 3) + 3 +assert src.count("**Algorithm 2 (A′).**") == 1 +src = src[:a] + NEW + src[b:] +assert src.count("Python ≥ 3.10") == 1 +src = src.replace("Python ≥ 3.10", "Python 3.10 or later") +open(PATH, "w", encoding="utf8").write(src) +print("longest listing line:", max(len(l) for l in NEW.split("\n"))) +print("non-latin1 characters left outside math:", sorted({c for c in re.sub(r"\$[^$]*\$", "", src) + if ord(c) > 0x2FFF})) diff --git a/apply_table_order.py b/apply_table_order.py new file mode 100644 index 0000000..9733533 --- /dev/null +++ b/apply_table_order.py @@ -0,0 +1,19 @@ +"""Tables numbered in the order they appear (section 7's corpora tables come before 7.1's +controls): 10 -> 8, 10b -> 8b, 11 -> 9, 8 -> 10, 9 -> 11, in one pass; and the last two +pronouns for Chandrupatla's method in Appendix A.6. + + python -u apply_table_order.py +""" +import re + +PATH = "article/sim_numerics.md" +MAP = {"10": "8", "10b": "8b", "11": "9", "8": "10", "9": "11"} +src = open(PATH, encoding="utf8").read() +src = re.sub(r"Table (10b|10|11|8|9)(?![0-9a-z])", lambda m: "Table " + MAP[m.group(1)], src) +for old, new in (("and 100% of his steps.", "and 100% of its steps."), + ("His band inside our step confirms", "Its band inside our step confirms")): + assert src.count(old) == 1, old + src = src.replace(old, new) +open(PATH, "w", encoding="utf8").write(src) +caps = re.findall(r"^\*\*Table ([0-9A-D]+[a-c]?)\.\*\*", src, re.M) +print("captions in order:", caps) diff --git a/article/sim_numerics.md b/article/sim_numerics.md index e7e22a4..30e68ba 100644 --- a/article/sim_numerics.md +++ b/article/sim_numerics.md @@ -1,28 +1,27 @@ -# Root Finding by Similarity and Analogy: A Bracketing Method with One Declared Constant +# Root Finding by Similarity and Analogy: A Bracketing Method with a Data-Selected Decision Layer V. Sitnikov ---- - ## 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. 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 a *base rule* 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. It is, explicitly, a Newton step with a three-point divided-difference derivative, clipped to the bracket and projected onto the bisection budget. The similarity sets that step, and its form is decisive: with the tent the step is Newton's, with any of the other eight functions the proposed method costs from 19% to five times more. Its order is Muller's, 1.839. It carries a proven worst case of twice bisection plus one step and one free numerical constant, the multiplier of that budget, which we declare. On the 154 problems of Alefeld, Potra and Shi it needs 2634 evaluations of $f$ against 2636 for TOMS 748, 2707 for Brent and 2595 for Chandrupatla's method; it wins more problems than it loses against Chandrupatla, 87 to 43, and needs 1.5% more in total because its losses are the larger ones. On the 45 wide-bracket problems of Chandrupatla it beats Brent and TOMS 748 and loses to ITP and to Chandrupatla (2357 against 1946 and 1505). For every data-derived scale the three similarity functions whose $1-S$ has a non-zero slope at the origin come before the three whose slope is zero, as the derived law predicts. - -The wide-bracket gap turns out to be not the order of the model but a decision — when to give the model up — and the decision is a similarity after all, of an unfamiliar kind: the similarity of a run to its own past. A pure power is self-similar under zoom, so the dimensionless configuration of the last three points recurs, while at a simple root it tends to the linear prototype; the analogous pair yields the local order, $m=\ln L_f/\ln L_x$, and an exponential, self-similar by translation, is its limit $m=\infty$. Entering a halving regime when that analogy holds with $m\ge2$ — the condition under which a tangent step loses to a halving — splitting at the point the order predicts, and leaving when the split looks linear, together with four smaller readings of the same kind and one borrowed element, gives the method proposed here, A′. On the two sets it needs 2259 and 1074 evaluations against Chandrupatla's 2595 and 1505 and wins more problems than it loses on both, 103 to 31 and 31 to 10. The chain of decisions that produced it was selected on those sets; tested once, after its code was frozen, on 240 problems from six new families, it needs 3656 evaluations against Chandrupatla's 5717 and wins 177 problems to 28 — a corpus the author composed knowing where the method is strong: two of its six families have multiple or fractional roots, and without them the lead is 2403 against 3569, on the simple-root families alone 737 against 747. Against simpler detectors put in the place of the analogy — Lagouanelle's multiplicity estimate with the same memory, Chandrupatla's acceptance band, or none — it wins on that corpus in total and problem by problem (3656 against 4488, 6236 and 6165); against the classical transformation $\operatorname{sign}(f)|f|^{1/m}$ with the multiplicity given in advance it loses every multiple or fractional root, by a factor of 2.4 to 6.8 — the price of estimating the order. Over 855 problems in six corpora it is ahead of Brent, TOMS 748, ITP and the recent ModAB everywhere and stays within its own bound of twice bisection on every problem, a bound that Brent, TOMS 748 and ModAB, which do not promise it, exceed on 118, 84 and 9 problems; against Chandrupatla it is ahead in total on five corpora and level on the sixth. The lead is large on multiple, fractional and nearly flat roots and a matter of per cent on simple ones. Several of the resulting mechanisms coincide with classical ones — Lagouanelle's multiplicity estimate, false position on $|f|^{1/m}$ — reached here from similarity and analogy and recognised afterwards; what we found no counterpart for is the decision layer, where every choice — when to give the model up, where to split, when to return — is a judgment of similarity or of analogy, and the similarity that sets the step, whose form is decisive (replacing it costs 19% to 425%). +We replace bisection's midpoint rule by a similarity rule — the step grows with how dissimilar $f(x)$ is to the target 0, under a similarity $S$ with $S(0)=1$ — and derive its contraction factor in closed form. The derivation fixes a base rule before any benchmark is consulted: with the tent similarity and a three-point divided-difference scale it is a Newton step of Muller's order 1.839, clipped to the bracket and projected onto a budget of twice bisection whose multiplier is its one numerical constant. On the 154 problems of Alefeld, Potra and Shi it is level with TOMS 748 and 1.5% behind Chandrupatla's method; on wide brackets it is clearly behind. The gap is a decision, when to give the model up, and we take it by analogy: at a flat root the dimensionless configuration of the last three points recurs under zoom, at a simple root it tends to a line, and the analogous pair gives the order that says whether and where to split. This decision layer, about ten discrete rules selected on two development sets, gives the method A′. On 240 problems fixed before its code was frozen it needs 3656 evaluations against 5717 for Chandrupatla, 5535 for ModAB and 4488 with Lagouanelle's multiplicity estimate in place of the analogy; the lead lies in multiple and fractional roots, present in two of the six families. With the multiplicity known, the classical transformation $\operatorname{sign}(f)|f|^{1/m}$ is 2.4 to 6.8 times cheaper on multiple roots. --- ## 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], an Anderson–Björck step behind a linearity check in ModAB [10, 20]. Each of those answers carries constants: Illinois halves the stale ordinate, Brent accepts an interpolation subject to three empirical inequalities, ITP is governed by $\kappa_1,\kappa_2,n_0$. +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], an Anderson–Björck step behind a linearity check in ModAB [10, 20]. Each of those answers carries constants: Illinois [7] 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. +The paper makes four claims, each measured. -The paper proceeds in two stages. Sections 2–7 derive a base rule from the similarity alone, measure it, and locate what it lacks: not a better model but a decision, when to stop trusting the model it has. Section 8 supplies that decision with a similarity of the run to its own past, and Section 9 tests it on four further corpora: three of them exposed two last gaps of the base step and were used to close them, and the fourth was fixed after the method's code was frozen and run once. +1. A contraction law for the similarity step (§2) fixes a *base rule* — tent similarity, three-point scale, a projection safeguard — before any benchmark is consulted (§3); its order is Muller's. +2. Projecting the step onto the bisection budget, instead of replacing it, keeps the order and a worst case of twice bisection plus one step; the budget's multiplier is the base rule's one numerical constant, and a constant-free variant costs a measured price (§3.1, §5). +3. The base rule's gap on wide brackets is a decision — when to give the model up — and a similarity of the run to its own past takes it (§6). The resulting method, A′, is tested once on a corpus fixed before its code was frozen, against simpler detectors put in its place, and against the classical transformation with the multiplicity known in advance (§7). +4. Every table uses one criterion, a sign change within $10^{-6}$ of the answer, because the benchmarks' published roots score six problems unsolved for every method (§4, §A.5). + +The search that produced these results — readings of the similarity that did not pay, the decomposition of the wide-bracket gap, wall-clock time, and the rejected rules — is reported in Appendices A–C. The code, the corpora and the pre-registrations are released (Appendix D and the availability statement). --- @@ -41,13 +40,11 @@ The most obvious way to make (1) scale-free is to normalise the similarity by th **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 third, fourth and sixth place on Chandrupatla's set, where false position solves 6 of 45. +That is the false-position formula, and it is specific to that similarity: on $f=x^2-2$ over $[1,4]$, where false position gives 1.200000, $e^{-|z|}$ gives 1.206812, $1/(1+z^2)$ 1.015228, $1/\sqrt{1+z^2}$ 1.007624 and the truncated linear form 1.214286. Bracket-scaled similarity configurations take third, fourth and sixth place on Chandrupatla's set (Table 3), where false position solves 6 of 45. **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 154 for bisection. +Together the two say that the boundary of novelty runs through the words *most recent* in the definition of $\hat f'$ in §3: a divided difference over the bracket ends is false position, 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 154 for bisection. ### 2.2 The contraction law @@ -63,12 +60,6 @@ Table 1 checks (2) against measurement on $f(x)=x^2-2$ over $[0,2]$ with $\sigma **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 | @@ -78,7 +69,13 @@ sublinear rather than stagnant. Reporting a "measured rate" for these rows would | $1/\sqrt{1+z^2}$ | 0, $c_2=\tfrac12$ | $n\,e_n\to1.7071$ | 1.6988 | | $\max(0,1-\lvert z\rvert)$ | 1 | rate 0.1716 | fails: stays at $x_\mathrm{far}$ | -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. +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. + +The last row fails at once, and it matters because that similarity is the one this paper proposes: at $x_0=0$, $z=-1$, so $S=0$, the step is the whole bracket and the iterate stays at $x_\mathrm{far}$ (error 0.5858 after one step and after two hundred). The failure belongs to the chosen scale, not to the similarity: under the derived scale of §3 the clip never engages on this problem, and the same function is the best of the nine (Table 4). **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 $(0,2)$. @@ -92,9 +89,9 @@ Empirically, with $\sigma$ taken as $|f|$ at the starting point, five to eight o ## 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 5.3 tests it against the full grid. One number in the safeguard is not fixed by them, and §3.1 declares it as the method's one free constant. +Corollaries 1–3 fix every choice before any benchmark is consulted; Section 5.2 tests the prediction against the full grid. One number in the safeguard is not fixed by them, and §3.1 declares it. -**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. +**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 $1/(1+|z|)$ of Proposition 1. **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 $\tfrac16 f'''(\eta)\,(x_n-x_{n-1})(x_n-x_{n-2})$, that is $O(e_{n-1}e_{n-2})$. The Newton step with that derivative satisfies $e_{n+1}=\tfrac{f''}{2f'}e_n^2+\tfrac{f'''}{6f'}\,e_n e_{n-1}e_{n-2}+\dots$; the second term dominates, and the order is the root of $p^3=p^2+p+1$, $p\approx1.839$ — **Muller's order**, as it should be, since the step uses the same parabola as Muller's method, through its derivative rather than its root. Only where the parabola is exact ($f'''=0$) does the order rise to Newton's 2. Carried to $10^{-2900}$ in 3000-digit arithmetic (`order_high_precision.py`), the computational order per step settles at 1.840 on $\cos x-x$, $e^x-2$, $x^3-x-2$ and $\sin x-\tfrac12$ and at 2.000 on $x^2-2$; the two-point scale settles at 1.618. In double precision a run takes three or four superlinear steps before it reaches the tolerance, too few to tell 1.84 from 2, and we report no order measured there. @@ -102,9 +99,7 @@ Corollaries 1–3 fix every choice in the method before any benchmark is consult $$x' \;=\; x + \operatorname{sign}(x_\mathrm{far}-x)\cdot\min\!\Bigl(\frac{|f(x)|}{|\hat f'|},\;|x_\mathrm{far}-x|\Bigr), \tag{4}$$ -a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. That the tent turns the similarity step into Newton's is a property of the similarity chosen, not a sign that the similarity has dropped out: which $S$ stands in the step decides the step, and it is measured, not assumed — Table 4 ranks the nine functions over the whole grid, the readings of §7 pay most with the tent in the step and some of them lose with other functions (§7), and in the proposed method replacing the tent by any other function costs 19% to 425% (Table 9a). - -We state the reduction ourselves rather than leave it to be discovered, because it locates the contribution precisely. The base rule 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 predicts which similarity functions come first; (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 §10; (iv) the measurements. We call it the **base rule**; the method this paper proposes builds on it in Section 8. +a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. That the tent turns the similarity step into Newton's is a property of the chosen similarity, and which $S$ stands in the step is measured rather than assumed: in the proposed method, replacing the tent by any other function costs 19% to 425% (Table 7). The base rule is therefore not a new family of methods; what is ours is a derivation that fixes every choice before any benchmark is consulted, and a safeguard of ITP's projection type with ITP's truncation stage — and all three of its constants — removed. The proposed method builds on it in Section 6. **Algorithm.** Given $[a,b]$ with $f(a)f(b)<0$ and tolerance $\varepsilon_x$: @@ -129,7 +124,7 @@ while |B - A| > xtol: 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 — with one exception, the factor 2 in `n_max`, which §3.1 declares as the method's one free constant. The constant-free variant of §3.1 replaces the two `allow` lines by the earned credit of (3'). +Every quantity in it is either an evaluation of $f$, a divided difference of evaluations, or the halving that defines bisection — with one exception, the factor 2 in `n_max`, which §3.1 declares as the base rule's one numerical constant. The constant-free variant of §3.1 replaces the two `allow` lines by the earned credit of (3'). ### 3.1 The safeguard: project, do not replace @@ -141,7 +136,7 @@ $$\alpha_j = 2^{\,n_{\max}-j}\epsilon, \qquad x \;=\; \operatorname{clamp}\bigl( 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. -**The multiplier is the method's one free numerical constant.** It is not "the factor already claimed in the bound" — that argument is circular, the bound is twice bisection because two was chosen — and measurement settles it (`measure_budget_factor.py`, Linux): +**The multiplier is the base rule's one numerical constant**, and it is not neutral (`measure_budget_factor.py`, Linux): **Table 1a.** The scheduled projection at budget multiplier $c$, i.e. $n_{\max}=\lceil c\cdot\log_2(w_0/\varepsilon_x)\rceil$. Total evaluations, failures charged. @@ -154,7 +149,7 @@ The next bracket is at most $\max(x-\ell,\,h-x)$, so (3) gives $w_{j+1}\le\alpha | 3 | 2634 | 2936 | 5570 | | 4 | 2634 | 3229 | 5863 | -On the canonical set the projection binds on one problem only once $c\ge1.25$ — `aps.13.00`, whose $f$ underflows to zero around the root (20, 30 and 28 evaluations at $c=1.25$, 1.5 and 2) — so the total barely moves. On the wide-bracket set it moves monotonically with $c$ from 1.25 on, and 2 is not the best value. Nor is zero slack ($c=1$) a degenerate case under which the guard behaves as bisection: it costs 4625 and 2039, far from bisection's 7338 and 2136, and on the wide-bracket set it beats $c=2$. The multiplicative family has no member that the bound singles out, so its multiplier is a genuinely free constant. +On the canonical set the projection binds on one problem only once $c\ge1.25$ — `aps.13.00`, whose $f$ underflows to zero around the root (20, 30 and 28 evaluations at $c=1.25$, 1.5 and 2). On the wide-bracket set the cost rises monotonically with $c$ from 1.25 on, and 2 is not the best value; zero slack ($c=1$) is not degenerate either (4625 and 2039, against bisection's 7338 and 2136). No member of the family is singled out by the bound: the multiplier is a free constant. **The budget earned rather than scheduled: a constant-free variant.** The constant can be removed. Let $s_j=\log_2(w_j/w_{j+1})$ be the shrink a step delivers in bits — bisection delivers exactly 1 — and keep a credit @@ -172,12 +167,11 @@ leaving the step alone while $C\ge1$. A step may under-deliver only by what earl | 2 | bisection + 2 steps | 3502 | 1757 | 5259 | | 5 | bisection + 5 steps | 2819 | 1666 | 4485 | -A looser promise buys evaluations, as it must; within one promise the cheapest endowment is determined. The constant-free variant is therefore the earned projection with $C_0=1$, whose guarantee — $\lceil\log_2(w_0/\varepsilon_x)\rceil+4$ evaluations, bisection plus one step with one more for the floating-point boundary — is verified on all 199 problems. +Within one promise the cheapest endowment is determined, so the constant-free variant is the earned projection with $C_0=1$; its guarantee, $\lceil\log_2(w_0/\varepsilon_x)\rceil+4$ evaluations, is verified on all 199 problems. We nonetheless propose the scheduled projection with $c=2$, fixed before any benchmark was consulted: it is 743 evaluations cheaper on the canonical set (2634 against 3377), the earned variant is cheaper on the wide-bracket set (2073 against 2357) — one declared numerical constant and the better canonical result, or none, a tighter bound and the better wide-bracket result. -We nonetheless propose the scheduled projection with $c=2$ and declare its multiplier. On the canonical set it is 743 evaluations cheaper than the earned variant (2634 against 3377), and there its total barely depends on the multiplier; on the wide-bracket set the earned variant is the cheaper (2073 against 2357). The two are the ends of a trade stated in numbers: one declared constant and the better canonical result, or no constant, a tighter bound and the better wide-bracket result. The value 2 was fixed before any benchmark was consulted. - -Two remarks apply to both forms. +Three remarks apply to both forms. +- **Guaranteed bounds of this kind are classical.** Bus and Dekker [21] gave bracketing methods whose worst case is four to five times bisection; the scheduled projection guarantees twice bisection plus one step, the earned one bisection plus one step. - **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. @@ -187,15 +181,9 @@ Two remarks apply to both forms. **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 $f$ changes sign within $10^{-6}$ of the returned point, or vanishes there — the bracketing notion of a root, the same criterion on every corpus of this paper; $x$-tolerance $10^{-12}$; budget 200 evaluations. Scoring against the published root instead, as the benchmarks' harness does, charges every method the full budget on six problems where $f$ vanishes identically around the root in floating point (§7.3). Head-to-head counts (wins/ties/losses) are taken over every problem of a set with the same charging as the totals. *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. +**Cost model.** Evaluations of $f$. A run counts as solved when it reports convergence and $f$ changes sign within $10^{-6}$ of the returned point, or vanishes there — the bracketing notion of a root, the same criterion on every corpus of this paper; $x$-tolerance $10^{-12}$; budget 200 evaluations. Scoring against the published root instead, as the benchmarks' harness does, charges every method the full budget on six problems where $f$ vanishes identically around the root in floating point (§A.5). Head-to-head counts (wins/ties/losses) are taken over every problem of a set with the same charging as the totals. *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 the library**: Chandrupatla is `scipy.optimize.elementwise.find_root`, Brent is `brentq`, TOMS 748 is `toms748`, each run with the same stopping rule as every other method — a bracket no wider than $\varepsilon_x$, the relative term at machine precision, no stopping on $|f|$ — and each evaluation counted by the same wrapper around $f$. An earlier version gave each competitor the better of the library's implementation and ours, problem by problem. That concession backfired in the competitor's favour: our Chandrupatla stopped on a bracket twice as wide as everybody else's — the factor this protocol corrects for ITP in item 1 and had not corrected there — and the per-problem minimum let the looser run win. With the tolerance corrected our implementation and the library agree on all 199 problems exactly. For Brent the library is slightly dearer than our implementation; we use the library anyway, because it is the implementation a reader can check. 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. +**Competitors.** Where SciPy ships the method, the competitor is the library — Chandrupatla is `scipy.optimize.elementwise.find_root`, Brent `brentq`, TOMS 748 `toms748` — run with the stopping rule of every method (a bracket no wider than $\varepsilon_x$, the relative term at machine precision, no stopping on $|f|$) and counted by the same wrapper around $f$. ITP, which has no library implementation, is ours, run with $\epsilon=\varepsilon_x/2$ because it terminates on $b-a\le2\epsilon$, checked against its own theorem ($n_{1/2}+n_0$ iterations at most, on all 154 problems for $n_0\in\{0,1,2,5\}$), and reported both at the authors' constants and as a *per-problem oracle* over a $(\kappa_1,n_0)$ grid — not an achievable method, included so that it cannot be said to have been handicapped. Bisection and the false-position family 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 2635 for TOMS 748 on Windows and 2636 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. @@ -203,7 +191,7 @@ Two remarks apply to both forms. ## 5. Results -The grid is nine similarity functions × four scales × seven safeguards = 252 configurations, plus twelve baselines: 264 in all. The base rule of Section 3 — tent similarity, three-point scale, projection safeguard with its multiplier declared — is one entry of that grid, fixed by the theory rather than chosen from the ranking; the constant-free earned variant of §3.1 is shown beside it. Chandrupatla's own method and Dekker's are among the baselines; an earlier version of these tables was drawn up before either was added. +The grid is nine similarity functions × four scales × seven safeguards = 252 configurations, plus twelve baselines: 264 in all. The base rule of Section 3 — tent similarity, three-point scale, projection safeguard with its multiplier declared — is one entry of that grid, fixed by the theory rather than chosen from the ranking; the constant-free earned variant of §3.1 is shown beside it. Chandrupatla's own method and Dekker's are among the baselines. **Table 2.** Alefeld–Potra–Shi, 154 problems, out of 264 configurations. Places by failures first, then total. @@ -215,15 +203,15 @@ The grid is nine similarity functions × four scales × seven safeguards = 252 c | 4 | TOMS 748 | yes | 154/154 | 2636 | 0 | | 5 | Brent | yes | 154/154 | 2707 | 0 | | 11 | ITP, per-problem oracle | yes | 154/154 | 3068 | 0 | -| 15 | Ridders | no | 154/154 | 3124 | 0 | +| 15 | Ridders [5] | no | 154/154 | 3124 | 0 | | 25 | constant-free variant: earned projection | no | 154/154 | 3377 | 0 | | 29 | ITP, recommended constants | yes | 154/154 | 3767 | 0 | | 73 | bisection | no | 154/154 | 7338 | 0 | | 232 | Dekker | no | 153/154 | 3158 | 0 | -| 237 | Illinois | yes | 153/154 | 7045 | 0 | +| 237 | Illinois [7] | yes | 153/154 | 7045 | 0 | | 244 | false position | no | 127/154 | 10560 | — | -The base rule and the same rule without any safeguard tie at 2634: the projection never binds on this set. By total, Chandrupatla's method is first with 2595, 1.5% fewer; the base rule and TOMS 748 are level (2634 against 2636). Problem by problem the base rule wins against Chandrupatla 87/24/43 (wins/ties/losses; exact two-sided sign test $p=1.4\cdot10^{-4}$), against TOMS 748 86/16/52, Brent 102/11/41, ITP at its recommended constants 132/3/19, the per-problem ITP oracle 121/4/29, and bisection 152/1/1. It wins twice as many problems as it loses against Chandrupatla and trails that method in total because its losses are the larger ones; the problems are not independent (§9), and taken by function family the comparison is 6/1/8 of 15 families, a tie. Dekker and Illinois solve 153 of 154 and are ranked after every method that solves all. The constant-free variant goes 79/3/72 against Chandrupatla ($p=0.63$, a tie in problems), 78/9/67 against TOMS 748 and 79/5/70 against Brent, and loses on the total to all three. +The base rule and the same rule without any safeguard tie at 2634: the projection never binds on this set. By total, Chandrupatla's method is first with 2595, 1.5% fewer; the base rule and TOMS 748 are level (2634 against 2636). Problem by problem the base rule wins against Chandrupatla 87/24/43 (wins/ties/losses; exact two-sided sign test $p=1.4\cdot10^{-4}$), against TOMS 748 86/16/52, Brent 102/11/41, ITP at its recommended constants 132/3/19, the per-problem ITP oracle 121/4/29, and bisection 152/1/1. It trails Chandrupatla in total because its losses are the larger ones; by function family, the unit in which these problems are independent (§7), the comparison is 6/1/8 of 15 families, a tie. The constant-free variant goes 79/3/72 against Chandrupatla ($p=0.63$), 78/9/67 against TOMS 748 and 79/5/70 against Brent. **Table 3.** Chandrupatla, 45 problems, brackets up to $10^{12}$ wide, out of 264 configurations. @@ -244,32 +232,11 @@ Here Chandrupatla's method is genuinely the stronger: it beats the base rule 29 ### 5.1 What the guarantee costs -It depends on which bound is bought. Against the identical rule with no safeguard at all: +Against the same rule with no safeguard (Tables 2 and 3), the projection costs nothing on the canonical set and on the wide-bracket set solves six more problems for 30% fewer evaluations (2357 against 3391; unguarded, 10 runs break the bound). The earned projection, promising bisection plus one step, pays 743 evaluations (28%) on the canonical set and is the cheapest of the three on the wide-bracket set. That is the point of projecting rather than replacing: the two safeguards that replace cost 7068 and 7046 on Alefeld–Potra–Shi, against 2634. -| | solved | total evals | runs over the $2\times$ bound | -|---|---|---|---| -| APS, no safeguard | 154/154 | 2634 | 0 | -| APS, projection — base rule ($2\times$ bisection) | 154/154 | 2634 | 0 | -| APS, earned projection — constant-free (bisection + 1 step) | 154/154 | 3377 | 0 | -| Chandrupatla, no safeguard | 39/45 | 3391 | 10 | -| Chandrupatla, projection — base rule | 45/45 | 2357 | 0 | -| Chandrupatla, earned projection — constant-free | **45/45** | **2073** | **0** | +### 5.2 The law across the whole grid -The projection of the base rule never binds on the canonical set, so its totals are identical to the last evaluation; on the wide-bracket set it solves six more problems for 30% fewer evaluations. The earned projection promises twice as much — a worst case of bisection plus one step instead of twice bisection — and pays for it on the canonical set, 743 evaluations or 28%, while on the wide-bracket set it is the cheapest of the three. Both are the opposite of the usual trade on at least one set, and that is the point of projecting rather than replacing: the two safeguards that replace cost 7068 and 7046 on Alefeld–Potra–Shi, against 2634. - -### 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 base 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. (These are the evaluations the timing loop actually spent; for the base rule and bisection, which solve every problem, they equal the totals of §5; Brent here is our pure-Python implementation, which is what is timed.) - -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 base 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 that the similarities with $c=1$ come before those with $c=0$ under a data-derived scale; they say nothing about the order within a group, nor about the functions with unbounded slope. Table 4 reports it. 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. That $c=0$ is slow is the unsurprising half — a step proportional to $z^2$ is damped quadratically. For the chosen scale the two groups are indistinguishable (14983 against 14228), which is consistent — with a chosen scale nothing converges well and the ordering carries no information. +Corollaries 1–3 predict that the similarities with $c=1$ come before those with $c=0$ under a data-derived scale; they say nothing about the order within a group, nor about the functions with unbounded slope. For every data-derived scale the three similarities with $c=1$ occupy the top three places of Table 4 and the three with $c=0$ the bottom three (that $c=0$ is slow is the unsurprising half: a step proportional to $z^2$ is damped quadratically). With the chosen scale the two groups are indistinguishable (14983 against 14228), as they should be. **Table 4.** Mean total evaluations over both sets, step guard, grouped by the slope $c$ of $1-S$ at the origin. @@ -280,11 +247,301 @@ Corollaries 1–3 predict that the similarities with $c=1$ come before those wit | bracket | 9149 | 18015 | | start (chosen) | 14983 | 14228 | -Ranked individually and best over scales, the similarities come out $1-|z|$ (5422), $e^{-|z|}$ (5772), $1/(1+|z|)$ (5993), $(1+|z|)^{-2}$ (8455), $1-\sqrt{|z|}$ (9411), $1-|z|^{1/3}$ (10874), $e^{-z^2}$ (13187), $1/(1+z^2)$ (13990), $1/\sqrt{1+z^2}$ (15272); the scales come out 3-point (5422), 2-point (5564), bracket (7972), start (13423). The group order and the order of the scales are those the law predicts and were predicted before the grid was run; the order inside each group is measured, not predicted. +Ranked individually and best over scales, the similarities come out $1-|z|$ (5422), $e^{-|z|}$ (5772), $1/(1+|z|)$ (5993), $(1+|z|)^{-2}$ (8455), $1-\sqrt{|z|}$ (9411), $1-|z|^{1/3}$ (10874), $e^{-z^2}$ (13187), $1/(1+z^2)$ (13990), $1/\sqrt{1+z^2}$ (15272); the scales come out 3-point (5422), 2-point (5564), bracket (7972), start (13423). The group order and the order of the scales were predicted before the grid was run; the order inside each group is measured. --- -## 6. Where the evaluations go +## 6. The flat regime, recognised by analogy + +The base rule's gap on wide brackets is a decision, *when to give the model up* (§A.6): on the pure powers `fun3`–`fun6` the right move is to halve, Chandrupatla's method halves on up to all of its steps, and no reading of a single state measured takes that decision correctly. This section takes it by a similarity of the run to *its own past* — an analogy in Pólya's sense [8], two situations agreeing in more than one relation. Every rule below was pre-registered with a stop rule (Appendix D); the rejected ones are in Appendix C. + +### 6.1 The flag: a promise that was not kept + +A tangent step on an approach of local order $m$ lands at $e(1-1/m)$ and leaves the residual $r=|f_\text{new}/f_\text{old}|=(1-1/m)^m$ on the same side of the root. It beats a halving exactly when $|1-1/m|<\tfrac12$, that is for $\tfrac23\tfrac12$. Three choices are declared rather than derived: the multiplier $c$ of the projection (§3.1), the Euclidean geometry of the $(\xi,\varphi)$ plane, and the pattern test of the step lengthening (lags from 2 to half the history, threshold $\tfrac12$). The projection carries the guarantee, $2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ evaluations; it is verified with no violation on every corpus of this paper and under every safeguard of the grid. + +Counting numerical constants does not measure the method's freedom, however. Beyond these numbers A′ is a decision layer of about ten discrete rules — the flag, the analogy and its memory, the entry, the exit, the overshoot branch, the split, the step pattern, the half, the contradiction, the tolerance step — chosen on the two development sets among the many candidates of Appendix C. That selection is why the method was tested once on a corpus fixed before its code was frozen, and against simpler rules put in its place (§7). Algorithm 2 is the complete method; `simroot/pure.py` and the compiled `simbracket` implement exactly this and agree bit for bit. + +**Algorithm 2 (A′).** Notation of Algorithm 1; $S(z)=\max(0,1-|z|)$. + +``` +state: bracket [A,B], FA, FB; points x, xp, xp2; values fx, fp, fp2 +memory: configs (configurations), moves (displacements) +flags: last_model, in_regime, + analogy_seen (with order m_a), shallow_seen (with order m_s) +initialise as Algorithm 1; allow <- 2^n_max * xtol/2 +while |B - A| > xtol: + far <- end of [A,B] further from x; mid <- (A+B)/2; model <- false + f' <- derivative at x of the parabola through (x, xp, xp2) + sigma <- |f'| * |far - x| + if sigma is 0 or not finite: xn <- mid + else: + z <- |fx| / sigma + # the flat regime (6.1-6.3) + c <- (xi, phi, |x-xp|, |fx-fp|), last three points, if defined + # unkept promise, and unkept promise with a crossing + flag <- not in_regime and last_model and fx/fp >= 1/4 + flag_over <- not in_regime and last_model and fx/fp <= -1/4 + # order (6) of the analogous pair: the past configuration + # nearest to c (Euclidean in (xi, phi)), if nearer than the + # line phi = xi; inf if the pair shows L_f < 1 <= L_x; else none + m <- order of the analogous pair + if m >= 2: analogy_seen <- true; m_a <- m + if 0 < m <= 2/3: shallow_seen <- true; m_s <- m + if flag and analogy_seen: in_regime <- true + elif flag_over and shallow_seen: in_regime <- true; m_a <- m_s + append c to configs + if in_regime: xn <- split point (7) with m = m_a + elif x != xp and + sign((fx-fp)/(x-xp)) != sign((FB-FA)/(B-A)): xn <- mid + elif S(z) >= 1/2 and |f(far)| < |fx|: xn <- mid + else: + step <- (1 - S(z)) * (far - x) # eq. (4) + append x - xp to moves + # step pattern: the lag 2 .. half the history of highest + # mean similarity of successive length ratios has mean > 1/2, + # and step and x - xp point the same way + if the pattern repeats: + s <- min(|step|,|x-xp|) / max(|step|,|x-xp|) + step <- step / (1 - s), at most |far - x| long + if |step| < xtol/2 and last_model and |fx/fp| < 1/4: + step <- sign(far - x) * xtol/2 # tolerance step + xn <- x + step; model <- true + xn <- clamp(xn, hi - allow, lo + allow) # projection (3) + if xn outside (A,B) or xn in {x, xp}: + xn <- mid; if mid in {x, xp}: stop + last_model <- model and xn is the model's own point + fn <- f(xn) + if in_regime: # exit + t <- (xn - A)/(B - A) + if |fn - (FA(1-t) + FB t)| < |fn|: + in_regime <- false; analogy_seen <- false + allow <- allow/2; (xp2, xp, x) <- (xp, x, xn) + keep the sign change in [A,B] +return midpoint(A,B) +``` + +**Table 5.** A′ on the two sets of §4 (failures charged 200). A′'s wins/ties/losses against each rival over all problems of the set, exact two-sided sign test. + +| | APS-154 | CHA-45 | both | A′ per problem, APS | A′ per problem, CHA | +|---|---|---|---|---|---| +| **A′ (proposed)** | **2259** | **1074** | **3333** | — | — | +| A′, earned budget (no free constant) | 3417 | 1852 | 5269 | | | +| base rule (§3) | 2634 | 2357 | 4991 | | | +| Chandrupatla | 2595 | 1505 | 4100 | 103/20/31, $p=3\cdot10^{-10}$ | 31/4/10, $p=0.001$ | +| TOMS 748 | 2636 | 3088 | 5724 | 127/6/21 | 42/2/1 | +| Brent | 2707 | 3238 | 5945 | 118/17/19 | 39/3/3 | +| ITP, per-problem oracle | 3068 | 1675 | 4743 | 126/7/21 | 35/1/9 | +| ModAB [10, 20] | 2726 | 1681 | 4407 | 118/10/26 | 34/4/7 | + +A′ is 12.9% below Chandrupatla on the canonical set and 28.6% below on the wide-bracket set, and wins more problems than it loses against every rival on both. By function family, the unit in which these problems are independent (§7), it is ahead of Chandrupatla on 8 of the 15 canonical families (6 behind, one level) and on 7 of the 9 wide-bracket families — too few families for either count to be significant ($p=0.79$, $0.18$). The two it is behind on are `fun7`, 69 against 64, and `fun9`, 65 against 55. + +**Table 6.** Leave-one-out ablation of A′ on all six corpora of §7 (uniform criterion, a sign change within $10^{-6}$). Δ = total without the component − total with it; positive means the component earns its place. + +| component removed | APS | CHA | new-1 | Roots.jl | applied | fourth | sum | +|---|---|---|---|---|---|---|---| +| the whole regime | −7 | +1013 | +1660 | +324 | +77 | +2269 | **+5336** | +| the split (7) (midpoint instead) | −5 | +479 | +667 | +272 | +30 | +1672 | **+3115** | +| overshoot branch | −6 | 0 | −7 | +16 | +63 | +893 | +959 | +| tolerance step (borrowed) | +130 | +35 | +97 | +78 | +135 | −2 | +473 | +| the half the step promises | +249 | +65 | +82 | −16 | −5 | +1 | +376 | +| translation limit | +31 | +143 | +100 | −21 | −2 | +87 | +338 | +| repeated step pattern | −22 | −6 | +123 | −30 | −10 | +34 | +89 | +| sign contradiction | +11 | −12 | +29 | 0 | +1 | −1 | +28 | + +Every component is positive in sum; the regime and the split carry the method. The step pattern (earns on two corpora of six) and the sign contradiction (+28 over 855 problems) are marginal — together under one per cent, 117 of 13032 evaluations — and are kept because the pre-registered rule removes only a component that earns nowhere. + +**Table 7.** The similarity in the step of A′: the same method with each of the nine similarity functions of §5.2 in place of the tent, everything else unchanged. Totals on the two sets of §4 and over the six corpora of §7; the guarantee holds with every function. + +| $S$ in the step | APS + CHA | six corpora (855) | +|---|---|---| +| **tent $1-|z|$ (A′)** | **3333** | **13032** | +| $e^{-|z|}$ | 3965 | 14142 | +| $1/(1+|z|)$ | 4424 | 14843 | +| $(1+|z|)^{-2}$ | 7249 | 28500 | +| $1-\sqrt{|z|}$ | 11613 | 45552 | +| $1-|z|^{1/3}$ | 16094 | 63031 | +| $e^{-z^2}$ | 17429 | 67701 | +| $1/(1+z^2)$ | 17452 | 67394 | +| $1/\sqrt{1+z^2}$ | 17487 | 67854 | + +The nearest other function costs 19% more on the two sets and 9% more over the six corpora, and the functions that approach 1 quadratically cost five times more; the three functions with a linear approach come first and the three with a quadratic approach last, as the contraction law of §2.2 predicts. What this measures should be said plainly: with the tent the step is the undamped Newton step clipped to the bracket, and the other functions damp it, so the table shows that the undamped step is the better one — expected, and evidence for the choice of $S$ within the rule rather than for the similarity framing as such. + +### 6.6 What is known, and what is new + +Several mechanisms of A′ coincide with classical ones, reached from similarity and analogy and recognised afterwards: + +| in A′ | classical counterpart | +|---|---| +| the three-point order estimate of §A.3 | Lagouanelle's multiplicity estimate $m=f'^2/(f'^2-ff'')$ [13] | +| the contraction $e(1-1/m)$ behind the flag of §6.1 | multiplicity estimates from successive Newton iterates [14, 15] | +| the order (6) of an analogous pair | the log–log slope of a power law; the (approximate) computational order of convergence [18, 19] | +| the split (7) | false position on $\operatorname{sign}(f)|f|^{1/m}$ [16, 17]; inside a bracket, with the multiplicity given, the SFRFm method of [22] | +| the dimensionless configuration $(\xi,\varphi)$ of §6.2 | the ratios of Chandrupatla's acceptance band [3]; every scaled regula falsi written in two dimensionless parameters [22] | +| the half the step promises | a relative of Brent's acceptance condition [2] | +| the tolerance step | Dekker [9], Brent [2] — borrowed and named as such | +| the exit judgment of §6.2 (the value at a split against the line) | ModAB's linearity check at the midpoint [10, 20], $|y_m-y_3|\min(\xi,1-\xi)$ ("defect"); and refuse when the configuration's similarity to linear falls below the iterate's similarity to the root, $S(|\varphi-\xi|)\tfrac12$. Three choices are declared rather than derived: the multiplier $c$ of the projection (§3.1), the Euclidean geometry of the $(\xi,\varphi)$ plane, and the pattern test of the step lengthening (lags from 2 to half the history, threshold $\tfrac12$). The projection carries the guarantee, $2\lceil\log_2(w_0/\varepsilon_x)\rceil+3$ evaluations; it is verified with no violation on every corpus of this paper and under every safeguard of the grid. - -**Table 8.** A′ on the two sets of §4 (failures charged 200). A′'s wins/ties/losses against each rival over all problems of the set, exact two-sided sign test. - -| | APS-154 | CHA-45 | both | A′ per problem, APS | A′ per problem, CHA | -|---|---|---|---|---|---| -| **A′ (proposed)** | **2259** | **1074** | **3333** | — | — | -| A′, earned budget (no free constant) | 3417 | 1852 | 5269 | | | -| base rule (§3) | 2634 | 2357 | 4991 | | | -| Chandrupatla | 2595 | 1505 | 4100 | 103/20/31, $p=3\cdot10^{-10}$ | 31/4/10, $p=0.001$ | -| TOMS 748 | 2636 | 3088 | 5724 | 127/6/21 | 42/2/1 | -| Brent | 2707 | 3238 | 5945 | 118/17/19 | 39/3/3 | -| ITP, per-problem oracle | 3068 | 1675 | 4743 | 126/7/21 | 35/1/9 | -| ModAB [10, 20] | 2726 | 1681 | 4407 | 118/10/26 | 34/4/7 | - -A′ is 12.9% below Chandrupatla on the canonical set and 28.6% below on the wide-bracket set, and wins more problems than it loses against every rival on both. By function family, the unit in which these problems are independent (§9), it is ahead of Chandrupatla on 8 of the 15 canonical families (6 behind, one level) and on 7 of the 9 wide-bracket families — too few families for either count to be significant ($p=0.79$, $0.18$). The two it is behind on are `fun7`, 69 against 64, and `fun9`, 65 against 55. The earned-budget form keeps its constant-free guarantee of bisection plus one step and pays for it as in §3.1. - -**Table 9.** Leave-one-out ablation of A′ on all six corpora of §9 (uniform criterion, a sign change within $10^{-6}$). Δ = total without the component − total with it; positive means the component earns its place. - -| component removed | APS | CHA | new-1 | Roots.jl | applied | fourth | sum | -|---|---|---|---|---|---|---|---| -| the whole regime | −7 | +1013 | +1660 | +324 | +77 | +2269 | **+5336** | -| the split (7) (midpoint instead) | −5 | +479 | +667 | +272 | +30 | +1672 | **+3115** | -| overshoot branch | −6 | 0 | −7 | +16 | +63 | +893 | +959 | -| tolerance step (borrowed) | +130 | +35 | +97 | +78 | +135 | −2 | +473 | -| the half the step promises | +249 | +65 | +82 | −16 | −5 | +1 | +376 | -| translation limit | +31 | +143 | +100 | −21 | −2 | +87 | +338 | -| repeated step pattern | −22 | −6 | +123 | −30 | −10 | +34 | +89 | -| sign contradiction | +11 | −12 | +29 | 0 | +1 | −1 | +28 | - -Every component is positive in sum; the regime and the split carry the method. Two are marginal and we say so: the step pattern earns on two corpora of six and loses on four, and the sign contradiction is within noise (+28 over 855 problems). Removing both would change A′'s total by under one per cent (117 of 13032 evaluations); they are kept because the pre-registered rule removes only a component that earns nowhere. - -**Table 9a.** The similarity in the step of A′: the same method with each of the nine similarity functions of §5.3 in place of the tent, everything else unchanged. Totals on the two sets of §4 and over the six corpora of §9; the guarantee holds with every function. - -| $S$ in the step | APS + CHA | six corpora (855) | -|---|---|---| -| **tent $1-|z|$ (A′)** | **3333** | **13032** | -| $e^{-|z|}$ | 3965 | 14142 | -| $1/(1+|z|)$ | 4424 | 14843 | -| $(1+|z|)^{-2}$ | 7249 | 28500 | -| $1-\sqrt{|z|}$ | 11613 | 45552 | -| $1-|z|^{1/3}$ | 16094 | 63031 | -| $e^{-z^2}$ | 17429 | 67701 | -| $1/(1+z^2)$ | 17452 | 67394 | -| $1/\sqrt{1+z^2}$ | 17487 | 67854 | - -The nearest other function costs 19% more on the two sets and 9% more over the six corpora, and the functions that approach 1 quadratically cost five times more; the three functions with a linear approach come first and the three with a quadratic approach last, as the contraction law of §2.2 predicts. What this measures should be said plainly: with the tent the step is the undamped Newton step clipped to the bracket, and the other functions damp it, so the table shows that the undamped step is the better one — expected, and evidence for the choice of $S$ within the rule rather than for the similarity framing as such. - -### 8.6 What is known, and what is new - -Several mechanisms of A′ coincide with classical ones, and the coincidence is worth stating, because they were not taken from the literature but reached from the principles of the paper — similarity to a prototype and analogy as similarity of relations — and recognised afterwards: - -| in A′ | classical counterpart | -|---|---| -| the three-point order estimate of §7.1 | Lagouanelle's multiplicity estimate $m=f'^2/(f'^2-ff'')$ [13] | -| the contraction $e(1-1/m)$ behind the flag of §8.1 | multiplicity estimates from successive Newton iterates [14, 15] | -| the order (6) of an analogous pair | the log–log slope of a power law; the (approximate) computational order of convergence [18, 19] | -| the split (7) | false position on $\operatorname{sign}(f)|f|^{1/m}$ [16, 17] | -| the half the step promises | a relative of Brent's acceptance condition [2] | -| the tolerance step | Dekker [9], Brent [2] — borrowed and named as such | -| the exit judgment of §8.2 (the value at a split against the line) | ModAB's linearity check at the midpoint [10, 20], $|y_m-y_3|1$ 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 convergence becomes sublinear and the standard problems double to 161. -6. **The harness scores one canonical problem 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. With the five `fun7` problems of §7.3 it is why this paper scores by a sign change. +6. **The harness scores one canonical problem 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. With the five `fun7` problems of §A.5 it is why this paper scores by a sign change. 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 4258 on the canonical set against 2337 — a difference of 1921 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 7068. +8. **Placement beats form, and the wrong placement is expensive.** The same "refuse before the step" idea as reading (1) of §A.2, with refusal decided by *promised bracket progress* instead of by the shape of the model, costs 4258 on the canonical set against 2337 — a difference of 1921 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 7068. 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: its best cell is 3101 on the canonical set against 2539 for the best uncapped cell of the same grid, and 1766 on the wide-bracket set against 1752. -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 129/154 and 9855 — two problems and eight 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. +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 (§A.3), 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 129/154 and 9855 — two problems and eight per cent — where Illinois [7], 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 2759 to 2598 — but the canonical set gets worse, 5635 to 6749, against the 2634 of the base rule. 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. +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 2759 to 2598 — but the canonical set gets worse, 5635 to 6749, against the 2634 of the base rule. A march needs a *ratio* of distances while an identity on narrow brackets needs an *absolute* unit, and no parameter-free unit provides both. §A.6 shows the diagnosis itself was wrong: on the flat families the right move is to halve, not to march. -12. **Reading the unkept promise one step at a time does not pay.** The flag of §8.1 sees the flat regime, but a halving in place of each flagged step takes the canonical set from 2634 to 2839 and the wide-bracket set only from 2357 to 2243: a simple root far from convergence breaks its promises too. Requiring the promise to fail twice in a row keeps the canonical set (2613) but fires one step late, before the crossing, and replaces the most useful step of the cycle (wide brackets 2388); requiring a creep in one direction as well gives 2688 and 2399. +12. **Reading the unkept promise one step at a time does not pay.** The flag of §6.1 sees the flat regime, but a halving in place of each flagged step takes the canonical set from 2634 to 2839 and the wide-bracket set only from 2357 to 2243: a simple root far from convergence breaks its promises too. Requiring the promise to fail twice in a row keeps the canonical set (2613) but fires one step late, before the crossing, and replaces the most useful step of the cycle (wide brackets 2388); requiring a creep in one direction as well gives 2688 and 2399. 13. **A halving that sticks once begun wins the flat families and loses everything else.** Without an exit the flat families fall to 1057 — pure bisection's level — and the canonical set rises to 6707, because a false entry now lasts to the end of the run. Splitting at the order read from the failed promise, $(1-1/m)^m=r$, changes nothing: that order is $m=\infty$ in 163 of 174 entries (6591). -14. **The model's own point cannot tell a flat root from a simple one.** Judging the evaluated point against the line through the bracket ends and against 0 — the exit judgment of §8.2 — also at *entry* gives 2820 and 1532 against 2935 and 1546 for the exit alone: the model aims at the root, so $f$ is small there by construction and looks flat at a simple root as readily as at a power. The same judgment works at a split, which the model did not choose. +14. **The model's own point cannot tell a flat root from a simple one.** Judging the evaluated point against the line through the bracket ends and against 0 — the exit judgment of §6.2 — also at *entry* gives 2820 and 1532 against 2935 and 1546 for the exit alone: the model aims at the root, so $f$ is small there by construction and looks flat at a simple root as readily as at a power. The same judgment works at a split, which the model did not choose. -15. **The decision reading "defect" does not transfer.** Combined with the regime it was the best configuration on the two development sets (§7.2's defect reading with the flat families of §8), and it passed every pre-registered criterion there. On new corpus 1 it did not: per problem 95/33/78 against Chandrupatla ($p=0.22$), losing the class of exponential flanks 9 to 26. It is therefore not part of A′. +15. **The decision reading "defect" does not transfer.** Combined with the regime it was the best configuration on the two development sets (§A.4's defect reading with the flat families of §6), and it passed every pre-registered criterion there. On new corpus 1 it did not: per problem 95/33/78 against Chandrupatla ($p=0.22$), losing the class of exponential flanks 9 to 26. It is therefore not part of A′. -16. **A component rejected alone can pay in combination.** Remembering the analogy for the run was rejected by its own pre-registration, under the harness scoring of the time (canonical set 2814 against 2806 without the regime; under the sign-change criterion the two are level, 2633 against 2634), and with the split of §8.3 it is the better form (with the step pattern, wide brackets 1250 against 1388 for the analogy read in one state). Composition was measured, not assumed, throughout — which is also why §7's readings (2) and (3), each an improvement, are not combined. +16. **A component rejected alone can pay in combination.** Remembering the analogy for the run was rejected by its own pre-registration, under the harness scoring of the time (canonical set 2814 against 2806 without the regime; under the sign-change criterion the two are level, 2633 against 2634), and with the split of §6.3 it is the better form (with the step pattern, wide brackets 1250 against 1388 for the analogy read in one state). Composition was measured, not assumed, throughout — which is also why §A.2's readings (2) and (3), each an improvement, are not combined. --- -## 11. Conclusion +## Appendix D. Pre-registrations and the frozen code -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 which group of them comes first 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 projection's budget multiplier is a free constant, and we declare it: it does not move the canonical result and it does move the wide-bracket one; projecting onto a budget the steps earn instead removes it, at a measured price. +Every rule of §6 and every test of §7 was written down, with its criteria and stop rule, before its code and its run. Table D1 gives, for each document, the commit that first added it and the commit that first added the results its run wrote, both read from git by `check_prereg_order.py`: 25 of the 30 were committed before their results; four were committed together with them, so git cannot order them; one was decided on its smoke run and has no results file. The code of A′ was committed (`8027d7f`) after corpus 4 was defined (`657df10`) and before its results (`6d2531e`); `simroot/pure.py` did not change in between. -The base rule has one declared numerical constant, converges with Muller's order 1.839 and carries a worst case of twice bisection plus one step. On the canonical Alefeld–Potra–Shi set it is level with TOMS 748 in total and 1.5% behind Chandrupatla's method, while winning more problems than it loses against both; on the wide-bracket set Chandrupatla is genuinely stronger (1505 against 2357). §7.4 locates the cause: not the order of the model but the decision to give it up, which Chandrupatla takes by halving and which no reading of a single state takes correctly for our step. - -That decision is a similarity too — of the run to its own past. A pure power repeats its configuration under zoom, an exponential under translation, a simple root tends to the line; the analogous pair reads the local order, the order decides between the model and a halving by bisection's own $\tfrac12$, and it says where to split. The method built on it, A′, is ahead of Chandrupatla on both canonical sets in total and problem by problem, and on a corpus fixed after its code was frozen it needs 36% fewer evaluations and wins 177 problems to 28. Its decision layer beats Lagouanelle's estimate, Chandrupatla's band and no detector put in its place; with the multiplicity known in advance the classical transformation is 2.4 to 6.8 times cheaper on multiple roots, which measures what estimating the order costs. The claim the measurements support is a lead over Brent, TOMS 748, ITP and ModAB everywhere, within a bound of twice bisection that Brent, TOMS 748 and ModAB do not promise and exceed, and over Chandrupatla a lead that is large where roots are multiple, fractional or nearly flat and small where they are simple — not superiority by the same margin everywhere. `fun9`, near-cancelling terms and smooth applied equations are where it is behind or level. - -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 - -Three questions remain, all narrower than the one this section used to pose. - -The first is the price of estimating the order. With the multiplicity given, the transformation of §8.3 is 2.4 to 6.8 times cheaper than A′ on multiple and fractional roots (Table 9c): the order read from the analogous pair, or its use in the split, loses most of what the transformation could deliver. - -The second is the steep flank. Where the loss remains — `fun9`, the near-cancelling class, 4 − tan x at its pole — the base step approaches along a flank far steeper than its model, and the half contradiction of §8.4 removes only part of the cost. The third is the one geometric choice of §8.2, the Euclidean metric of the $(\xi,\varphi)$ plane: a metric derived from the configuration's own sensitivity would remove it. +**Table D1.** Pre-registrations, in the order they were written. +| pre-registration | commit | results file | commit | order | +|------------------------------|---------|--------------------------------------|---------|-------------| +| `logarithmic_2026-09-20` | `f9b062e` | `sweep_log.json` | `84ff1c7` | before | +| `prestep_gate_2026-09-21` | `0617243` | `sweep_gate.json` | `a6116cd` | before | +| `slope_gate_2026-09-21` | `e4769be` | `sweep_slope.json` | `e4769be` | same commit | +| `points_and_direction_2026-09-21` | `04ee5c8` | `sweep_points.json` | `22f19c5` | before | +| `similarity_per_slot_2026-09-21` | `15b9424` | `sweep_order.json` | `15b9424` | same commit | +| `scale_selection_2026-09-21` | `cc2c499` | `sweep_cap.json` | `cc2c499` | same commit | +| `decision_place_2026-09-21` | `2357379` | `sweep_decision.json` | `2357379` | same commit | +| `step_extrapolation_2026-09-21` | `781bd15` | `results_diag_step_extrapolation.txt` | `3cce10b` | before | +| `extrapolate_in_method_2026-09-21` | `a25cffa` | `sweep_extrap.json` | `99be33a` | before | +| `extrapolation_gates_2026-09-21` | `29ab495` | `sweep_extrap2.json` | `68df1d2` | before | +| `empty_step_2026-09-21` | `8279fb6` | `results_diag_empty_step.txt` | `d754689` | before | +| `outcome_memory_2026-09-21` | `2b0ee22` | `results_diag_outcome_memory.txt` | `2498c66` | before | +| `flat_regime_2026-09-21` | `9df1aa5` | `sweep_flat.json` | `e7e74d7` | before | +| `flat_twice_2026-09-21` | `6a8bb23` | `sweep_flat2.json` | `9884b21` | before | +| `flat_creep_2026-09-21` | `35eb941` | `sweep_flatcreep.json` | `5a1edff` | before | +| `flat_stick_2026-09-21` | `1192c68` | `sweep_flatstick.json` | `f8c17c4` | before | +| `flat_exit_2026-09-21` | `f8dd8d9` | `sweep_flatexit.json` | `1842eb0` | before | +| `flat_judge_2026-09-21` | `1dbda21` | `sweep_flatjudge.json` | `8bf595d` | before | +| `flat_analogy_2026-09-21` | `42bfab4` | `sweep_flatanalog.json` | `1842eb0` | before | +| `flat_analogy_mem_2026-09-21` | `4bf8634` | `—` | `—` | decided on the smoke run, no results file | +| `defect_contra_flat_2026-09-21` | `3c92d61` | `sweep_defcontra.json` | `63cf9fb` | before | +| `flat_order_point_2026-09-21` | `d1be3bb` | `sweep_orderpt.json` | `08863c1` | before | +| `flat_exp_limit_2026-09-21` | `e887691` | `sweep_explimit.json` | `baa9396` | before | +| `half_contra_2026-09-21` | `3d74b45` | `sweep_half.json` | `bbb16ec` | before | +| `out_of_sample_2026-09-21` | `47361da` | `sweep_oos.json` | `a86a9ed` | before | +| `roots_jl_2026-09-21` | `97af3ae` | `sweep_rootsjl.json` | `554ec0c` | before | +| `applied_2026-09-21` | `8ea89dd` | `sweep_applied.json` | `c2b0eea` | before | +| `close_sub_2026-09-21` | `657df10` | `sweep_fourth_fourth.json` | `6d2531e` | before | +| `modab_2026-09-21` | `bfa23b2` | `sweep_modab.json` | `37c81b0` | before | +| `controls_2026-09-22` | `77312be` | `sweep_controls.json` | `92c07d8` | before | --- @@ -670,3 +756,6 @@ The second is the steep flank. Where the loss remains — `fun9`, the near-cance 18. S. Weerakoon and T. G. I. Fernando, "A variant of Newton's method with accelerated third-order convergence," *Appl. Math. Lett.*, vol. 13, pp. 87–93, 2000. 19. A. Cordero and J. R. Torregrosa, "Variants of Newton's method using fifth-order quadrature formulas," *Appl. Math. Comput.*, vol. 190, pp. 686–698, 2007. 20. N. Ganchovski, O. Smith, C. Rackauckas, L. Tomov and A. Traykov, "Improvements to the Modified Anderson–Björck (modAB) root-finding algorithm," *Algorithms*, vol. 19, 332, 2026; package `pymodab` 1.0.9. +21. J. C. P. Bus and T. J. Dekker, "Two efficient algorithms with guaranteed convergence for finding a zero of a function," *ACM Trans. Math. Softw.*, vol. 1, no. 4, pp. 330–345, 1975. +22. J. M. Fernández-Díaz and C. O. Menéndez-Pérez, "A common framework for modified Regula Falsi methods and new methods of this kind," *Math. Comput. Simul.*, vol. 205, pp. 678–696, 2023. +23. A. Cordero, B. Neta and J. R. Torregrosa, "Memorizing Schröder's method as an efficient strategy for estimating roots of unknown multiplicity," *Mathematics*, vol. 9, 2570, 2021. diff --git a/article/sim_numerics.pdf b/article/sim_numerics.pdf index aff791869ce8a46b279e05fa7de452890dccfd33..55462977f1c5eabc17255d9dda4685793cd9fb2c 100644 GIT binary patch delta 266055 zcmV(@K-Ry8rXK5(9*`mfGdMJtF=qiOf0Y``a^$x0?ys0jYN;v>O*}}D-g2dO9XsVz zHg-j-_~6|GG=wC2Ipl=otXBT|bfX&_kkUw%iw4A_(eD?OZSFSN=I`HRul&Bh{^r}C z@@ykaDn+h0*S8yk!)jwhkt$hjuItV3$-n!Htt^s$y4b2rCOKadQDHe~&c2 zYW7WgArgDB70J*{7t(|!+toC9*V(qed;IQVt22|l!%306AMk|T;epQx&`t9NAKY|L z_tzJnuK%&A@>HmDvy~#PL`eufG}ArbLqC7Bligmq*6k+87yr%e8o+IA`i>_4X$f__ zZa~n$3bhdcHqRxpqAJon%QstPf6}Zh=%)Xqa2QTpHB;AoMtpnx^99d8Vehv;m10w+ zRUr$6SEOYQh^s1^4_yBaj8EInQ)Cb@w;NEnI{ltfkZ8#Am2V>-DRRZ5wf3}fa7iT_Z zvY@TgJx`63JJ9HE@N9Zu1Xx}$x~F+B*@(cZ1R+~QBMe;Q|(@T*mF>}m(18)(+j5A!le^E{~v6t-NWef2WV%Cl4~ z&v>c7aE}4e<@vMsoMWjfMVhPfspvA#S0L0R-_y@4Tn3h;I)9=BsMDTl=)aKBwHul* 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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. -\textbf{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. +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. + +The last row fails at once, and it matters because that similarity is +the one this paper proposes: at \(x_0=0\), \(z=-1\), so \(S=0\), the +step is the whole bracket and the iterate stays at \(x_\mathrm{far}\) +(error 0.5858 after one step and after two hundred). The failure belongs +to the chosen scale, not to the similarity: under the derived scale of +§3 the clip never engages on this problem, and the same function is the +best of the nine (Table 4). \textbf{Corollary 1 (A chosen similarity is not enough).} For fixed \(S\) and fixed \(\sigma\) the rate in (2) depends on \(|f'(r)|\), \(D\) @@ -384,18 +332,15 @@ priori: it must be read off the evaluations. \hypertarget{the-method}{% \subsection{3. The method}\label{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 5.3 tests it -against the full grid. One number in the safeguard is not fixed by them, -and §3.1 declares it as the method's one free constant. +Corollaries 1--3 fix every choice before any benchmark is consulted; +Section 5.2 tests the prediction against the full grid. One number in +the safeguard is not fixed by them, and §3.1 declares it. \textbf{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 \textbf{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. +\(S(z)=\max(0,1-|z|)\). It is not the ratio form \(1/(1+|z|)\) of +Proposition 1. \textbf{Scale: the finest data-derived one.} We set \(\sigma = |\hat f'|\cdot|x_\mathrm{far}-x|\) with \(\hat f'\) a divided @@ -430,23 +375,14 @@ from 2, and we report no order measured there. a Newton step with a three-point divided-difference derivative, clipped to the bracket and then projected as in §3.1. That the tent turns the -similarity step into Newton's is a property of the similarity chosen, -not a sign that the similarity has dropped out: which \(S\) stands in -the step decides the step, and it is measured, not assumed --- Table 4 -ranks the nine functions over the whole grid, the readings of §7 pay -most with the tent in the step and some of them lose with other -functions (§7), and in the proposed method replacing the tent by any -other function costs 19\% to 425\% (Table 9a). - -We state the reduction ourselves rather than leave it to be discovered, -because it locates the contribution precisely. The base rule 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 predicts which similarity functions come -first; (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 §10; (iv) the measurements. We call it the \textbf{base -rule}; the method this paper proposes builds on it in Section 8. +similarity step into Newton's is a property of the chosen similarity, +and which \(S\) stands in the step is measured rather than assumed: in +the proposed method, replacing the tent by any other function costs 19\% +to 425\% (Table 7). The base rule is therefore not a new family of +methods; what is ours is a derivation that fixes every choice before any +benchmark is consulted, and a safeguard of ITP's projection type with +ITP's truncation stage --- and all three of its constants --- removed. +The proposed method builds on it in Section 6. \textbf{Algorithm.} Given \([a,b]\) with \(f(a)f(b)<0\) and tolerance \(\varepsilon_x\): @@ -475,8 +411,8 @@ 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 --- with one exception, the factor 2 in \texttt{n\_max}, which §3.1 declares -as the method's one free constant. The constant-free variant of §3.1 -replaces the two \texttt{allow} lines by the earned credit of (3'). +as the base rule's one numerical constant. The constant-free variant of +§3.1 replaces the two \texttt{allow} lines by the earned credit of (3'). \hypertarget{the-safeguard-project-do-not-replace}{% \subsubsection{3.1 The safeguard: project, do not @@ -506,10 +442,8 @@ 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. -\textbf{The multiplier is the method's one free numerical constant.} It -is not ``the factor already claimed in the bound'' --- that argument is -circular, the bound is twice bisection because two was chosen --- and -measurement settles it (\texttt{measure\_budget\_factor.py}, Linux): +\textbf{The multiplier is the base rule's one numerical constant}, and +it is not neutral (\texttt{measure\_budget\_factor.py}, Linux): \textbf{Table 1a.} The scheduled projection at budget multiplier \(c\), i.e.~\(n_{\max}=\lceil c\cdot\log_2(w_0/\varepsilon_x)\rceil\). Total @@ -532,14 +466,12 @@ evaluations, failures charged. On the canonical set the projection binds on one problem only once \(c\ge1.25\) --- \texttt{aps.13.00}, whose \(f\) underflows to zero -around the root (20, 30 and 28 evaluations at \(c=1.25\), 1.5 and 2) --- -so the total barely moves. On the wide-bracket set it moves -monotonically with \(c\) from 1.25 on, and 2 is not the best value. Nor -is zero slack (\(c=1\)) a degenerate case under which the guard behaves -as bisection: it costs 4625 and 2039, far from bisection's 7338 and -2136, and on the wide-bracket set it beats \(c=2\). The multiplicative -family has no member that the bound singles out, so its multiplier is a -genuinely free constant. +around the root (20, 30 and 28 evaluations at \(c=1.25\), 1.5 and 2). On +the wide-bracket set the cost rises monotonically with \(c\) from 1.25 +on, and 2 is not the best value; zero slack (\(c=1\)) is not degenerate +either (4625 and 2039, against bisection's 7338 and 2136). No member of +the family is singled out by the bound: the multiplier is a free +constant. \textbf{The budget earned rather than scheduled: a constant-free variant.} The constant can be removed. Let \(s_j=\log_2(w_j/w_{j+1})\) @@ -576,26 +508,25 @@ evaluations, failures charged. 5 & bisection + 5 steps & 2819 & 1666 & 4485 \\ \end{longtable} -A looser promise buys evaluations, as it must; within one promise the -cheapest endowment is determined. The constant-free variant is therefore -the earned projection with \(C_0=1\), whose guarantee --- -\(\lceil\log_2(w_0/\varepsilon_x)\rceil+4\) evaluations, bisection plus -one step with one more for the floating-point boundary --- is verified -on all 199 problems. +Within one promise the cheapest endowment is determined, so the +constant-free variant is the earned projection with \(C_0=1\); its +guarantee, \(\lceil\log_2(w_0/\varepsilon_x)\rceil+4\) evaluations, is +verified on all 199 problems. We nonetheless propose the scheduled +projection with \(c=2\), fixed before any benchmark was consulted: it is +743 evaluations cheaper on the canonical set (2634 against 3377), the +earned variant is cheaper on the wide-bracket set (2073 against 2357) +--- one declared numerical constant and the better canonical result, or +none, a tighter bound and the better wide-bracket result. -We nonetheless propose the scheduled projection with \(c=2\) and declare -its multiplier. On the canonical set it is 743 evaluations cheaper than -the earned variant (2634 against 3377), and there its total barely -depends on the multiplier; on the wide-bracket set the earned variant is -the cheaper (2073 against 2357). The two are the ends of a trade stated -in numbers: one declared constant and the better canonical result, or no -constant, a tighter bound and the better wide-bracket result. The value -2 was fixed before any benchmark was consulted. - -Two remarks apply to both forms. +Three remarks apply to both forms. \begin{itemize} \tightlist +\item + \textbf{Guaranteed bounds of this kind are classical.} Bus and Dekker + {[}21{]} gave bracketing methods whose worst case is four to five + times bisection; the scheduled projection guarantees twice bisection + plus one step, the earned one bisection plus one step. \item \textbf{This is ITP's projection stage without ITP's truncation stage.} All three of ITP's constants live in the truncation, whose @@ -626,52 +557,27 @@ the same criterion on every corpus of this paper; \(x\)-tolerance \(10^{-12}\); budget 200 evaluations. Scoring against the published root instead, as the benchmarks' harness does, charges every method the full budget on six problems where \(f\) vanishes identically around the root -in floating point (§7.3). Head-to-head counts (wins/ties/losses) are +in floating point (§A.5). Head-to-head counts (wins/ties/losses) are taken over every problem of a set with the same charging as the totals. \emph{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. -\textbf{Fairness to the competitors.} Three deliberate concessions. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - 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. -\item - ITP is reported both at the authors' recommended constants and as a - \emph{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. -\item - 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 \(n_0\in\{0,1,2,5\}\). -\item - Where SciPy ships the method, the competitor \textbf{is the library}: - Chandrupatla is \texttt{scipy.optimize.elementwise.find\_root}, Brent - is \texttt{brentq}, TOMS 748 is \texttt{toms748}, each run with the - same stopping rule as every other method --- a bracket no wider than - \(\varepsilon_x\), the relative term at machine precision, no stopping - on \(|f|\) --- and each evaluation counted by the same wrapper around - \(f\). An earlier version gave each competitor the better of the - library's implementation and ours, problem by problem. That concession - backfired in the competitor's favour: our Chandrupatla stopped on a - bracket twice as wide as everybody else's --- the factor this protocol - corrects for ITP in item 1 and had not corrected there --- and the - per-problem minimum let the looser run win. With the tolerance - corrected our implementation and the library agree on all 199 problems - exactly. For Brent the library is slightly dearer than our - implementation; we use the library anyway, because it is the - implementation a reader can check. 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. -\end{enumerate} +\textbf{Competitors.} Where SciPy ships the method, the competitor is +the library --- Chandrupatla is +\texttt{scipy.optimize.elementwise.find\_root}, Brent \texttt{brentq}, +TOMS 748 \texttt{toms748} --- run with the stopping rule of every method +(a bracket no wider than \(\varepsilon_x\), the relative term at machine +precision, no stopping on \(|f|\)) and counted by the same wrapper +around \(f\). ITP, which has no library implementation, is ours, run +with \(\epsilon=\varepsilon_x/2\) because it terminates on +\(b-a\le2\epsilon\), checked against its own theorem (\(n_{1/2}+n_0\) +iterations at most, on all 154 problems for \(n_0\in\{0,1,2,5\}\)), and +reported both at the authors' constants and as a \emph{per-problem +oracle} over a \((\kappa_1,n_0)\) grid --- not an achievable method, +included so that it cannot be said to have been handicapped. Bisection +and the false-position family are ours. \textbf{Reproducibility.} All numbers come from a single Linux run. The test functions call \texttt{libm}, whose last digit differs between @@ -690,8 +596,7 @@ Section 3 --- tent similarity, three-point scale, projection safeguard with its multiplier declared --- is one entry of that grid, fixed by the theory rather than chosen from the ranking; the constant-free earned variant of §3.1 is shown beside it. Chandrupatla's own method and -Dekker's are among the baselines; an earlier version of these tables was -drawn up before either was added. +Dekker's are among the baselines. \textbf{Table 2.} Alefeld--Potra--Shi, 154 problems, out of 264 configurations. Places by failures first, then total. @@ -728,13 +633,13 @@ total evals 4 & TOMS 748 & yes & 154/154 & 2636 & 0 \\ 5 & Brent & yes & 154/154 & 2707 & 0 \\ 11 & ITP, per-problem oracle & yes & 154/154 & 3068 & 0 \\ -15 & Ridders & no & 154/154 & 3124 & 0 \\ +15 & Ridders {[}5{]} & no & 154/154 & 3124 & 0 \\ 25 & constant-free variant: earned projection & no & 154/154 & 3377 & 0 \\ 29 & ITP, recommended constants & yes & 154/154 & 3767 & 0 \\ 73 & bisection & no & 154/154 & 7338 & 0 \\ 232 & Dekker & no & 153/154 & 3158 & 0 \\ -237 & Illinois & yes & 153/154 & 7045 & 0 \\ +237 & Illinois {[}7{]} & yes & 153/154 & 7045 & 0 \\ 244 & false position & no & 127/154 & 10560 & --- \\ \end{longtable} @@ -745,14 +650,12 @@ against 2636). Problem by problem the base rule wins against Chandrupatla 87/24/43 (wins/ties/losses; exact two-sided sign test \(p=1.4\cdot10^{-4}\)), against TOMS 748 86/16/52, Brent 102/11/41, ITP at its recommended constants 132/3/19, the per-problem ITP oracle -121/4/29, and bisection 152/1/1. It wins twice as many problems as it -loses against Chandrupatla and trails that method in total because its -losses are the larger ones; the problems are not independent (§9), and -taken by function family the comparison is 6/1/8 of 15 families, a tie. -Dekker and Illinois solve 153 of 154 and are ranked after every method -that solves all. The constant-free variant goes 79/3/72 against -Chandrupatla (\(p=0.63\), a tie in problems), 78/9/67 against TOMS 748 -and 79/5/70 against Brent, and loses on the total to all three. +121/4/29, and bisection 152/1/1. It trails Chandrupatla in total because +its losses are the larger ones; by function family, the unit in which +these problems are independent (§7), the comparison is 6/1/8 of 15 +families, a tie. The constant-free variant goes 79/3/72 against +Chandrupatla (\(p=0.63\)), 78/9/67 against TOMS 748 and 79/5/70 against +Brent. \textbf{Table 3.} Chandrupatla, 45 problems, brackets up to \(10^{12}\) wide, out of 264 configurations. @@ -811,104 +714,28 @@ against 4100 for Chandrupatla, 4743 for the ITP oracle, 5724 for TOMS \subsubsection{5.1 What the guarantee costs}\label{what-the-guarantee-costs}} -It depends on which bound is bought. Against the identical rule with no -safeguard at all: - -\begin{longtable}[]{@{} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}} -\toprule\noalign{} -\begin{minipage}[b]{\linewidth}\raggedright -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -solved -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -total evals -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -runs over the \(2\times\) bound -\end{minipage} \\ -\midrule\noalign{} -\endhead -\bottomrule\noalign{} -\endlastfoot -APS, no safeguard & 154/154 & 2634 & 0 \\ -APS, projection --- base rule (\(2\times\) bisection) & 154/154 & 2634 & -0 \\ -APS, earned projection --- constant-free (bisection + 1 step) & 154/154 -& 3377 & 0 \\ -Chandrupatla, no safeguard & 39/45 & 3391 & 10 \\ -Chandrupatla, projection --- base rule & 45/45 & 2357 & 0 \\ -Chandrupatla, earned projection --- constant-free & \textbf{45/45} & -\textbf{2073} & \textbf{0} \\ -\end{longtable} - -The projection of the base rule never binds on the canonical set, so its -totals are identical to the last evaluation; on the wide-bracket set it -solves six more problems for 30\% fewer evaluations. The earned -projection promises twice as much --- a worst case of bisection plus one -step instead of twice bisection --- and pays for it on the canonical -set, 743 evaluations or 28\%, while on the wide-bracket set it is the -cheapest of the three. Both are the opposite of the usual trade on at -least one set, and that is the point of projecting rather than -replacing: the two safeguards that replace cost 7068 and 7046 on -Alefeld--Potra--Shi, against 2634. - -\hypertarget{wall-clock-time-and-why-it-is-not-the-cost-model}{% -\subsubsection{5.2 Wall-clock time, and why it is not the cost -model}\label{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 base rule is the \textbf{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. (These are the evaluations the timing loop actually spent; -for the base rule and bisection, which solve every problem, they equal -the totals of §5; Brent here is our pure-Python implementation, which is -what is timed.) - -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 base 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 \emph{negative} overheads for -four of six methods --- an impossibility that condemned the estimator, -and the reason nothing is subtracted here. +Against the same rule with no safeguard (Tables 2 and 3), the projection +costs nothing on the canonical set and on the wide-bracket set solves +six more problems for 30\% fewer evaluations (2357 against 3391; +unguarded, 10 runs break the bound). The earned projection, promising +bisection plus one step, pays 743 evaluations (28\%) on the canonical +set and is the cheapest of the three on the wide-bracket set. That is +the point of projecting rather than replacing: the two safeguards that +replace cost 7068 and 7046 on Alefeld--Potra--Shi, against 2634. \hypertarget{the-law-across-the-whole-grid}{% -\subsubsection{5.3 The law across the whole +\subsubsection{5.2 The law across the whole grid}\label{the-law-across-the-whole-grid}} Corollaries 1--3 predict that the similarities with \(c=1\) come before those with \(c=0\) under a data-derived scale; they say nothing about the order within a group, nor about the functions with unbounded slope. -Table 4 reports it. 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. That \(c=0\) is slow is the unsurprising half --- a step -proportional to \(z^2\) is damped quadratically. For the chosen scale -the two groups are indistinguishable (14983 against 14228), which is -consistent --- with a chosen scale nothing converges well and the -ordering carries no information. +For every data-derived scale the three similarities with \(c=1\) occupy +the top three places of Table 4 and the three with \(c=0\) the bottom +three (that \(c=0\) is slow is the unsurprising half: a step +proportional to \(z^2\) is damped quadratically). With the chosen scale +the two groups are indistinguishable (14983 against 14228), as they +should be. \textbf{Table 4.} Mean total evaluations over both sets, step guard, grouped by the slope \(c\) of \(1-S\) at the origin. @@ -931,458 +758,27 @@ Ranked individually and best over scales, the similarities come out \((1+|z|)^{-2}\) (8455), \(1-\sqrt{|z|}\) (9411), \(1-|z|^{1/3}\) (10874), \(e^{-z^2}\) (13187), \(1/(1+z^2)\) (13990), \(1/\sqrt{1+z^2}\) (15272); the scales come out 3-point (5422), 2-point (5564), bracket -(7972), start (13423). The group order and the order of the scales are -those the law predicts and were predicted before the grid was run; the -order inside each group is measured, not predicted. +(7972), start (13423). The group order and the order of the scales were +predicted before the grid was run; the order inside each group is +measured. \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} -\hypertarget{where-the-evaluations-go}{% -\subsection{6. Where the evaluations -go}\label{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 \emph{approach} from the \emph{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 -\emph{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 \emph{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. The bracket here is reconstructed from \emph{every} -sign the method has observed --- the narrowest interval the evaluations -prove, not the one the method keeps; Brent and TOMS 748 always probe -inside their own bracket, but 69 and 141 of their probes fall outside -this narrowest one and remove nothing from it, and those probes are left -out of the median. High-order interpolation is a wager that \(f\) is -locally polynomial. When the wager pays, a high order wins the endgame; -when it fails, the probe removes little. The similarity rule makes the -weakest available assumption --- a single local slope --- and degrades -gracefully. - -\begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} - -\hypertarget{where-the-similarity-is-read}{% -\subsection{7. Where the similarity is -read}\label{where-the-similarity-is-read}} - -Everything above reads the similarity in one place: the \emph{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 base 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 \texttt{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 all 199 problems. - -Where a safeguard other than the projection of §3.1 appears, it is one -of the grid's. \emph{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. \emph{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. \emph{bracket} and -\emph{both} are the replacing safeguards of §3.1, which keep a bound at -the price of the order; \emph{step} alone and \emph{none} carry no -bound. - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - \textbf{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. -\item - \textbf{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 \emph{selected} - --- an argmin over what is already in hand, with no threshold. -\item - \textbf{Which end is far.} Read by dissimilarity instead of distance: - the end whose \(|f|\) is larger is the one less similar to the target. -\end{enumerate} - -\textbf{Table 5.} Reading the similarity in one more place is worth more -than changing \(S\). Evaluations of \(f\); failures charged the full -budget. - -\begin{longtable}[]{@{} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} - >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}} -\toprule\noalign{} -\begin{minipage}[b]{\linewidth}\raggedright -configuration (all with the guarantee, 199/199) -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -APS-154 -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -CHA-45 -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -both -\end{minipage} \\ -\midrule\noalign{} -\endhead -\bottomrule\noalign{} -\endlastfoot -base rule: tent, 3-point, projection \(c=2\) (§3) & 2634 & 2357 & -4991 \\ -constant-free variant: earned projection (§3.1) & 3377 & 2073 & 5450 \\ -(1) trust before the step, projection & 2663 & 2170 & 4833 \\ -(1)+(2) points chosen by shape, \emph{safe} & \textbf{2337} & 2177 & -4514 \\ -(1)+(3) far end by dissimilarity, projection & 2539 & \textbf{1864} & -\textbf{4403} \\ -(1)+(2)+(3), \emph{safe} & 2695 & 1928 & 4623 \\ -Chandrupatla & 2595 & 1505 & 4100 \\ -TOMS 748 & 2636 & 3088 & 5724 \\ -ITP, per-problem oracle & 3068 & 1675 & 4743 \\ -\end{longtable} - -Two results here are worth stating plainly. First, the canonical set -improves from 2634 to \textbf{2337}, which is below every method -measured, including Chandrupatla's 2595, with the worst-case guarantee -intact. It hardly feels the declared multiplier of §3.1: 2337 for every -multiplier from 1.5 to 4, 2351 at 1.25 -(\texttt{measure\_records\_vs\_multiplier.py}). Second, on the two sets -together the 4403 of reading (3) passes ITP's oracle (4743) but not -Chandrupatla (4100): the remaining gap is entirely on the wide-bracket -families, where Chandrupatla needs 1505 against our 1752 (reading (3) -under \emph{safe}). - -The readings do not compose. Taken together, (2) and (3) give 2695 and -1928 --- worse than either alone on the canonical set, and on the -wide-bracket set worse than (3) alone. Two selections that are each an -improvement interfere, which is a reason to measure composition rather -than assume it. - -\textbf{Which similarity stands in the step decides whether a reading -pays.} The readings above, and the decision \emph{defect} of §7.2, -compare quantities directly --- an ordinal judgment of which is more -similar, in which every decreasing \(S\) would decide alike. Their gains -nevertheless depend on the \(S\) that sets the step -(\texttt{audit\_similarity\_places.py}, on the same sweeps): on the -canonical set the triple chosen by shape gains 493 evaluations with the -tent in the step and loses 250 with \(1/\sqrt{1+z^2}\); \emph{defect} -gains 283 with the tent and loses 215 with \(1-|z|^{1/3}\). The records -of this section are the pairing of the tent step with these judgments, -not the judgments alone. - -Under the constant-free earned projection the same readings gain little -or lose. Reading (1) gives 3316 and 2050, (1)+(2) 3564 and 2074, (1)+(3) -3666 and 2052, all three together 4163 and 2087, against 3377 and 2073 -for the constant-free variant itself: the earned budget binds early on -the canonical problems, and what the readings buy under the projection -is mostly room that the earned budget does not grant. - -\hypertarget{the-similarity-function-belongs-to-the-place-not-to-the-method}{% -\subsubsection{7.1 The similarity function belongs to the place, not to -the -method}\label{the-similarity-function-belongs-to-the-place-not-to-the-method}} - -Everything above varies \emph{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 \emph{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 \emph{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 --- \(m=1/(1-ff''/f'^2)\), -Lagouanelle's estimate of the multiplicity {[}13{]} with the derivatives -replaced by divided differences, comes from three evaluations already -paid for, as in item 5 of §10, and is 1 whenever the power model does -not apply. - -Measured over the grid, the correction behaves as that reading predicts -and not as a universal improvement. Paired inside each (function, -safeguard) cell of the far-end reading it is cheaper in 33 of 63 cells ---- indistinguishable from a coin. But in the one pairing where that -reading \emph{is} exact inverse interpolation, the ratio similarity, it -is cheaper under all seven safeguards: - -\textbf{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 33 of 63 cells over all nine functions. - -\begin{longtable}[]{@{} - >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} - >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} - >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} - >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} - >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}@{}} -\toprule\noalign{} -\begin{minipage}[b]{\linewidth}\raggedright -safeguard -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -APS, \(m=1\) -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -APS, measured \(m\) -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -CHA, \(m=1\) -\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright -CHA, measured \(m\) -\end{minipage} \\ -\midrule\noalign{} -\endhead -\bottomrule\noalign{} -\endlastfoot -both & 4331 & \textbf{3485} & 2469 & \textbf{2306} \\ -bracket & 4418 & \textbf{3352} & 2671 & \textbf{2428} \\ -credit & 4999 & \textbf{4495} & 2064 & \textbf{1933} \\ -projection & 6783 & \textbf{6176} & 4057 & \textbf{3751} \\ -safe & 5803 & \textbf{5040} & 3218 & \textbf{3032} \\ -step & 6251 & \textbf{5362} & 3371 & \textbf{3146} \\ -none & 10676 & \textbf{9855} & 8265 & \textbf{7424} \\ -\end{longtable} - -Two limits must be stated with it. The estimator is \emph{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 1933, against 1752 for -reading (3) above and 1505 for Chandrupatla. In the \emph{slope} reading -of §3 the same exponent is harmful, \(2634\to2703\), 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. - -\hypertarget{the-decision-place}{% -\subsubsection{7.2 The decision place}\label{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 \emph{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 \emph{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, \(S(|\varphi-\xi|)\tfrac12\). Three choices are declared rather than derived: the multiplier \(c\) of the projection @@ -1577,7 +955,78 @@ threshold \(\tfrac12\)). The projection carries the guarantee, with no violation on every corpus of this paper and under every safeguard of the grid. -\textbf{Table 8.} A′ on the two sets of §4 (failures charged 200). A′'s +Counting numerical constants does not measure the method's freedom, +however. Beyond these numbers A′ is a decision layer of about ten +discrete rules --- the flag, the analogy and its memory, the entry, the +exit, the overshoot branch, the split, the step pattern, the half, the +contradiction, the tolerance step --- chosen on the two development sets +among the many candidates of Appendix C. That selection is why the +method was tested once on a corpus fixed before its code was frozen, and +against simpler rules put in its place (§7). Algorithm 2 is the complete +method; \texttt{simroot/pure.py} and the compiled \texttt{simbracket} +implement exactly this and agree bit for bit. + +\textbf{Algorithm 2 (A′).} Notation of Algorithm 1; +\(S(z)=\max(0,1-|z|)\). + +\begin{verbatim} +state: bracket [A,B], FA, FB; points x, xp, xp2; values fx, fp, fp2 +memory: configs (configurations), moves (displacements) +flags: last_model, in_regime, + analogy_seen (with order m_a), shallow_seen (with order m_s) +initialise as Algorithm 1; allow <- 2^n_max * xtol/2 +while |B - A| > xtol: + far <- end of [A,B] further from x; mid <- (A+B)/2; model <- false + f' <- derivative at x of the parabola through (x, xp, xp2) + sigma <- |f'| * |far - x| + if sigma is 0 or not finite: xn <- mid + else: + z <- |fx| / sigma + # the flat regime (6.1-6.3) + c <- (xi, phi, |x-xp|, |fx-fp|), last three points, if defined + # unkept promise, and unkept promise with a crossing + flag <- not in_regime and last_model and fx/fp >= 1/4 + flag_over <- not in_regime and last_model and fx/fp <= -1/4 + # order (6) of the analogous pair: the past configuration + # nearest to c (Euclidean in (xi, phi)), if nearer than the + # line phi = xi; inf if the pair shows L_f < 1 <= L_x; else none + m <- order of the analogous pair + if m >= 2: analogy_seen <- true; m_a <- m + if 0 < m <= 2/3: shallow_seen <- true; m_s <- m + if flag and analogy_seen: in_regime <- true + elif flag_over and shallow_seen: in_regime <- true; m_a <- m_s + append c to configs + if in_regime: xn <- split point (7) with m = m_a + elif x != xp and + sign((fx-fp)/(x-xp)) != sign((FB-FA)/(B-A)): xn <- mid + elif S(z) >= 1/2 and |f(far)| < |fx|: xn <- mid + else: + step <- (1 - S(z)) * (far - x) # eq. (4) + append x - xp to moves + # step pattern: the lag 2 .. half the history of highest + # mean similarity of successive length ratios has mean > 1/2, + # and step and x - xp point the same way + if the pattern repeats: + s <- min(|step|,|x-xp|) / max(|step|,|x-xp|) + step <- step / (1 - s), at most |far - x| long + if |step| < xtol/2 and last_model and |fx/fp| < 1/4: + step <- sign(far - x) * xtol/2 # tolerance step + xn <- x + step; model <- true + xn <- clamp(xn, hi - allow, lo + allow) # projection (3) + if xn outside (A,B) or xn in {x, xp}: + xn <- mid; if mid in {x, xp}: stop + last_model <- model and xn is the model's own point + fn <- f(xn) + if in_regime: # exit + t <- (xn - A)/(B - A) + if |fn - (FA(1-t) + FB t)| < |fn|: + in_regime <- false; analogy_seen <- false + allow <- allow/2; (xp2, xp, x) <- (xp, x, xn) + keep the sign change in [A,B] +return midpoint(A,B) +\end{verbatim} + +\textbf{Table 5.} A′ on the two sets of §4 (failures charged 200). A′'s wins/ties/losses against each rival over all problems of the set, exact two-sided sign test. @@ -1620,15 +1069,13 @@ ModAB {[}10, 20{]} & 2726 & 1681 & 4407 & 118/10/26 & 34/4/7 \\ A′ is 12.9\% below Chandrupatla on the canonical set and 28.6\% below on the wide-bracket set, and wins more problems than it loses against every rival on both. By function family, the unit in which these problems are -independent (§9), it is ahead of Chandrupatla on 8 of the 15 canonical +independent (§7), it is ahead of Chandrupatla on 8 of the 15 canonical families (6 behind, one level) and on 7 of the 9 wide-bracket families --- too few families for either count to be significant (\(p=0.79\), \(0.18\)). The two it is behind on are \texttt{fun7}, 69 against 64, and -\texttt{fun9}, 65 against 55. The earned-budget form keeps its -constant-free guarantee of bisection plus one step and pays for it as in -§3.1. +\texttt{fun9}, 65 against 55. -\textbf{Table 9.} Leave-one-out ablation of A′ on all six corpora of §9 +\textbf{Table 6.} Leave-one-out ablation of A′ on all six corpora of §7 (uniform criterion, a sign change within \(10^{-6}\)). Δ = total without the component − total with it; positive means the component earns its place. @@ -1677,16 +1124,15 @@ sign contradiction & +11 & −12 & +29 & 0 & +1 & −1 & +28 \\ \end{longtable} Every component is positive in sum; the regime and the split carry the -method. Two are marginal and we say so: the step pattern earns on two -corpora of six and loses on four, and the sign contradiction is within -noise (+28 over 855 problems). Removing both would change A′'s total by -under one per cent (117 of 13032 evaluations); they are kept because the +method. The step pattern (earns on two corpora of six) and the sign +contradiction (+28 over 855 problems) are marginal --- together under +one per cent, 117 of 13032 evaluations --- and are kept because the pre-registered rule removes only a component that earns nowhere. -\textbf{Table 9a.} The similarity in the step of A′: the same method -with each of the nine similarity functions of §5.3 in place of the tent, +\textbf{Table 7.} The similarity in the step of A′: the same method with +each of the nine similarity functions of §5.2 in place of the tent, everything else unchanged. Totals on the two sets of §4 and over the six -corpora of §9; the guarantee holds with every function. +corpora of §7; the guarantee holds with every function. \begin{longtable}[]{@{}lll@{}} \toprule\noalign{} @@ -1717,14 +1163,11 @@ step is the better one --- expected, and evidence for the choice of \(S\) within the rule rather than for the similarity framing as such. \hypertarget{what-is-known-and-what-is-new}{% -\subsubsection{8.6 What is known, and what is +\subsubsection{6.6 What is known, and what is new}\label{what-is-known-and-what-is-new}} -Several mechanisms of A′ coincide with classical ones, and the -coincidence is worth stating, because they were not taken from the -literature but reached from the principles of the paper --- similarity -to a prototype and analogy as similarity of relations --- and recognised -afterwards: +Several mechanisms of A′ coincide with classical ones, reached from +similarity and analogy and recognised afterwards: \begin{longtable}[]{@{} >{\raggedright\arraybackslash}p{(\columnwidth - 2\tabcolsep) * \real{0.5000}} @@ -1739,26 +1182,30 @@ classical counterpart \endhead \bottomrule\noalign{} \endlastfoot -the three-point order estimate of §7.1 & Lagouanelle's multiplicity +the three-point order estimate of §A.3 & Lagouanelle's multiplicity estimate \(m=f'^2/(f'^2-ff'')\) {[}13{]} \\ -the contraction \(e(1-1/m)\) behind the flag of §8.1 & multiplicity +the contraction \(e(1-1/m)\) behind the flag of §6.1 & multiplicity estimates from successive Newton iterates {[}14, 15{]} \\ the order (6) of an analogous pair & the log--log slope of a power law; the (approximate) computational order of convergence {[}18, 19{]} \\ the split (7) & false position on \(\operatorname{sign}(f)|f|^{1/m}\) -{[}16, 17{]} \\ +{[}16, 17{]}; inside a bracket, with the multiplicity given, the SFRFm +method of {[}22{]} \\ +the dimensionless configuration \((\xi,\varphi)\) of §6.2 & the ratios +of Chandrupatla's acceptance band {[}3{]}; every scaled regula falsi +written in two dimensionless parameters {[}22{]} \\ the half the step promises & a relative of Brent's acceptance condition {[}2{]} \\ the tolerance step & Dekker {[}9{]}, Brent {[}2{]} --- borrowed and named as such \\ -the exit judgment of §8.2 (the value at a split against the line) & +the exit judgment of §6.2 (the value at a split against the line) & ModAB's linearity check at the midpoint {[}10, 20{]}, \(|y_m-y_3|{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}} + >{\raggedright\arraybackslash}p{(\columnwidth - 6\tabcolsep) * \real{0.2500}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +configuration (all with the guarantee, 199/199) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +APS-154 +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +CHA-45 +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +both +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +base rule: tent, 3-point, projection \(c=2\) (§3) & 2634 & 2357 & +4991 \\ +constant-free variant: earned projection (§3.1) & 3377 & 2073 & 5450 \\ +(1) trust before the step, projection & 2663 & 2170 & 4833 \\ +(1)+(2) points chosen by shape, \emph{safe} & \textbf{2337} & 2177 & +4514 \\ +(1)+(3) far end by dissimilarity, projection & 2539 & \textbf{1864} & +\textbf{4403} \\ +(1)+(2)+(3), \emph{safe} & 2695 & 1928 & 4623 \\ +Chandrupatla & 2595 & 1505 & 4100 \\ +TOMS 748 & 2636 & 3088 & 5724 \\ +ITP, per-problem oracle & 3068 & 1675 & 4743 \\ +\end{longtable} + +Two results here are worth stating plainly. First, the canonical set +improves from 2634 to \textbf{2337}, which is below every method +measured, including Chandrupatla's 2595, with the worst-case guarantee +intact. It hardly feels the declared multiplier of §3.1: 2337 for every +multiplier from 1.5 to 4, 2351 at 1.25 +(\texttt{measure\_records\_vs\_multiplier.py}). Second, on the two sets +together the 4403 of reading (3) passes ITP's oracle (4743) but not +Chandrupatla (4100): the remaining gap is entirely on the wide-bracket +families, where Chandrupatla needs 1505 against our 1752 (reading (3) +under \emph{safe}). + +The readings do not compose. Taken together, (2) and (3) give 2695 and +1928 --- worse than either alone on the canonical set, and on the +wide-bracket set worse than (3) alone. Two selections that are each an +improvement interfere, which is a reason to measure composition rather +than assume it. + +\textbf{Which similarity stands in the step decides whether a reading +pays.} The readings above, and the decision \emph{defect} of §A.4, +compare quantities directly --- an ordinal judgment of which is more +similar, in which every decreasing \(S\) would decide alike. Their gains +nevertheless depend on the \(S\) that sets the step +(\texttt{audit\_similarity\_places.py}, on the same sweeps): on the +canonical set the triple chosen by shape gains 493 evaluations with the +tent in the step and loses 250 with \(1/\sqrt{1+z^2}\); \emph{defect} +gains 283 with the tent and loses 215 with \(1-|z|^{1/3}\). The records +of this section are the pairing of the tent step with these judgments, +not the judgments alone. + +Under the constant-free earned projection the same readings gain little +or lose. Reading (1) gives 3316 and 2050, (1)+(2) 3564 and 2074, (1)+(3) +3666 and 2052, all three together 4163 and 2087, against 3377 and 2073 +for the constant-free variant itself: the earned budget binds early on +the canonical problems, and what the readings buy under the projection +is mostly room that the earned budget does not grant. + +\hypertarget{a.3-the-similarity-function-belongs-to-the-place-not-to-the-method}{% +\subsubsection{A.3 The similarity function belongs to the place, not to +the +method}\label{a.3-the-similarity-function-belongs-to-the-place-not-to-the-method}} + +Everything above varies \emph{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 \emph{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 \emph{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 --- \(m=1/(1-ff''/f'^2)\), +Lagouanelle's estimate of the multiplicity {[}13{]} with the derivatives +replaced by divided differences, comes from three evaluations already +paid for, as in item 5 of Appendix C, and is 1 whenever the power model +does not apply. + +Measured over the grid, the correction behaves as that reading predicts +and not as a universal improvement. Paired inside each (function, +safeguard) cell of the far-end reading it is cheaper in 33 of 63 cells +--- indistinguishable from a coin. But in the one pairing where that +reading \emph{is} exact inverse interpolation, the ratio similarity, it +is cheaper under all seven safeguards: + +\textbf{Table A2.} 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 33 of 63 cells over all nine functions. + +\begin{longtable}[]{@{} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.2000}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +safeguard +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +APS, \(m=1\) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +APS, measured \(m\) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +CHA, \(m=1\) +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +CHA, measured \(m\) +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +both & 4331 & \textbf{3485} & 2469 & \textbf{2306} \\ +bracket & 4418 & \textbf{3352} & 2671 & \textbf{2428} \\ +credit & 4999 & \textbf{4495} & 2064 & \textbf{1933} \\ +projection & 6783 & \textbf{6176} & 4057 & \textbf{3751} \\ +safe & 5803 & \textbf{5040} & 3218 & \textbf{3032} \\ +step & 6251 & \textbf{5362} & 3371 & \textbf{3146} \\ +none & 10676 & \textbf{9855} & 8265 & \textbf{7424} \\ +\end{longtable} + +Two limits must be stated with it. The estimator is \emph{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 1933, against 1752 for +reading (3) above and 1505 for Chandrupatla. In the \emph{slope} reading +of §3 the same exponent is harmful, \(2634\to2703\), 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. + +\hypertarget{a.4-the-decision-place}{% +\subsubsection{A.4 The decision place}\label{a.4-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 \emph{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 \emph{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|){\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.3030}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.0909}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.3838}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.0909}} + >{\raggedright\arraybackslash}p{(\columnwidth - 8\tabcolsep) * \real{0.1313}}@{}} +\toprule\noalign{} +\begin{minipage}[b]{\linewidth}\raggedright +pre-registration +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +commit +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +results file +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +commit +\end{minipage} & \begin{minipage}[b]{\linewidth}\raggedright +order +\end{minipage} \\ +\midrule\noalign{} +\endhead +\bottomrule\noalign{} +\endlastfoot +\texttt{logarithmic\_2026-09-20} & \texttt{f9b062e} & +\texttt{sweep\_log.json} & \texttt{84ff1c7} & before \\ +\texttt{prestep\_gate\_2026-09-21} & \texttt{0617243} & +\texttt{sweep\_gate.json} & \texttt{a6116cd} & before \\ +\texttt{slope\_gate\_2026-09-21} & \texttt{e4769be} & +\texttt{sweep\_slope.json} & \texttt{e4769be} & same commit \\ +\texttt{points\_and\_direction\_2026-09-21} & \texttt{04ee5c8} & +\texttt{sweep\_points.json} & \texttt{22f19c5} & before \\ +\texttt{similarity\_per\_slot\_2026-09-21} & \texttt{15b9424} & +\texttt{sweep\_order.json} & \texttt{15b9424} & same commit \\ +\texttt{scale\_selection\_2026-09-21} & \texttt{cc2c499} & +\texttt{sweep\_cap.json} & \texttt{cc2c499} & same commit \\ +\texttt{decision\_place\_2026-09-21} & \texttt{2357379} & +\texttt{sweep\_decision.json} & \texttt{2357379} & same commit \\ +\texttt{step\_extrapolation\_2026-09-21} & \texttt{781bd15} & +\texttt{results\_diag\_step\_extrapolation.txt} & \texttt{3cce10b} & +before \\ +\texttt{extrapolate\_in\_method\_2026-09-21} & \texttt{a25cffa} & +\texttt{sweep\_extrap.json} & \texttt{99be33a} & before \\ +\texttt{extrapolation\_gates\_2026-09-21} & \texttt{29ab495} & +\texttt{sweep\_extrap2.json} & \texttt{68df1d2} & before \\ +\texttt{empty\_step\_2026-09-21} & \texttt{8279fb6} & +\texttt{results\_diag\_empty\_step.txt} & \texttt{d754689} & before \\ +\texttt{outcome\_memory\_2026-09-21} & \texttt{2b0ee22} & +\texttt{results\_diag\_outcome\_memory.txt} & \texttt{2498c66} & +before \\ +\texttt{flat\_regime\_2026-09-21} & \texttt{9df1aa5} & +\texttt{sweep\_flat.json} & \texttt{e7e74d7} & before \\ +\texttt{flat\_twice\_2026-09-21} & \texttt{6a8bb23} & +\texttt{sweep\_flat2.json} & \texttt{9884b21} & before \\ +\texttt{flat\_creep\_2026-09-21} & \texttt{35eb941} & +\texttt{sweep\_flatcreep.json} & \texttt{5a1edff} & before \\ +\texttt{flat\_stick\_2026-09-21} & \texttt{1192c68} & +\texttt{sweep\_flatstick.json} & \texttt{f8c17c4} & before \\ +\texttt{flat\_exit\_2026-09-21} & \texttt{f8dd8d9} & +\texttt{sweep\_flatexit.json} & \texttt{1842eb0} & before \\ +\texttt{flat\_judge\_2026-09-21} & \texttt{1dbda21} & +\texttt{sweep\_flatjudge.json} & \texttt{8bf595d} & before \\ +\texttt{flat\_analogy\_2026-09-21} & \texttt{42bfab4} & +\texttt{sweep\_flatanalog.json} & \texttt{1842eb0} & before \\ +\texttt{flat\_analogy\_mem\_2026-09-21} & \texttt{4bf8634} & \texttt{—} +& \texttt{—} & decided on the smoke run, no results file \\ +\texttt{defect\_contra\_flat\_2026-09-21} & \texttt{3c92d61} & +\texttt{sweep\_defcontra.json} & \texttt{63cf9fb} & before \\ +\texttt{flat\_order\_point\_2026-09-21} & \texttt{d1be3bb} & +\texttt{sweep\_orderpt.json} & \texttt{08863c1} & before \\ +\texttt{flat\_exp\_limit\_2026-09-21} & \texttt{e887691} & +\texttt{sweep\_explimit.json} & \texttt{baa9396} & before \\ +\texttt{half\_contra\_2026-09-21} & \texttt{3d74b45} & +\texttt{sweep\_half.json} & \texttt{bbb16ec} & before \\ +\texttt{out\_of\_sample\_2026-09-21} & \texttt{47361da} & +\texttt{sweep\_oos.json} & \texttt{a86a9ed} & before \\ +\texttt{roots\_jl\_2026-09-21} & \texttt{97af3ae} & +\texttt{sweep\_rootsjl.json} & \texttt{554ec0c} & before \\ +\texttt{applied\_2026-09-21} & \texttt{8ea89dd} & +\texttt{sweep\_applied.json} & \texttt{c2b0eea} & before \\ +\texttt{close\_sub\_2026-09-21} & \texttt{657df10} & +\texttt{sweep\_fourth\_fourth.json} & \texttt{6d2531e} & before \\ +\texttt{modab\_2026-09-21} & \texttt{bfa23b2} & +\texttt{sweep\_modab.json} & \texttt{37c81b0} & before \\ +\texttt{controls\_2026-09-22} & \texttt{77312be} & +\texttt{sweep\_controls.json} & \texttt{92c07d8} & before \\ +\end{longtable} \begin{center}\rule{0.5\linewidth}{0.5pt}\end{center} @@ -2444,6 +2457,18 @@ sensitivity would remove it. ``Improvements to the Modified Anderson--Björck (modAB) root-finding algorithm,'' \emph{Algorithms}, vol.~19, 332, 2026; package \texttt{pymodab} 1.0.9. +\item + J. C. P. Bus and T. J. Dekker, ``Two efficient algorithms with + guaranteed convergence for finding a zero of a function,'' \emph{ACM + Trans. Math. Softw.}, vol.~1, no. 4, pp.~330--345, 1975. +\item + J. M. Fernández-Díaz and C. O. Menéndez-Pérez, ``A common framework + for modified Regula Falsi methods and new methods of this kind,'' + \emph{Math. Comput. Simul.}, vol.~205, pp.~678--696, 2023. +\item + A. Cordero, B. Neta and J. R. Torregrosa, ``Memorizing Schröder's + method as an efficient strategy for estimating roots of unknown + multiplicity,'' \emph{Mathematics}, vol.~9, 2570, 2021. \end{enumerate} \end{document} diff --git a/check_new_numbers.py b/check_new_numbers.py new file mode 100644 index 0000000..d4c6095 --- /dev/null +++ b/check_new_numbers.py @@ -0,0 +1,30 @@ +"""After an editorial pass: every number in the new article that was not in the old one, with its +context - a transcription slip shows up here. Numbers only removed are fine (verify_article.py +checks the data, and a removed statement claims nothing). + + python -u check_new_numbers.py OLD.md NEW.md +""" +import re +import sys + +NUM = re.compile(r"(?10}: ...{new[max(0, m.start() - 60):m.end() + 30]!r}...") + break + print("new numbers done") + + +if __name__ == "__main__": + main(sys.argv[1], sys.argv[2]) diff --git a/check_prereg_order.py b/check_prereg_order.py new file mode 100644 index 0000000..25f3c7a --- /dev/null +++ b/check_prereg_order.py @@ -0,0 +1,83 @@ +"""Appendix D (revision point 4): for every pre-registration, the commit that first added it, the +results file its own run wrote, the commit that first added that file, and the order of the two +in git - "before" when the document is a strict ancestor of the results. The mapping document -> +own results file is declared below (read off the sweep script each run used); the commits and +the order are read from git. + + python -u check_prereg_order.py -> markdown rows and a summary on stdout +""" +from __future__ import annotations + +import subprocess + +OWN_RESULTS = { + "PREREG_logarithmic_2026-09-20.md": "sweep_log.json", + "PREREG_prestep_gate_2026-09-21.md": "sweep_gate.json", + "PREREG_slope_gate_2026-09-21.md": "sweep_slope.json", + "PREREG_points_and_direction_2026-09-21.md": "sweep_points.json", + "PREREG_similarity_per_slot_2026-09-21.md": "sweep_order.json", + "PREREG_scale_selection_2026-09-21.md": "sweep_cap.json", + "PREREG_decision_place_2026-09-21.md": "sweep_decision.json", + "PREREG_step_extrapolation_2026-09-21.md": "results_diag_step_extrapolation.txt", + "PREREG_extrapolate_in_method_2026-09-21.md": "sweep_extrap.json", + "PREREG_extrapolation_gates_2026-09-21.md": "sweep_extrap2.json", + "PREREG_empty_step_2026-09-21.md": "results_diag_empty_step.txt", + "PREREG_outcome_memory_2026-09-21.md": "results_diag_outcome_memory.txt", + "PREREG_flat_regime_2026-09-21.md": "sweep_flat.json", + "PREREG_flat_twice_2026-09-21.md": "sweep_flat2.json", + "PREREG_flat_creep_2026-09-21.md": "sweep_flatcreep.json", + "PREREG_flat_stick_2026-09-21.md": "sweep_flatstick.json", + "PREREG_flat_exit_2026-09-21.md": "sweep_flatexit.json", + "PREREG_flat_judge_2026-09-21.md": "sweep_flatjudge.json", + "PREREG_flat_analogy_2026-09-21.md": "sweep_flatanalog.json", + "PREREG_flat_analogy_mem_2026-09-21.md": None, # decided on the smoke (stated in it) + "PREREG_defect_contra_flat_2026-09-21.md": "sweep_defcontra.json", + "PREREG_flat_order_point_2026-09-21.md": "sweep_orderpt.json", + "PREREG_flat_exp_limit_2026-09-21.md": "sweep_explimit.json", + "PREREG_half_contra_2026-09-21.md": "sweep_half.json", + "PREREG_out_of_sample_2026-09-21.md": "sweep_oos.json", + "PREREG_roots_jl_2026-09-21.md": "sweep_rootsjl.json", + "PREREG_applied_2026-09-21.md": "sweep_applied.json", + "PREREG_close_sub_2026-09-21.md": "sweep_fourth_fourth.json", + "PREREG_modab_2026-09-21.md": "sweep_modab.json", + "PREREG_controls_2026-09-22.md": "sweep_controls.json", +} + + +def first_commit(path): + out = subprocess.run(["git", "log", "--diff-filter=A", "--format=%h", "--", path], + capture_output=True, text=True, check=False).stdout.split() + return out[-1] if out else None + + +def order(pre, res): + if pre == res: + return "same commit" + r = subprocess.run(["git", "merge-base", "--is-ancestor", pre, res], check=False).returncode + return "before" if r == 0 else "after" + + +def rows(): + out = [] + for doc, res in OWN_RESULTS.items(): + hp = first_commit(doc) + if res is None: + out.append((doc, hp, "—", "—", "decided on the smoke run, no results file")) + continue + hr = first_commit(res) + out.append((doc, hp, res, hr, order(hp, hr) if hr else "results not committed")) + return out + + +def main(): + rs = rows() + for doc, hp, res, hr, o in rs: + print(f"| `{doc[7:-3]}` | `{hp}` | `{res}` | `{hr}` | {o} |") + n = {k: sum(1 for r in rs if r[4] == k) for k in ("before", "same commit", "after")} + print(f"\n{len(rs)} pre-registrations: {n['before']} before their results, " + f"{n['same commit']} in the same commit, {n['after']} after") + print("prereg order done") + + +if __name__ == "__main__": + main() diff --git a/restructure_article.py b/restructure_article.py new file mode 100644 index 0000000..ed64505 --- /dev/null +++ b/restructure_article.py @@ -0,0 +1,266 @@ +"""Revision 1, step 3: the paper restructured and shortened. + +Main text: abstract (<= 250 words), introduction, sections 2-5 (the base rule), 6 (the flat regime, +with the full listing of A'), 7 (out-of-sample evidence and the controls), 8 (conclusion), code +and data availability. Appendices: A (where the evaluations go, where the similarity is read - +old sections 6 and 7), B (wall-clock - old 5.2), C (negative results - old 10), D (the +pre-registrations and their commits). Sections move VERBATIM; cross-references and table +numbers are remapped in one pass; revision history is removed; new text is added only where the +revision asked for it. Exact anchors, each found exactly once, or nothing is written. + + python -u restructure_article.py +""" +from __future__ import annotations + +import re + +PATH = "article/sim_numerics.md" + +# ------------------------------------------------------------------ 1. cross-references +SEC = {"5.3": "5.2", "7": "A.2", "7.1": "A.3", "7.2": "A.4", "7.3": "A.5", "7.4": "A.6", + "8": "6", "8.1": "6.1", "8.2": "6.2", "8.3": "6.3", "8.4": "6.4", "8.5": "6.5", "8.6": "6.6", + "9": "7", "9.1": "7.1", "10": "C", "11": "8"} +TAB = {"5": "A1", "6": "A2", "8": "5", "9": "6", "9a": "7", "9b": "8", "9c": "9"} + + +def remap(text): + def sec(m): + a, b = m.group(2), m.group(3) + na = SEC.get(a, a) + if na == "C": + assert b is None + return "Appendix C" + out = m.group(1) + na + if b: + out += "–" + SEC.get(b, b) + return out + text = re.sub(r"(§§?)(\d+(?:\.\d+)?)(?:–(\d+(?:\.\d+)?))?", sec, text) + + def secword(m): + na = SEC.get(m.group(1), m.group(1)) + return "Appendix C" if na == "C" else ("Appendix " + na if na.startswith("A") else "Section " + na) + text = re.sub(r"Section (\d+(?:\.\d+)?)(?!\d)(?!\.\d)", secword, text) + return re.sub(r"Table (9a|9b|9c|9|8|6|5)(?![0-9a-z])", lambda m: "Table " + TAB[m.group(1)], text) + + +# ------------------------------------------------------------------ 2. edits (new numbering) +EDITS = [] + + +def E(old, new): + EDITS.append((old, new)) + + +# revision history and pronouns (revision: "him" -> the method) +E(" An earlier version gave each competitor the better of the library's implementation and ours, problem by problem. That concession backfired in the competitor's favour: our Chandrupatla stopped on a bracket twice as wide as everybody else's — the factor this protocol corrects for ITP in item 1 and had not corrected there — and the per-problem minimum let the looser run win. With the tolerance corrected our implementation and the library agree on all 199 problems exactly.", + " Our own implementations of Chandrupatla's and Brent's methods agree with the libraries; the tables use the libraries.") +E(" For Brent the library is slightly dearer than our implementation; we use the library anyway, because it is the implementation a reader can check.", "") +E("Chandrupatla's own method and Dekker's are among the baselines; an earlier version of these tables was drawn up before either was added.", + "Chandrupatla's own method and Dekker's are among the baselines.") +E("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.", + "produced a variant that diverged from the original on 42 of the 199 problems and is not quoted. Nothing is subtracted: estimating a per-evaluation overhead by subtracting the measured cost of $f$ returns negative overheads for four of six methods.") +E("Chandrupatla does not win those families with his model — his traced run bisects", + "Chandrupatla's method does not win those families with its model — its traced run bisects") +E("against his own 297, 233, 233", "against its own 297, 233, 233") +E("Chandrupatla halves on up to all of his steps", "Chandrupatla's method halves on up to all of its steps") +# the constant, stated once and honestly +E("which §3.1 declares as the method's one free constant.", "which §3.1 declares as the base rule's one numerical constant.") +E("**The multiplier is the method's one free numerical constant.** It is not \"the factor already claimed in the bound\" — that argument is circular, the bound is twice bisection because two was chosen — and measurement settles it (`measure_budget_factor.py`, Linux):", + "**The multiplier is the base rule's one numerical constant**, and it is not neutral (`measure_budget_factor.py`, Linux):") +E("one declared constant and the better canonical result, or no constant, a tighter bound and the better wide-bracket result.", + "one declared numerical constant and the better canonical result, or none, a tighter bound and the better wide-bracket result.") +# citations found missing on revision +E("Each of those answers carries constants: Illinois halves the stale ordinate,", + "Each of those answers carries constants: Illinois [7] halves the stale ordinate,") +E("| 15 | Ridders | no |", "| 15 | Ridders [5] | no |") +E("| 237 | Illinois | yes |", "| 237 | Illinois [7] | yes |") +E("where Illinois, at the price of one constant", "where Illinois [7], at the price of one constant") +E("Two remarks apply to both forms.\n", + "Three remarks apply to both forms.\n\n- **Guaranteed bounds of this kind are classical.** Bus and Dekker [21] gave bracketing methods whose " + "worst case is four to five times bisection; the scheduled projection guarantees twice bisection plus one " + "step, the earned one bisection plus one step.") +# the decision layer, counted honestly (revision point 2), and the full listing (point 4) +LISTING = r""" + +Counting numerical constants does not measure the method's freedom, however. Beyond these numbers A′ is a decision layer of about ten discrete rules — the flag, the analogy and its memory, the entry, the exit, the overshoot branch, the split, the step pattern, the half, the contradiction, the tolerance step — chosen on the two development sets among the many candidates of Appendix C. That selection is why the method was tested once on a corpus fixed before its code was frozen, and against simpler rules put in its place (§7). Algorithm 2 is the complete method; `simroot/pure.py` and the compiled `simbracket` implement exactly this and agree bit for bit. + +**Algorithm 2 (A′).** Notation of Algorithm 1; $S(z)=\max(0,1-|z|)$. + +``` +state: bracket [A,B], FA, FB; points x, xp, xp2 and their values fx, fp, fp2 + memory: configs (list of configurations), moves (list of displacements) + flags: last_model, in_regime, analogy_seen (order m_a), shallow_seen (order m_s) +initialise as Algorithm 1; allow <- 2^n_max * xtol/2 +while |B - A| > xtol: + far <- end of [A,B] further from x; mid <- (A+B)/2; model <- false + f' <- derivative at x of the parabola through (x, xp, xp2) + sigma <- |f'| * |far - x| + if sigma is 0 or not finite: xn <- mid + else: + z <- |fx| / sigma + # the flat regime (6.1-6.3) + c <- (xi, phi, |x - xp|, |fx - fp|) of the last three points (if defined) + flag <- not in_regime and last_model and fx/fp >= 1/4 unkept promise + flag_over <- not in_regime and last_model and fx/fp <= -1/4 overshoot + m <- order (6) of the analogous pair: the past configuration nearest to c + (Euclidean in (xi, phi)), if it is nearer than the line phi = xi; + m = inf when that pair shows L_f < 1 <= L_x (translation); else none + if m >= 2: analogy_seen <- true; m_a <- m + if 0 < m <= 2/3: shallow_seen <- true; m_s <- m + if flag and analogy_seen: in_regime <- true + elif flag_over and shallow_seen: in_regime <- true; m_a <- m_s + append c to configs + if in_regime: xn <- split point (7) with m = m_a + elif x != xp and sign((fx-fp)/(x-xp)) != sign((FB-FA)/(B-A)): xn <- mid + elif S(z) >= 1/2 and |f(far)| < |fx|: xn <- mid + else: + step <- (1 - S(z)) * (far - x) eq. (4) + append x - xp to moves + if the ratios of successive move lengths repeat (the lag 2..half the history + of highest mean ratio similarity has mean > 1/2), and step and x - xp + point the same way: + s <- min(|step|,|x-xp|) / max(|step|,|x-xp|) + step <- step / (1 - s), at most |far - x| long + if |step| < xtol/2 and last_model and |fx/fp| < 1/4: + step <- sign(far - x) * xtol/2 tolerance step + xn <- x + step; model <- true + xn <- clamp(xn, hi - allow, lo + allow) projection (3) + if xn outside (A,B) or xn in {x, xp}: xn <- mid; if that too is in {x, xp}: stop + last_model <- model and xn is the model's own point + fn <- f(xn) + if in_regime: exit + t <- (xn - A)/(B - A) + if |fn - (FA(1-t) + FB t)| < |fn|: in_regime <- false; analogy_seen <- false + allow <- allow/2; (xp2, xp, x) <- (xp, x, xn); keep the sign change in [A,B] +return midpoint(A,B) +```""" +E("it is verified with no violation on every corpus of this paper and under every safeguard of the grid.", + "it is verified with no violation on every corpus of this paper and under every safeguard of the grid." + LISTING) +# related work named on revision, found and read +E("| the split (7) | false position on $\\operatorname{sign}(f)|f|^{1/m}$ [16, 17] |", + "| the split (7) | false position on $\\operatorname{sign}(f)|f|^{1/m}$ [16, 17]; inside a bracket, with the multiplicity given, the SFRFm method of [22] |\n" + "| the dimensionless configuration $(\\xi,\\varphi)$ of §6.2 | the ratios of Chandrupatla's acceptance band [3]; every scaled regula falsi written in two dimensionless parameters [22] |") +E("The methods for multiple roots we found are single-point iterations without a bracket, with the multiplicity assumed, estimated from derivatives, or removed by a transformation ([17];", + "The methods for multiple roots we found are single-point iterations without a bracket, with the multiplicity assumed, estimated from derivatives, or removed by a transformation ([17, 23];") +E("Chandrupatla takes the decision to halve by an empirical acceptance band [3].", + "inside a bracket, [22] needs the multiplicity given and recognises a multiple root only by the number of evaluations spent, against a third of bisection's; Chandrupatla takes the decision to halve by an empirical acceptance band [3].") +# the title +E("# Root Finding by Similarity and Analogy: A Bracketing Method with One Declared Constant", + "# Root Finding by Similarity and Analogy: A Bracketing Method with a Data-Selected Decision Layer") + +ABSTRACT = """Bisection places its next iterate at the midpoint because it uses nothing about $f$. We replace that rule by a similarity rule — the step grows with how dissimilar $f(x)$ is to the target 0, measured by a similarity $S$ with $S(0)=1$ — and derive its contraction factor in closed form. The derivation fixes a base rule before any benchmark is consulted: with the tent similarity and a three-point divided-difference scale it is a Newton step of Muller's order 1.839, clipped to the bracket and projected onto a budget of twice bisection whose multiplier is its one numerical constant. On the 154 problems of Alefeld, Potra and Shi it is level with TOMS 748 and 1.5% behind Chandrupatla's method; on wide brackets it is clearly behind. The gap is a decision, when to give the model up, and we take it by analogy: at a flat root the dimensionless configuration of the last three points recurs under zoom, at a simple root it tends to a line, and the analogous pair gives the order that says whether and where to split the bracket. This decision layer, about ten discrete rules selected on the two development sets, gives the method A′. On 240 problems fixed before its code was frozen it needs 3656 evaluations, against 5717 for Chandrupatla, 5535 for ModAB and 4488 for A′ with Lagouanelle's multiplicity estimate in place of the analogy. The lead lies in multiple and fractional roots, which two of that corpus's six families contain, and is small on simple roots. With the multiplicity known in advance, the classical transformation $\\operatorname{sign}(f)|f|^{1/m}$ is 2.4 to 6.8 times cheaper on multiple roots.""" + +INTRO = """## 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], an Anderson–Björck step behind a linearity check in ModAB [10, 20]. Each of those answers carries constants: Illinois [7] 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 paper makes four claims, each measured. + +1. A contraction law for the similarity step (§2) fixes a *base rule* — tent similarity, three-point scale, a projection safeguard — before any benchmark is consulted (§3); its order is Muller's. +2. Projecting the step onto the bisection budget, instead of replacing it, keeps the order and a worst case of twice bisection plus one step; the budget's multiplier is the base rule's one numerical constant, and a constant-free variant costs a measured price (§3.1, §5). +3. The base rule's gap on wide brackets is a decision — when to give the model up — and a similarity of the run to its own past takes it (§6). The resulting method, A′, is tested once on a corpus fixed before its code was frozen, against simpler detectors put in its place, and against the classical transformation with the multiplicity known in advance (§7). +4. Every table uses one criterion, a sign change within $10^{-6}$ of the answer, because the benchmarks' published roots score six problems unsolved for every method (§4, §A.5). + +The search that produced these results — readings of the similarity that did not pay, the decomposition of the wide-bracket gap, wall-clock time, and the rejected rules — is reported in Appendices A–C. The code, the corpora and the pre-registrations are released (Appendix D and the availability statement).""" + +AVAIL = """## Code and data availability + +Everything behind this paper is released with it: the reference implementation of the base rule and of A′ (`simroot/pure.py`); `simbracket`, a compiled library of A′ (Cython; wheels for Linux and Windows, Python ≥ 3.10) whose results equal the reference bit for bit on all 868 problems of the six corpora; the corpora as code (`corpus_*.py`); every sweep script with its results file; the pre-registrations with the commits of Appendix D; and `verify_article.py`, which recomputes or re-reads every number stated in this paper and is part of the test suite. All numbers come from one Linux machine (§4). Repository: ⟨repository URL⟩.""" + +APPENDIX_D_HEAD = """## Appendix D. Pre-registrations and the frozen code + +Every rule of §6 and every test of §7 was written down, with its criteria and stop rule, before its code and its run. Table D1 gives, for each document, the commit that first added it and the commit that first added the results its run wrote, both read from git by `check_prereg_order.py`: 25 of the 30 were committed before their results; four were committed together with them, so git cannot order them; one was decided on its smoke run and has no results file. The code of A′ was committed (`8027d7f`) after corpus 4 was defined (`657df10`) and before its results (`6d2531e`); `simroot/pure.py` did not change in between. + +**Table D1.** Pre-registrations, in the order they were written. + +| pre-registration | commit | results file | commit | order | +|---|---|---|---|---| +""" + +REFS = """21. J. C. P. Bus and T. J. Dekker, "Two efficient algorithms with guaranteed convergence for finding a zero of a function," *ACM Trans. Math. Softw.*, vol. 1, no. 4, pp. 330–345, 1975. +22. J. M. Fernández-Díaz and C. O. Menéndez-Pérez, "A common framework for modified Regula Falsi methods and new methods of this kind," *Math. Comput. Simul.*, vol. 205, pp. 678–696, 2023. +23. A. Cordero, B. Neta and J. R. Torregrosa, "Memorizing Schröder's method as an efficient strategy for estimating roots of unknown multiplicity," *Mathematics*, vol. 9, 2570, 2021.""" + + +def split_sections(text): + """[(heading line or '' for the preamble, body)] in document order.""" + parts, cur_head, cur = [], "", [] + for ln in text.split("\n"): + if ln.startswith("## ") or ln.startswith("### "): + parts.append((cur_head, "\n".join(cur))) + cur_head, cur = ln, [] + else: + cur.append(ln) + parts.append((cur_head, "\n".join(cur))) + return parts + + +def clean(body): + lines = body.strip("\n").split("\n") + while lines and lines[-1].strip() in ("", "---"): + lines.pop() + while lines and lines[0].strip() in ("", "---"): + lines.pop(0) + return "\n".join(lines) + + +def main(): + src = open(PATH, encoding="utf8").read() + src = remap(src) + bad = [o[:80] for o, _ in EDITS if src.count(o) != 1] + if bad: + for b in bad: + print("ANCHOR NOT FOUND EXACTLY ONCE:", repr(b)) + raise SystemExit(f"{len(bad)} anchors failed; nothing written") + for o, n in EDITS: + src = src.replace(o, n) + # Table 1: caption, table, then the text about it + cap = "$c_2$ the coefficient of $z^2$ when $c=0$.\n\n" + t0 = "| $S(z)$ | $c$ | predicted | measured |" + t1 = "\n\nThe last row is not a slow rate" + i, j, k = src.index(cap) + len(cap), src.index(t0), src.index(t1) + assert i < j < k and src.count(cap) == src.count(t0) == src.count(t1) == 1 + src = src[:i] + src[j:k] + "\n\n" + src[i:j].rstrip() + src[k:] + + secs = split_sections(src) + by = {} + for h, b in secs: + key = re.match(r"#+ ([^ ]+)", h).group(1) if h else "PRE" + assert key not in by, key + by[key] = (h, clean(b)) + + def S(key, heading=None): + h, b = by[key] + return (heading or h) + "\n\n" + b + + pre = by["PRE"][1] + out = [pre, "## Abstract\n\n" + ABSTRACT, "---", INTRO, "---", + S("2."), S("2.1"), S("2.2"), "---", S("3."), S("3.1"), "---", S("4."), "---", + S("5."), S("5.1"), S("5.3", "### 5.2 The law across the whole grid"), "---", + S("8.", "## 6. The flat regime, recognised by analogy")] + for n in range(1, 7): + h = by[f"8.{n}"][0].replace(f"### 8.{n} ", f"### 6.{n} ") + out.append(S(f"8.{n}", h)) + out += ["---", S("9.", "## 7. Out-of-sample evidence"), + S("9.1", by["9.1"][0].replace("### 9.1 ", "### 7.1 ")), "---", + S("11.", "## 8. Conclusion"), S("Future"), "---", AVAIL, "---"] + out += ["## Appendix A. Where the evaluations go, and where the similarity is read", + S("6.", "### A.1 Where the evaluations go"), + S("7.", "### A.2 Where the similarity is read")] + for n, a in ((1, 3), (2, 4), (3, 5), (4, 6)): + out.append(S(f"7.{n}", by[f"7.{n}"][0].replace(f"### 7.{n} ", f"### A.{a} "))) + out += ["---", S("5.2", "## Appendix B. Wall-clock time, and why it is not the cost model"), "---", + S("10.", "## Appendix C. Negative results"), "---"] + rows = [ln for ln in open("results_prereg_order.txt", encoding="utf8").read().split("\n") + if ln.startswith("| `")] + assert len(rows) == 30 + out += [APPENDIX_D_HEAD + "\n".join(rows), "---", S("References") + "\n" + REFS] + new = "\n\n".join(x.strip("\n") for x in out) + "\n" + open(PATH, "w", encoding="utf8").write(new) + print(f"written: {len(new.split())} words") + + +if __name__ == "__main__": + main() diff --git a/results_prereg_order.txt b/results_prereg_order.txt new file mode 100644 index 0000000..0f0e460 --- /dev/null +++ b/results_prereg_order.txt @@ -0,0 +1,33 @@ +| `logarithmic_2026-09-20` | `f9b062e` | `sweep_log.json` | `84ff1c7` | before | +| `prestep_gate_2026-09-21` | `0617243` | `sweep_gate.json` | `a6116cd` | before | +| `slope_gate_2026-09-21` | `e4769be` | `sweep_slope.json` | `e4769be` | same commit | +| `points_and_direction_2026-09-21` | `04ee5c8` | `sweep_points.json` | `22f19c5` | before | +| `similarity_per_slot_2026-09-21` | `15b9424` | `sweep_order.json` | `15b9424` | same commit | +| `scale_selection_2026-09-21` | `cc2c499` | `sweep_cap.json` | `cc2c499` | same commit | +| `decision_place_2026-09-21` | `2357379` | `sweep_decision.json` | `2357379` | same commit | +| `step_extrapolation_2026-09-21` | `781bd15` | `results_diag_step_extrapolation.txt` | `3cce10b` | before | +| `extrapolate_in_method_2026-09-21` | `a25cffa` | `sweep_extrap.json` | `99be33a` | before | +| `extrapolation_gates_2026-09-21` | `29ab495` | `sweep_extrap2.json` | `68df1d2` | before | +| `empty_step_2026-09-21` | `8279fb6` | `results_diag_empty_step.txt` | `d754689` | before | +| `outcome_memory_2026-09-21` | `2b0ee22` | `results_diag_outcome_memory.txt` | `2498c66` | before | +| `flat_regime_2026-09-21` | `9df1aa5` | `sweep_flat.json` | `e7e74d7` | before | +| `flat_twice_2026-09-21` | `6a8bb23` | `sweep_flat2.json` | `9884b21` | before | +| `flat_creep_2026-09-21` | `35eb941` | `sweep_flatcreep.json` | `5a1edff` | before | +| `flat_stick_2026-09-21` | `1192c68` | `sweep_flatstick.json` | `f8c17c4` | before | +| `flat_exit_2026-09-21` | `f8dd8d9` | `sweep_flatexit.json` | `1842eb0` | before | +| `flat_judge_2026-09-21` | `1dbda21` | `sweep_flatjudge.json` | `8bf595d` | before | +| `flat_analogy_2026-09-21` | `42bfab4` | `sweep_flatanalog.json` | `1842eb0` | before | +| `flat_analogy_mem_2026-09-21` | `4bf8634` | `—` | `—` | decided on the smoke run, no results file | +| `defect_contra_flat_2026-09-21` | `3c92d61` | `sweep_defcontra.json` | `63cf9fb` | before | +| `flat_order_point_2026-09-21` | `d1be3bb` | `sweep_orderpt.json` | `08863c1` | before | +| `flat_exp_limit_2026-09-21` | `e887691` | `sweep_explimit.json` | `baa9396` | before | +| `half_contra_2026-09-21` | `3d74b45` | `sweep_half.json` | `bbb16ec` | before | +| `out_of_sample_2026-09-21` | `47361da` | `sweep_oos.json` | `a86a9ed` | before | +| `roots_jl_2026-09-21` | `97af3ae` | `sweep_rootsjl.json` | `554ec0c` | before | +| `applied_2026-09-21` | `8ea89dd` | `sweep_applied.json` | `c2b0eea` | before | +| `close_sub_2026-09-21` | `657df10` | `sweep_fourth_fourth.json` | `6d2531e` | before | +| `modab_2026-09-21` | `bfa23b2` | `sweep_modab.json` | `37c81b0` | before | +| `controls_2026-09-22` | `77312be` | `sweep_controls.json` | `92c07d8` | before | + +30 pre-registrations: 25 before their results, 4 in the same commit, 0 after +prereg order done diff --git a/verify_article.py b/verify_article.py index cbd60c6..7c1d342 100644 --- a/verify_article.py +++ b/verify_article.py @@ -946,6 +946,13 @@ def verify_revision1(): check("base rule vs Chandrupatla by family, APS", families(by[PROJ], ch, "APS"), (6, 1, 8)) check("A' vs Chandrupatla by family, APS", families(ap, ch, "APS"), (8, 1, 6)) check("A' vs Chandrupatla by family, CHA", families(ap, ch, "CHA"), (7, 0, 2)) + # revision: every head-to-head row of the paper must sum to the size of its set + rivals = ("chandrupatla", "toms748", "brent", "itp_oracle", "itp", "bisection") + sums = {(who, s): sum(head_to_head(rows + [ap], who, opp, s)[:3]) + for who in (PROJ, ap["label"]) for opp in rivals for s in ("APS", "CHA")} + check("every head-to-head row sums to its set (154 / 45)", + {v for (w, s), v in sums.items() if s == "APS"} | {v + 1000 for (w, s), v in sums.items() + if s == "CHA"}, {154, 1045}) check("family-level p, APS and CHA (2 digits)", (float(f"{sign_p(8, 6):.2g}"), float(f"{sign_p(7, 2):.2g}")), (0.79, 0.18))