vvsandClaude Opus 5
1b6c626f67
(2) The proposed method is again the scheduled projection (2806/3220/6026), its budget multiplier DECLARED as the one free constant: the canonical total is the same for every multiplier 1.25-4, the wide-bracket one is not. The earned projection is the constant-free variant (3568/3020/6588). What the old tables hid: problem by problem the proposed rule beats Chandrupatla on the canonical set 87/24/42, exact sign test p = 9e-5, while trailing him by 0.7% in total; on the wide-bracket set he is genuinely stronger (26 to 9, p = 0.006). Gap decomposition recomputed with the library: 244 (was 262), 160 of it on the flat families (two thirds), fun1 58 (a quarter, not a third as the abstract said). Also corrected on the way: the 'subsumed by Chandrupatla' pin claimed he is better on the worst single problem - true only with the looser tolerance (84 against the credit rule's 83 once fixed); §8.3 still called zero slack degenerate, which Table 1a refutes; §5.2's Brent figure is our pure-Python implementation and now says so. verify_article.py 286/286, 802 tests pass twice. Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
278 lines
11 KiB
Python
278 lines
11 KiB
Python
"""Stage 16 - similarity used to IMPROVE an existing method, not to replace it.
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Everything that worked in this project worked that way. The credit safeguard
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did not produce a new root finder that beats the field; it took regula falsi,
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the secant rule and inverse quadratic interpolation and made each of them
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solve 193 of 199 with a near-bisection worst case. Building a method from
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similarity landed second behind Chandrupatla 1997; grafting a measured
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quantity onto existing methods was the part that paid.
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This stage tries the one graft that comes straight from the similarity
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literature and has not been tried here: **similarity replaces a hard
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partition by a graded weight**.
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Every strong bracketing method contains such a partition:
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Brent three empirical inequalities decide interpolation OR bisection
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Chandrupatla the criterion 1-sqrt(1-xi) < phi < sqrt(xi) decides the same
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In both, the quantity being tested is continuous and the decision taken on it
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is a step function. The similarity move is to keep the quantity and drop the
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step:
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r = (how far outside the safe region the interpolation falls), 0 = safe
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w = S(r), S the tent similarity, S(0) = 1, no constant
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d = w * d_interpolation + (1 - w) * d_bisection
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At r = 0 this is exactly the original method. At r >= 1 it is exactly
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bisection. In between, where the original flips discontinuously, it blends.
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The comparison is against the SAME implementation with the blend switched
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off, so the difference can only be the blend. Our Brent is separately
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validated against scipy.optimize.brentq to 0.2% of evaluations.
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"""
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from __future__ import annotations
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import math
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from scipy.optimize._tstutils import (_APS_TESTS_DICTS,
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_CHANDRUPATLA_TESTS_DICTS)
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from simroot.baselines import Counter, _Budget, brent, chandrupatla
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XTOL, MAXFEV, ACC = 1e-12, 200, 1e-6
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# the tent similarity of the paper: S(0) = 1, S(1) = 0, no constant
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TENT = lambda r: max(0.0, 1.0 - r)
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def brent_blend(f, a, b, xtol, maxfev, mode="blend"):
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"""Brent's method with its accept/reject replaced by a similarity blend.
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The only change from simroot.baselines.brent is the four lines marked
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below; with mode="off" the two are the same algorithm. mode is a name, not a
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boolean carrying three states."""
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g = Counter(f, maxfev)
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b_ = b
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try:
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fa, fb = g(a), g(b)
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if fa == 0.0:
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return a, g.n, True
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if fb == 0.0:
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return b, g.n, True
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c, fc = a, fa
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d = e = b - a
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while True:
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if fb * fc > 0.0:
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c, fc = a, fa
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d = e = b - a
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if abs(fc) < abs(fb):
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a, fa = b, fb
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b, fb = c, fc
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c, fc = a, fa
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tol1 = 2.0 * 2.22e-16 * abs(b) + 0.5 * xtol
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xm = 0.5 * (c - b)
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if abs(xm) <= tol1 or fb == 0.0:
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return b, g.n, True
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if abs(e) >= tol1 and abs(fa) > abs(fb):
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s = fb / fa
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if a == c:
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p = 2.0 * xm * s
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q = 1.0 - s
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else:
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q = fa / fc
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r = fb / fc
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p = s * (2.0 * xm * q * (q - r) - (b - a) * (r - 1.0))
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q = (q - 1.0) * (r - 1.0) * (s - 1.0)
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if p > 0.0:
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q = -q
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p = abs(p)
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bound = min(3.0 * xm * q - abs(tol1 * q), abs(e * q))
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if mode == "project":
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# The lesson from the root-finding safeguard, applied to
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# somebody else's method: when the interpolation violates
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# the safety condition, do not discard it and do not
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# average it with the midpoint - keep as much of it as the
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# condition allows, by clamping it to the boundary.
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if 2.0 * p < bound:
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e, d = d, p / q
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elif bound > 0.0 and q != 0.0:
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e, d = d, math.copysign(0.5 * bound / abs(q),
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p / q if q else xm)
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else:
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d = e = xm
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elif mode == "off":
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if 2.0 * p < bound:
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e, d = d, p / q
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else:
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d = e = xm
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else:
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# ---- the graft -------------------------------------
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# Brent's own test is 2p < bound. The ratio of the two
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# is continuous; the decision on it is a step. Keep the
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# ratio, weight by similarity, blend the two candidate
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# steps. r = 0 reproduces Brent exactly, r >= 1 gives
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# bisection exactly.
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r_trust = (2.0 * p / bound) if bound > 0.0 else math.inf
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w = TENT(r_trust) if math.isfinite(r_trust) else 0.0
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d_int = p / q if q != 0.0 else xm
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e, d = d, w * d_int + (1.0 - w) * xm
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# ----------------------------------------------------
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else:
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d = e = xm
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a, fa = b, fb
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b = b + (d if abs(d) > tol1 else math.copysign(tol1, xm))
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b_ = b
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fb = g(b)
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except _Budget:
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return b_, g.n, False
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def chandrupatla_blend(f, a, b, xtol, maxfev, mode="blend"):
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"""Chandrupatla's method with its acceptance region softened the same way.
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Its test is 1 - sqrt(1-xi) < phi < sqrt(xi). The distance outside that
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interval, scaled by its own width, is the continuous quantity; the
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decision on it is a step."""
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g = Counter(f, maxfev)
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x1, x2 = b, a
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f1 = f2 = 0.0
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try:
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f1, f2 = g(x1), g(x2)
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if f1 == 0.0:
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return x1, g.n, True
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if f2 == 0.0:
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return x2, g.n, True
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x3, f3 = x2, f2
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t = 0.5
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while True:
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x = x1 + t * (x2 - x1)
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fx = g(x)
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if (fx > 0.0) == (f1 > 0.0):
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x3, f3 = x1, f1
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x1, f1 = x, fx
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else:
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x3, f3 = x2, f2
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x2, f2 = x1, f1
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x1, f1 = x, fx
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xm, fm = (x1, f1) if abs(f1) < abs(f2) else (x2, f2)
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if fm == 0.0:
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return xm, g.n, True
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tol = 2.0 * 2.22e-16 * abs(xm) + 0.5 * xtol # 2026-09-21 fix, see baselines.chandrupatla
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span = abs(x2 - x1)
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if span == 0.0:
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return xm, g.n, True
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tl = tol / span
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if tl > 0.5:
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return xm, g.n, True
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t = 0.5
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if x3 != x2 and f3 != f2:
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xi = (x1 - x2) / (x3 - x2)
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ph = (f1 - f2) / (f3 - f2)
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if 0.0 < xi < 1.0:
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lo = 1.0 - math.sqrt(1.0 - xi)
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hi = math.sqrt(xi)
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d1, d2, d3 = f1 - f2, f1 - f3, f2 - f3
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inside = lo < ph < hi
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if d1 != 0.0 and d2 != 0.0 and d3 != 0.0:
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t_iqi = ((f1 / d1) * (f3 / (-d3))
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+ ((x3 - x1) / (x2 - x1)) * (f1 / d2)
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* (f2 / d3))
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if mode == "off":
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if inside:
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t = t_iqi
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else:
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# distance outside the safe interval, in units of
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# the interval's own width
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width = hi - lo
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if width > 0.0:
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out = max(0.0, lo - ph, ph - hi) / width
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w = TENT(out)
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t = w * t_iqi + (1.0 - w) * 0.5
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elif inside:
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t = t_iqi
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if not math.isfinite(t):
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t = 0.5
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t = min(max(t, tl), 1.0 - tl)
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except _Budget:
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return (x1 if abs(f1) < abs(f2) else x2), g.n, False
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def load():
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out = []
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for d in _APS_TESTS_DICTS:
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a, b = d["bracket"]
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args = d.get("args", ())
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out.append(("APS", (lambda ff, aa: lambda t: ff(t, *aa))(d["f"], args),
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float(a), float(b), float(d["root"])))
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for d in _CHANDRUPATLA_TESTS_DICTS:
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a, b = d["bracket"]
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out.append(("CHA", (lambda ff: lambda t: ff(t))(d["f"]),
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float(a), float(b), float(d["root"])))
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return out
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def measure(method, probs):
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t = {"APS": 0, "CHA": 0}
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ok = {"APS": 0, "CHA": 0}
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worst = 0
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for s, f, a, b, root in probs:
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try:
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x, n, conv = method(f, a, b, XTOL, MAXFEV)
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except Exception:
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x, n, conv = float("nan"), MAXFEV, False
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good = bool(conv and math.isfinite(x) and abs(x - root) <= ACC)
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t[s] += n if good else MAXFEV
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ok[s] += good
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if good:
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worst = max(worst, n)
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return t, ok, worst
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def main():
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probs = load()
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rows = [
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("brent, as published", lambda f, a, b, xt, mx: brent(f, a, b, xt, mx)),
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("brent, blend off (same code)",
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lambda f, a, b, xt, mx: brent_blend(f, a, b, xt, mx, mode="off")),
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("brent + similarity blend",
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lambda f, a, b, xt, mx: brent_blend(f, a, b, xt, mx, mode="blend")),
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("brent + projection onto the bound",
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lambda f, a, b, xt, mx: brent_blend(f, a, b, xt, mx,
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mode="project")),
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("chandrupatla, as published",
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lambda f, a, b, xt, mx: chandrupatla(f, a, b, xt, mx)),
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("chandrupatla, blend off",
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lambda f, a, b, xt, mx: chandrupatla_blend(f, a, b, xt, mx,
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mode="off")),
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("chandrupatla + similarity blend",
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lambda f, a, b, xt, mx: chandrupatla_blend(f, a, b, xt, mx,
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mode="blend")),
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]
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print(f"{'method':34s}{'APS':>8s}{'CHA':>8s}{'both':>8s}"
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f"{'solved':>10s}{'worst':>8s}")
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print("-" * 78)
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base = {}
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for name, m in rows:
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t, ok, worst = measure(m, probs)
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both = t["APS"] + t["CHA"]
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base[name] = both
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print(f"{name:34s}{t['APS']:>8d}{t['CHA']:>8d}{both:>8d}"
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f"{ok['APS'] + ok['CHA']:>7d}/199{worst:>8d}", flush=True)
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print("\nThe two 'blend off' rows must reproduce the published methods -"
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"\nif they do not, the blend is being compared against a different"
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"\nalgorithm and the experiment is void.")
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for pub, off, on in (("brent, as published", "brent, blend off (same code)",
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"brent + similarity blend"),
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("chandrupatla, as published", "chandrupatla, blend off",
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"chandrupatla + similarity blend")):
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drift = abs(base[off] - base[pub]) / base[pub]
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gain = (base[pub] - base[on]) / base[pub]
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print(f"\n {pub.split(',')[0]:14s} control drift {drift * 100:5.2f}%"
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f" blend changes the total by {gain * 100:+6.2f}%")
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if __name__ == "__main__":
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main()
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