178 lines
6.9 KiB
Python
178 lines
6.9 KiB
Python
"""Pins the corrections the revision made.
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Each of these replaced a statement in the paper that was wrong or too broad.
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They are asserted here so a later edit cannot quietly restore the old claim.
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"""
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from __future__ import annotations
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import math
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import os
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import sys
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import pytest
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sys.path.insert(0, os.path.dirname(os.path.dirname(os.path.abspath(__file__))))
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from simroot.pure import PURE_SIMS, pure_root
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from stage5_three_point import PROBLEMS, order_of
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R2 = math.sqrt(2.0)
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F = lambda x: x * x - 2.0
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XTOL, MAXFEV = 1e-12, 200
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def iterate(S, n, sigma=2.0, x0=0.0, xfar=2.0):
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x, errs = x0, []
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for _ in range(n):
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x = x + (1.0 - S(F(x) / sigma)) * (xfar - x)
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errs.append(abs(x - R2))
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return errs
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# --------------------------------------------------------------------------
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# the naming contradiction: the proposed similarity is the tent, and its
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# degeneracy is a property of the CHOSEN scale only
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# --------------------------------------------------------------------------
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def test_the_tent_is_captured_under_the_chosen_scale():
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"""Table 1's last row. Not a slow rate - capture at the far end on the
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first step."""
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e = iterate(PURE_SIMS["ratio"], 200)
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assert abs(e[0] - e[-1]) < 1e-15, (e[0], e[-1])
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assert abs(e[0] - 0.5858) < 1e-3, e[0]
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def test_the_tent_is_the_best_similarity_under_the_derived_scale():
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"""The same function, the other scale. If these two tests ever agree in
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sign, the paper's central thesis - that the content is in the scale - has
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been broken."""
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tot = {}
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for sim in PURE_SIMS:
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s = 0
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for _n, f, a, b, root in PROBLEMS:
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x, n, conv = pure_root(f, a, b, XTOL, MAXFEV, similarity=sim,
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scale="derived3", guard="project")
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s += n if (conv and abs(x - root) <= 1e-8) else MAXFEV
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tot[sim] = s
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assert min(tot, key=lambda k: tot[k]) == "ratio", tot
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def test_the_tent_never_clips_under_the_derived_scale():
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"""Why the degeneracy disappears: |f|/sigma stays below 1, so the
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truncation in max(0, 1-|z|) never engages."""
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from simroot.pure import _slope3
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a, b = 0.0, 2.0
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fa, fb = F(a), F(b)
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A, FA, B = a, fa, b
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x, fx, xp, fp = a, fa, b, fb
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xp2 = fp2 = None
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clipped = steps = 0
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for _ in range(40):
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if abs(B - A) <= XTOL:
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break
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far = B if abs(B - x) >= abs(A - x) else A
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sc = abs(_slope3(x, fx, xp, fp, xp2, fp2)) * abs(far - x)
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if sc == 0.0:
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break
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steps += 1
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clipped += abs(fx) / sc >= 1.0
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xn = x + min(1.0, abs(fx) / sc) * (far - x)
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if not (min(A, B) < xn < max(A, B)):
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xn = 0.5 * (A + B)
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fn = F(xn)
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xp2, fp2, xp, fp, x, fx = xp, fp, x, fx, xn, fn
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if FA * fn < 0.0:
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B = xn
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else:
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A, FA = xn, fn
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assert steps > 0
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assert clipped == 0, f"{clipped} of {steps} steps clipped"
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# --------------------------------------------------------------------------
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# Proposition 1 is specific; Proposition 1' locates the boundary
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# --------------------------------------------------------------------------
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def test_only_the_ratio_form_reproduces_regula_falsi():
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a, b = 1.0, 4.0
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fa, fb = F(a), F(b)
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falsi = (a * fb - b * fa) / (fb - fa)
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matches = []
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for name, S in (("rational", PURE_SIMS["rational"]),
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("exp_abs", PURE_SIMS["exp_abs"]),
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("rat_sq", PURE_SIMS["rat_sq"]),
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("cosine", PURE_SIMS["cosine"]),
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("ratio", PURE_SIMS["ratio"])):
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pt = a + (1.0 - S(F(a) / abs(fb))) * (b - a)
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if abs(pt - falsi) < 1e-12:
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matches.append(name)
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assert matches == ["rational"], matches
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def test_the_bracket_end_difference_gives_regula_falsi_exactly():
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"""Proposition 1'. This is why 'most recent' in the definition of the
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divided difference is load-bearing."""
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a, b = 1.0, 4.0
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fa, fb = F(a), F(b)
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falsi = (a * fb - b * fa) / (fb - fa)
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sigma = abs((fb - fa) / (b - a)) * abs(b - a)
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pt = a + min(1.0, abs(fa) / sigma) * (b - a)
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assert pt == falsi
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# --------------------------------------------------------------------------
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# Table 1's c = 0 rows: a sublinear law with a predicted coefficient
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# --------------------------------------------------------------------------
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@pytest.mark.parametrize("sim,want", [("rat_sq", 0.8536), ("exp_sq", 0.8536),
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("cosine", 1.7071)])
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def test_the_quadratic_similarities_are_sublinear_with_a_predicted_constant(sim, want):
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S = PURE_SIMS[sim]
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h = 1e-4
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c2 = (1.0 - S(h)) / (h * h)
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fp, D, sigma = 2.0 * R2, 2.0 - R2, 2.0
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predicted = sigma * sigma / (c2 * fp * fp * D)
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assert abs(predicted - want) < 1e-3, predicted
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e = iterate(S, 4001)
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assert abs(4000 * e[4000] - predicted) < 0.02, (4000 * e[4000], predicted)
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def test_there_is_no_rate_to_measure_for_those_rows():
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"""The old table reported 0.93-0.94 as a 'measured rate'; the ratio
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creeps to 1, so that number was the window's end, not a rate."""
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e = iterate(PURE_SIMS["rat_sq"], 4001)
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ratios = [e[n] / e[n - 1] for n in (10, 100, 1000, 4000)]
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assert ratios == sorted(ratios), ratios
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assert ratios[0] < 0.94 < ratios[-1]
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assert ratios[-1] > 0.999
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# --------------------------------------------------------------------------
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# the order is 2, not Muller's 1.84
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# --------------------------------------------------------------------------
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def test_the_three_point_order_is_two():
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ps = [order_of(lambda f_, a_, b_, xt, mx: pure_root(
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f_, a_, b_, XTOL, MAXFEV, scale="derived3", guard="project"),
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f, a, b, root) for _n, f, a, b, root in PROBLEMS]
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ps = [p for p in ps if math.isfinite(p) and 0.5 < p < 5.0]
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med = sorted(ps)[len(ps) // 2]
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assert abs(med - 2.0) < 0.1, med
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# --------------------------------------------------------------------------
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# the two evaluation-count conventions must both be reported and must differ
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# in the direction the paper says
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# --------------------------------------------------------------------------
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@pytest.mark.skipif(not os.path.exists(os.path.join(
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os.path.dirname(os.path.dirname(os.path.abspath(__file__))),
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"sweep_full.json")), reason="sweep not present")
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def test_charged_totals_are_what_section_5_reports():
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import json
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root = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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with open(os.path.join(root, "sweep_full.json"), encoding="utf8") as fh:
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rows = {r["label"]: r for r in json.load(fh)}
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# brent is now run at the better of ours and scipy.brentq per problem,
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# which is why it is 6988 rather than the 6991 of our implementation alone
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for lbl, want in (("pure.ratio.derived3.project", 6026),
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("brent", 6988), ("bisection", 10620),
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("chandrupatla", 5208)):
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got = rows[lbl]["APS"]["total"] + rows[lbl]["CHA"]["total"]
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assert got == want, (lbl, got, want)
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