complex op• 資料種類:signal → labels2d
• 呼叫: import fullseye as fs; fs.ledger.cplx_newton_basins(coeffs, centre=0j, half_width=2.0, shape=(256, 256), max_iter=64, tol=1e-10)(要直接呼叫實作,import mathops; mathops.cplx_newton_basins(coeffs, centre=0j, half_width=2.0, shape=(256, 256), max_iter=64, tol=1e-10);從台帳取用則 opsmath.get("cplx_newton_basins"))
從每一點出發的牛頓法, 會落到多項式的哪一個根?
> 以下的詳細說明為原文 —— 摘要與標題已翻譯。
Labels the window 1..len(roots) by the root reached, and 0 where the iteration has not converged within max_iter (the Julia set and its neighbourhood). Roots come from numpy.roots — the same routine behind this family's poly_roots — and are sorted by (Re, Im) so the label of a given root does not change between runs.
★Degree 2 has a closed-form answer, so the operator can be checked exactly rather than plausibly. Cayley (1879): for z**2 - 1 the basins are the two open half-planes Re z > 0 and Re z < 0, and the boundary is the imaginary axis — no fractal. The famous fractal boundary appears at degree 3, which Cayley could not settle; there the honest checks are structural (every root's basin is non-empty; z**3 - 1 is invariant under rotation by 2*pi/3, and so is its labelling, up to the cyclic relabelling of the roots).
coeffs is highest-degree-first, as numpy.roots and poly_roots take it.
Raises ValueError: text, non-finite or all-zero coefficients; a non-zero constant (no roots); max_iter < 1; tol <= 0; degenerate window. Points where the derivative vanishes are left unconverged (label 0) rather than divided by — a critical point is genuinely undecided.
HALCON: no operator.
mathops 的每個運算子都先檢驗輸入再計算(不讓任何東西無聲通過):
• complex 輸入一律 ValueError —— 強制轉成 float64 會無聲丟掉虛部(numpy 只發一個 ComplexWarning,然後回傳一個「看似合理卻是錯的」實數)。請明確寫出 .real/.imag/abs(),或改用支援複數的 complexops。
• 含被遮罩元素的 masked array 一律 ValueError —— 拒絕「剝掉遮罩直接使用下面原值」的隱式轉換。請明確選擇填補還是丟棄。
• 所有輸入中的 NaN/Inf 一律 ValueError(明確給出個數後拒絕 —— 它會汙染整個結果)。
• 形狀嚴格:不對 1-D 與 2-D 做隱式提升或廣播(向量槽位收到矩陣、矩陣槽位收到向量都是 ValueError;請明確 reshape)。
• 尺寸上限:接受矩陣的運算子與 stat_histogram 的 bins,超過 mathops.MAX_ELEMENTS(2^26 ≈ 6700 萬個元素)即 ValueError。
• 範例資料目錄(下載 URL / 授權) —— 2-D 用 skimage.data(BSD/公有領域)加合成圖,3-D 給出真實資料源(Stanford/PDS 等)的下載 URL。
• 運算子來歷與參考文獻 —— 該運算子族所依據的研究/方法出處。
• 演算法的正典(作者、年份)與用途見上面的族使用指南。
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
labels2d 作為輸入)—
complex)cplx_contour_circle · cplx_poly_eval · cplx_contour_integral · cplx_winding_number · cplx_cauchy_value · cplx_argument_principle · cplx_laurent_coeffs · cplx_joukowski
*Provenance: mathops.py — MATH 運算子登記表。本條目由 tools/opdocs.py md 自動產生(請勿手動編輯)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.