cplx_newton_basins — MATH `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"))

用法

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.

該族通用的輸入契約(fail-closed)

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。

詳細使用指南

• math_metrology 族使用指南

參考(範例資料・文獻)

• 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。

可執行的範例(實際呼叫該運算子並已驗證的樣例)

• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py

型別可銜接的下一個運算子(可接受 `labels2d` 作為輸入)

—

同類別(`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)。*

cplx_newton_basins — MATH complex op

用法