cplx_newton_basins — MATH `complex` op

• 数据种类:signal → labels2d

• 调用: 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)(要直接调用实现,import mathops; mathops.cplx_newton_basins(coeffs, centre=0j, half_width=2.0, shape=(256, 256), max_iter=64, tol=1e-10);从台账取用则 opsmath.get("cplx_newton_basins"))

用法

从每一点出发的牛顿法, 会落到多项式的哪一个根?

> 以下的详细说明为原文 —— 摘要与标题已翻译。

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.

该族通用的输入契约(fail-closed)

mathops 的每个算子都先校验输入再计算(不让任何东西无声通过):

• **complex 输入一律 ValueError** —— 强制转成 float64 会无声丢掉虚部(numpy 只发一个 ComplexWarning,然后返回一个「看着合理却是错的」实数)。请显式写出 .real/.imag/abs(),或改用支持复数的 complexops。

• **含被掩元素的 masked array 一律 ValueError** —— 拒绝「剥掉掩码直接使用下面原值」的隐式转换。请显式选择填充还是丢弃。

• **所有输入中的 NaN/Inf 一律 ValueError**(明确给出个数后拒绝 —— 它会污染整个结果)。

• 形状严格:不对 1-D 与 2-D 做隐式提升或广播(向量槽位收到矩阵、矩阵槽位收到向量都是 ValueError;请显式 reshape)。

• 尺寸上限:接受矩阵的算子与 stat_histogram 的 bins,超过 mathops.MAX_ELEMENTS(2^26 ≈ 6700 万个元素)即 ValueError。

详细使用指南

• math_metrology 族使用指南

参考(示例数据・文献)

• 示例数据目录(下载 URL / 许可证) —— 2-D 用 skimage.data(BSD/公有领域)加合成图,3-D 给出真实数据源(Stanford/PDS 等)的下载 URL。

• 算子来历与参考文献 —— 该算子族所依据的研究/方法出处。

• 算法的正典(作者・年份)与用途见上面的族使用指南。

可运行的示例(实际调用该算子并已验证的样例)

• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py

类型可衔接的下一个算子(可接受 `labels2d` 作为输入)

—

同类别(`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 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*

cplx_newton_basins — MATH complex op

用法