complex op• Data kinds: signal → labels2d
• Call: 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) (to call the implementation directly, import mathops; mathops.cplx_newton_basins(coeffs, centre=0j, half_width=2.0, shape=(256, 256), max_iter=64, tol=1e-10); from the registry, opsmath.get("cplx_newton_basins"))
Which root of a polynomial does Newton's method fall into, from each point?
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.
Every mathops op validates its input before computing (nothing slips through silently):
• **complex input raises ValueError** — coercing to float64 silently discards the imaginary part (numpy only emits a ComplexWarning and returns a plausible-looking wrong real number). State .real/.imag/abs() explicitly, or use complexops, which handles complex data.
• **masked arrays with masked elements raise ValueError** — the implicit conversion that peels off the mask and uses the raw values underneath is refused. Say explicitly whether to fill or to drop.
• **NaN/Inf raises ValueError on every input** (refused with the count stated — it propagates through the whole result).
• Shapes are strict: 1-D and 2-D are never implicitly promoted or broadcast (a matrix in a vector slot, or a vector in a matrix slot, raises ValueError; reshape explicitly).
• Size cap: ops that take a matrix, and the stat_histogram bins, raise ValueError beyond mathops.MAX_ELEMENTS (2^26 ≈ 67 million elements).
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
labels2d as input)—
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 operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.