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.