complex op• 数据种类:cimage → measurement
• 调用: import fullseye as fs; fs.ledger.cplx_cr_residual(f, spacing=1.0)(要直接调用实现,import mathops; mathops.cplx_cr_residual(f, spacing=1.0);从台账取用则 opsmath.get("cplx_cr_residual"))
采样复场的柯西-黎曼残差 —— 把「这个场全纯吗?」变成一个数。
> 以下的详细说明为原文 —— 摘要与标题已翻译。
With f = u + i v sampled on a uniform grid, holomorphy means u_x = v_y and u_y = -v_x (Cauchy-Riemann). This returns the relative residual max(|u_x - v_y|, |u_y + v_x|) / max|grad| (central differences, numpy.gradient): 0 = the samples satisfy CR to the discretisation limit, 2 = the field is the conjugate of a holomorphic one (conj(z) gives exactly 2), values in between = partly analytic or noisy.
Grid convention (it decides the sign of the answer): f[i, j] is the field at z = x0 + j*spacing + i*spacing*1j — rows index the *increasing imaginary* axis, columns the real axis. Image arrays usually run rows *downward*; feeding one directly measures the conjugate field, whose residual is 2, not 0. Flip rows (f[::-1]) to use image data.
Discretisation, honestly: central differences are exact for polynomials of degree <= 2, so f = z**2 returns exactly 0; for higher order the residual floors at O(h^2 * |f'''|) (measured: f = z**3 on a [-1,1]^2 grid returns 1.7e-3 at h and 4.2e-4 at h/2 — a factor 4.00, the expected second order). Read a small value as "consistent with holomorphic at this resolution", never as proof.
A constant field returns 0.0 (it is holomorphic; the 0/0 of the normalisation is resolved by that limit, and stated here rather than left to numpy).
Raises ValueError: not a 2-D array, either dimension below 3 (no central difference exists), non-finite/masked input, over-cap size, non-finite or non-positive *spacing*.
HALCON: no operator (derivate_gauss supplies the real-valued derivatives one would build this from).
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。
• 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。
• math_complex — py -3.11 examples/math_complex.py
• poc_complex_plane_fields — py -3.11 examples/poc_complex_plane_fields.py
measurement 作为输入)—
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 算子登记表。本条目由 tools/opdocs.py md 自动生成(请勿手工编辑)。*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.