Polynomial Coordinate Functions

The coordinates module provides lightweight helpers for manipulating 6-D complex phase-space vectors.

_clean_coordinates()

The _clean_coordinates() function removes tiny numerical artefacts from a complex coordinate vector.

hiten.algorithms.polynomial.coordinates._clean_coordinates()

Remove tiny numerical artefacts from a complex coordinate vector.

Parameters:

• coords (numpy.ndarray) – Input array of shape (6,) interpreted as complex coordinates. The function accepts any real- or complex-typed array that can be cast to np.complex128.

• tol (float, default 1e-30) – Absolute threshold below which real or imaginary components are set to zero. The default is conservative and tailored to IEEE-754 double precision.

Returns:

Cleaned copy of coords with the same dtype as numpy.complex128.

Return type:

numpy.ndarray

Notes

The routine operates element-wise. If any changes are made, a warning is emitted via logger().

Examples

>>> import numpy as np
>>> from hiten.algorithms.polynomial.coordinates import _clean_coordinates
>>> c = np.array([1+1e-40j, 0+0j, 1e-32+2j, 0j, -3e-31, 5], dtype=complex)
>>> _clean_coordinates(c, tol=1e-30)
array([1.+0.j, 0.+0.j, 0.+2.j, 0.+0.j, 0.+0.j, 5.+0.j])

_substitute_coordinates()

The _substitute_coordinates() function applies a linear substitution matrix to a coordinate vector.

hiten.algorithms.polynomial.coordinates._substitute_coordinates()

Apply a linear substitution matrix to a coordinate vector.

The operation performed is

y_i = sum_{j=1}^6 M_ij * x_j

where M is matrix, x is coords and the result y is returned.

Parameters:

• coords (numpy.ndarray) – Source coordinates with shape (6,).

• matrix (numpy.ndarray) – Transformation matrix with shape (6, 6). Only non-zero entries are accessed so sparse structures are inexpensive although matrix itself must support NumPy indexing.

Returns:

Transformed coordinates with dtype numpy.complex128.

Return type:

numpy.ndarray

Raises:

ValueError – If matrix is not of shape (6, 6).

Notes

The implementation avoids temporary allocations by accumulating directly into the output array. Real inputs are seamlessly promoted to complex.

Examples

>>> import numpy as np
>>> from hiten.algorithms.polynomial.coordinates import _substitute_coordinates
>>> M = np.eye(6)
>>> M[0, 1] = 2  # simple shear: q1 -> q1 + 2*q2
>>> _substitute_coordinates(np.arange(6), M)
array([ 2.+0.j,  1.+0.j,  2.+0.j,  3.+0.j,  4.+0.j,  5.+0.j])