conatus.expert.HillBlocks

class conatus.expert.HillBlocks(*, diag, couplings, Nh, size, label, omega=1.0, notes=(), mass=None)[source]

Bases: object

Block-banded Hill matrix M(μ) with half-width M = max |m| of the couplings.

diag(μ, n) -> (m, m) block of harmonic n (includes (μ + i n ω) M); couplings[m] couples x_{n-m} into block row n (m ≠ 0, constant in μ). Nh >= M is required.

Parameters:

  • diag (Callable[[complex, int], np.ndarray])

  • couplings (Mapping[int, np.ndarray])

  • Nh (int)

  • size (int)

  • label (str)

  • omega (float)

  • notes (tuple[str, ...])

  • mass (np.ndarray | None)

property harmonics: list[int]
factor(mu)[source]

Block-Thomas on super-blocks: S_k = D_k - L_k S_{k-1}⁻¹ U_{k-1} (partial-pivot LU inside each super-block, no pivoting across them).

Block-Thomas needs every intermediate Schur complement S_k (k < last) to be well conditioned: a regular Hill matrix can have a singular S_k (finding 11: det -2j, cond 4.98, and slogdet / solve came back NaN). When the reciprocal condition of an intermediate S_k is below THOMAS_RCOND_MIN, the whole matrix is factored instead by a banded LU with partial pivoting across super-blocks (LAPACK gbtrf, still linear in Nh); factorization.method says which one ran. The last S_k may be singular: then M(μ) is (μ is an exponent).

Parameters:

mu (complex)

Return type:

HillFactorization | BandedHillFactorization

slogdet(mu)[source]
Parameters:

mu (complex)

Return type:

tuple[complex, float]

solve(mu, rhs)[source]

M(μ) x = rhs with its normwise backward error checked: above SOLVE_BACKWARD_ERROR_MAX the banded pivoted LU is tried, then NumericalError (never a NaN or an inaccurate solution passed on).

Parameters:

  • mu (complex)

  • rhs (ndarray)

Return type:

ndarray

backward_error(mu, x, rhs)[source]

‖M x - b‖∞ / (‖M‖∞ ‖x‖∞ + ‖b‖∞) (normwise backward error; NaN -> inf).

Parameters:

  • mu (complex)

  • x (ndarray)

  • rhs (ndarray)

Return type:

float

matvec(mu, x)[source]

M(μ) x for x of shape (2Nh+1, size).

Parameters:

  • mu (complex)

  • x (ndarray)

Return type:

ndarray

dense(mu)[source]

Assembled M(μ), small problems and oracles only.

Parameters:

mu (complex)

Return type:

ndarray

winding(center, radius, n_points)[source]

Winding number of det M on |μ - center| = radius (an integer, certified by the adaptive walk of conatus.floquet.roots.adaptive_winding(); ConvergenceError when the walk cannot certify it). n_points: initial points on the circle.

Parameters:

  • center (complex)

  • radius (float)

  • n_points (int)

Return type:

float

explain()[source]
Return type:

str