numerics op• Data kinds: signal × signal → table
• Call: import fullseye as fs; fs.ledger.integrate_hamiltonian(q0, p0, dt: 'float' = 0.05, n_steps: 'int' = 1000, *, system: 'str' = 'harmonic', method: 'str' = 'verlet', **params) -> 'dict' (to call the implementation directly, import mathnumerics; mathnumerics.integrate_hamiltonian(q0, p0, dt: 'float' = 0.05, n_steps: 'int' = 1000, *, system: 'str' = 'harmonic', method: 'str' = 'verlet', **params) -> 'dict'; from the registry, opsmath.get("integrate_hamiltonian"))
Integrates a named Hamiltonian system (mass 1) in time. Return value {"t","q","p","energy","energy_drift"}.
method:
• "verlet" —— velocity Verlet (symplectic, time-reversal symmetric). The energy error stays bounded at O(dt²) and oscillates; it does not grow even over long times. Running with negative dt returns to the original state (up to rounding error).
• "euler" —— explicit Euler (a failure example for comparison). For the harmonic oscillator the energy grows by a factor (1 + ω²dt²) every step, giving an outward spiral in phase space.
• "rk4" —— classical 4th-order Runge–Kutta. Very accurate over short times, but since it is not symplectic the energy keeps slowly decreasing over long times (not bounded).
system: "harmonic" (omega) / "pendulum" (omega) / "kepler" (mu, 2-D).
Gate: Verlet's energy error is bounded / time reversal returns / Euler's amplification factor is (1 + ω²dt²)^n.
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.
• numerics_tour — py -3.11 examples/numerics_tour.py
table as input)dynsys_poincare_section · dwt_inverse · chebyshev_eval_nd
numerics)erf · erfc · bessel · gauss_quadrature · gauss_cubature · chebyshev_coeffs_nd · chebyshev_eval_nd · low_discrepancy
*Provenance: mathnumerics.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.