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mathematics
nonlinear-systems
model-reduction
state-representation
exterior-algebra
exact-arithmetic
| """A small exact demonstration. Run from the research-package root.""" | |
| import sympy as sp | |
| from src.continuation import (separation_family, compile_exterior, affine_closure, | |
| snapshot_space, normalize) | |
| def main() -> None: | |
| n = 8 | |
| model = separation_family(n) | |
| cert = compile_exterior([model.A, model.B], [model.F[1, :]]) | |
| affine = affine_closure([model.A, model.B], model.F[[0, 1], :]) | |
| print("Certificate integrity:", cert.verify()) | |
| print("Nonlinear coordinates in the explicit optimal realization:", 2) | |
| print("Initial affine snapshot scalars:", snapshot_space(cert).rows - 1) | |
| print("Fixed self-updating affine scalars:", affine.rows - 1) | |
| x = sp.Matrix(range(1, n + 1)) / (n * (n + 1) // 2) | |
| compact = model.signature(x) | |
| for label in [0, 0, 1, 0]: | |
| x = normalize([model.A, model.B][label], x) | |
| compact = model.compact_step(compact, label) | |
| assert compact == model.signature(x) | |
| print("Exact continuation after A,A,B,A:", list(compact)) | |
| revised = separation_family(n, sp.Rational(1, 1000)) | |
| nu = sp.ones(n, 1) / n | |
| left = nu + model.F[3, :].T / (4 * n) | |
| right = nu - model.F[3, :].T / (4 * n) | |
| assert model.signature(left) == model.signature(right) | |
| first = (model.F[1, :] * normalize(revised.B, left))[0] | |
| second = (model.F[1, :] * normalize(revised.B, right))[0] | |
| assert first != second | |
| print("Old-identical states require different post-rewrite outputs:", first, second) | |
| print("The old memory is therefore exactly insufficient for this rewrite.") | |
| if __name__ == "__main__": | |
| main() | |