Transient amplification profile of a linear update

What it takes and gives

Runs with

Python 3.10 or later

Entry points

transient_profile, spectral_norm, spectral_radius

Known answers

9 cases its tests reproduce

Limits and scope

Limits

Dense powers by repeated multiplication: meant for matrices up to about 60 rows and horizons up to a few thousand steps. Power iteration converges slowly when the top two singular values nearly coincide; the result reports converged. Eigenvalues come from the characteristic polynomial, so they are limited to 8 rows and a repeated eigenvalue is accurate to about 1e-8. Linear time-invariant updates only; it does not model nonlinear or time-varying dynamics.

Guarantees

Each norm is the Rayleigh-quotient limit of power iteration on B^T B from two deterministic starts; the tests match closed forms to 1e-12. gelfand_bound = min over h >= 1 of ||A^h||^(1/h) is an upper bound on the spectral radius. settles_below_one_at is the first step from which every norm up to the horizon is below 1, or null.

Assumes

The update is linear and time-invariant: the same A at every step. Norms are Euclidean (spectral) norms; another norm gives another profile.

Where it is shown

• Statistics for comparing resultsDeterministic skills

Related components

• Brier score decomposition and expected calibration error
• Split conformal prediction intervals
• Anytime-valid confidence sequence for a bounded mean
• Probabilistic and deflated Sharpe ratio
• Ledoit-Wolf covariance shrinkage
• Maximum mean discrepancy with an RBF kernel