Lp_error()#
- Lp_error(Qtrue, Qapprox, t=None, p=2)[source]#
Compute the absolute and relative Lp-norm error (with respect to time) between the snapshot sets Qtrue and Qapprox, where Qapprox approximates Qtrue:
absolute_error = ||Qtrue - Qapprox||_{L^p}, relative_error = ||Qtrue - Qapprox||_{L^p} / ||Qtrue||_{L^p},
where
||Z||_{L^p} = (int_{t} ||z(t)||_{p} dt)^{1/p}, p < infinity, ||Z||_{L^p} = sup_{t}||z(t)||_{p}, p = infinity.
The trapezoidal rule is used to approximate the integrals (for finite p). This error measure is only consistent for data sets where each snapshot represents function values, i.e.,
Qtrue[:, j] = [q(t1), q(t2), …, q(tk)]^T.
- Parameters
- Qtrue(n, k) or (k,) ndarray
“True” data corresponding to time t. Each column is one snapshot, i.e., Qtrue[:, j] is the data at time t[j]. If one-dimensional, each entry is one snapshot.
- Qapprox(n, k) or (k,) ndarray
An approximation to Qtrue, i.e., Qapprox[:, j] approximates Qtrue[:, j] and corresponds to time t[j]. If one-dimensional, each entry is one snapshot.
- t(k,) ndarray
Time domain of the data Qtrue and the Qapprox. Required unless p == np.inf.
- pfloat > 0
Order of the Lp norm. May be infinite (np.inf).
- Returns
- abs_errfloat
Absolute error ||Qtrue - Qapprox||_{L^p}.
- rel_errfloat
Relative error ||Qtrue - Qapprox||_{L^p} / ||Qtrue||_{L^p}.