Notation¶
felcu implements the Sobolev-regularised, reflected and arcsine-Taylor
Fourier extensions of
P. Brearley, T. L. Howarth and B. Adcock, Fourier extensions for matrix-function block encodings with error-independent subnormalization bounds, arXiv:2601.18024.
These docs use the paper's notation. The table maps each symbol to the corresponding argument or result field.
Matrices and spectra¶
| Symbol | Meaning | In felcu |
|---|---|---|
| \(\mathcal H\) | Hermitian input matrix | matrix= |
| \(A = U\Sigma V^\dagger\) | general input matrix (acted on through its singular values) | matrix= |
| \(\lambda_{\min}, \lambda_{\max}\) | eigenvalue endpoints, \(\operatorname{spec}(\mathcal H) \subseteq [\lambda_{\min}, \lambda_{\max}]\) | sigma_min, sigma_max with eigenvalues |
| \(\Sigma_{\min}, \Sigma_{\max}\) | singular-value endpoints | sigma_min, sigma_max with singular values |
| \(a < 0 < b\) | eigenvalues either side of 0, \(\operatorname{spec}(\mathcal H) \subseteq [\lambda_{\min}, a] \cup [b, \lambda_{\max}]\) (indefinite inversion) | gap=(a, b), result.gap |
| \(\mu = (\lambda_{\max} + \lambda_{\min})/2\) | spectral midpoint; \(0\) for singular values | result.mu |
| \(\delta = (\lambda_{\max} - \lambda_{\min})/2\) | spectral half-width; \(\Sigma_{\max}\) for singular values | derived |
| \(\kappa\) | condition number, ratio of largest to smallest \(\lvert\lambda\rvert\) or \(\Sigma\) | derived |
sigma_min and sigma_max hold whichever endpoints the construction uses
(see Spectral information);
result.spectral_kind_used says which.
Scaling and the fitted function¶
| Symbol | Meaning | In felcu |
|---|---|---|
| \(\mathbb T = [-\pi, \pi)\) | one period | |
| \(\eta > 1\) | domain extension factor | eta, result.eta |
| \(\Omega_\eta = [-\pi/\eta, \pi/\eta]\) | fitted interval | result.x_max \(= \pi/\eta\) |
| \([-x_{\max}, \tau(a - \mu)] \cup [\tau(b - \mu), x_{\max}]\) | two-interval fitted set (indefinite or singular-value inversion); \(\Omega^{\rm indef}_{\eta,\kappa}\) when symmetric | result.x_gap, result.x_max |
| \(\Omega\) | the fitted set in general | fitted_domain_segments(result) |
| \(\tau\) | scale, \(\mathcal G = \tau(\mathcal H - \mu I)\) | result.tau |
| \(\mathcal G\) | scaled matrix with \(\operatorname{spec}(\mathcal G) \subseteq \Omega\) | |
| \(f\) | the matrix function: \(x\), \(x^{-1}\), \(e^x\), \(\sqrt x\) or \(\log x\) | problem= |
| \(g(x) = f(\mu + x/\tau)\) | fitted (scaled) target, \(f(\mathcal H) = g(\mathcal G)\) | evaluate_target_function(x, result) |
For singular values, \(\mathcal H\) is the Hermitian dilation of \(A\), whose spectrum \(\pm\operatorname{sv}(A)\) is centred at \(\mu = 0\), so \(g(x) = f(x/\tau)\) and the series is odd.
The series¶
| Symbol | Meaning | In felcu |
|---|---|---|
| \(m\) | truncation order | m, m_values, result.m |
| \(g_m(x) = \sum_{k=-m}^{m} c_k e^{ikx}\) | the Fourier extension | evaluate_fourier_series(x, result.a0, result.ak, result.bk) |
| \(a_0, a_k, b_k\) | real coefficients, \(g_m = a_0 + \sum_k [a_k \sin kx + b_k \cos kx]\) | result.a0, result.ak, result.bk |
| \(c_k\), with \(c_0 = a_0\), \(c_{\pm k} = (b_k \mp i a_k)/2\) | complex (LCU) coefficients | result.complex_coefficients \(= [c_{-m}, \dots, c_m]\); result.circuit_coefficients \(= [c_0, 0, c_1, c_{-1}, \dots, c_m, c_{-m}]\) |
| \(\alpha_m = \sum_k \lvert c_k\rvert\) | subnormalisation | result.alpha |
| \(\epsilon_m = \|g - g_m\|_{L^\infty(\Omega)}\) | approximation (and spectral) error | result.achieved_linf_error |
| \(\epsilon_{\rm t}\) | target tolerance | target_error |
| refl, asin, reg | reflected, arcsine-Taylor and Sobolev-regularised strategies | method="reflected", "arcsine_taylor", "sobolev"; result.method |
Explicit methods¶
| Symbol | Meaning | In felcu |
|---|---|---|
| \(F(x) = g(\arcsin(\sin x))\) | reflected extension of \(g\) about \(\pm\pi/2\) | method="reflected" |
| \(d_k\) | Taylor coefficients, \(g(\arcsin z) = \sum_k d_k z^k\) | method="arcsine_taylor" |
| \(\rho_\eta = \sin(\pi/\eta)\) | arcsine-Taylor convergence factor | |
| \(\Omega^{\rm asin}_\eta = [\mu - \eta\delta/2, \mu + \eta\delta/2]\) | interval on which both explicit methods evaluate \(f\) |
Regularisation¶
These apply to method="sobolev" only; the explicit methods report them as
None.
| Symbol | Meaning | In felcu |
|---|---|---|
| \(H^w_r(\mathbb T)\) | weighted Sobolev norm, weight \(P_w(r\lvert k\rvert) = \sum_{j=0}^{w} (r\lvert k\rvert)^{2j}\) | |
| \(0 < r \le 1\) | Sobolev scale | sobolev_r, result.sobolev_r |
| \(w > 1/2\) | Sobolev order | sobolev_w, result.sobolev_w |
| \(\gamma > 0\) | regularisation weight | lambdas, result.lambda_used hold \(\sqrt{2\pi\gamma}\) |
| \(H^1(\Omega)\) | fitting norm | order_fit=1 (default); order_fit=0 for \(L^2(\Omega)\) |
lambda is not an eigenvalue
For historical reasons the regularisation strength is called
lambda in the code (lambdas, lambda_used). It is
\(\sqrt{2\pi\gamma}\), the parameter that enters the augmented
least-squares system; it is unrelated to the eigenvalues
\(\lambda_j\).