NumCosmoMathSpectral

Computes the $C^{(2)}k$ coefficients of $f”(s)$ for $f(s) = \sum_n c^n T_n(s)$, $g^k = 2(k + 2)\,c^{k+2}$, from $T_n” = 2n\,C^{(2)}{n-2}$. For $N$ coefficients in c, g has $N - 2$, or one zero when $N \le 2$. The derivative is in $s$; on $[a, b]$ multiply by $(2/(b - a))^2$.

Computes the $C^{(2)}k$ coefficients of $f’(s)$ for $f(s) = \sum_n c^n T_n(s)$, $g^k = c^{k+1} - c^{k+3}$, from $T_n’ = n\,U{n-1}$ and $U_m = (C^{(2)}m - C^{(2)}{m-2})/(m + 1)$. For $N$ coefficients in c, g has $N - 1$, or one zero when $N \le 1$. The derivative is in $s$; on $[a, b]$ multiply by $2/(b - a)$.

Converts the $T_n$ coefficients c of a series to its $C^{(1)}_n = U_n$ coefficients, of the same length.

Converts the $T_n$ coefficients c of a series to its $C^{(2)}_k$ coefficients, of the same length.

Evaluates the derivative in $s$ in one backward pass that builds the coefficients of the derivative series and sums them by the Clenshaw recurrence.

Same as ncm_spectral_chebyshev_deriv() at $s$ given by ncm_spectral_x_to_s(), times $\mathrm{d}s/\mathrm{d}x = 2/(b - a)$.

Evaluates the series by the Clenshaw recurrence for $|s| < 0.9$ and by Reinsch’s modification of it closer to the endpoints.

Same as ncm_spectral_chebyshev_eval() at $s$ given by ncm_spectral_x_to_s().

Integrates the series over its interval, from $\int_{-1}^{1} T_k(s)\,\mathrm{d}s = 2/(1 - k^2)$ for even $k$ and zero for odd $k$. Applied to the coefficients of ncm_spectral_compute_chebyshev_coeffs() or its adaptive variants, this is Clenshaw-Curtis quadrature.

If spectral is not NULL, decreases its reference count by one and sets spectral to NULL.

Adds coeff times row $k$ of ncm_spectral_get_d2_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_d_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_proj_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_s2_d2_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_s2_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_s_d2_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_s_d_matrix() to row_data. Its nonzero entries are.

Adds coeff times row $k$ of ncm_spectral_get_s_matrix() to row_data. Its nonzero entries are.

Evaluates the series by the forward recurrence of $C^{(1)}_n = U_n$, with the closed form $U_n(\pm 1) = (\pm 1)^n (n + 1)$ at the endpoints.

Same as ncm_spectral_gegenbauer_alpha1_eval() at $s$ given by ncm_spectral_x_to_s().

Evaluates the series by the forward recurrence of $C^{(2)}_n$, with the closed form $C^{(2)}_n(\pm 1) = (\pm 1)^n \binom{n+3}{3}$ at the endpoints.

Same as ncm_spectral_gegenbauer_alpha2_eval() at $s$ given by ncm_spectral_x_to_s().

Multiplies $\sum_n g^n C^{(2)}n(s)$ by $\alpha s + \beta$, using $s\,C^{(2)}_n = [(n + 1)\,C^{(2)}{n+1} + (n + 3)\,C^{(2)}_{n-1}]/(2(n + 2))$. For $N$ coefficients in g, out has $N + 1$. out must not alias g.

Builds the $N \times N$ matrix of the second derivative from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $f”$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of the derivative from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $f’$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of the identity from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $f$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of $s^2\,\mathrm{d}^2/\mathrm{d}s^2$ from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $s^2 f”$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of multiplication by $s^2$ from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $s^2 f$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of $s\,\mathrm{d}^2/\mathrm{d}s^2$ from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $s f”$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of $s\,\mathrm{d}/\mathrm{d}s$ from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $s f’$, truncated to the first $N$ columns.

Builds the $N \times N$ matrix of multiplication by $s$ from the $T_n$ coefficients of $f$ to the $C^{(2)}_k$ coefficients of $s f$, truncated to the first $N$ columns.

Maps the Chebyshev variable s to $x = [(b - a)s + (a + b)]/2$, the inverse of ncm_spectral_x_to_s().

Maps x to the Chebyshev variable, $s = (2x - (a + b))/(b - a)$.

Expresses the same polynomial as a Chebyshev series on [a_out, b_out]. The Chebyshev variable $s_\mathrm{in}$ of [a_in, b_in] is $s_\mathrm{in} = \alpha s_\mathrm{out} + \beta$ in terms of the one of [a_out, b_out], and each $T_k(s_\mathrm{in})$ is expanded in the $T_j(s_\mathrm{out})$ by the Chebyshev recurrence, at a cost of $O(n^2)$.

Computes the $N$ Chebyshev coefficients of $F$ on $[a, b]$ from its values at the $N$ Chebyshev-Lobatto nodes. Aborts if $N < 2$.

Computes the Chebyshev coefficients of $F$ on $[a, b]$, doubling the nested Chebyshev-Lobatto grid from level level_min until, at level $r$, the $\ell_2$ norm of the change of the first $2^{r-1} + 1$ coefficients is below tol times the norm of the level-$r$ coefficients. Each doubling evaluates $F$ only at the new nodes. Reaching NcmSpectral:max-level without converging is an error.

As ncm_spectral_compute_chebyshev_coeffs_adaptive(), with the change compared to the larger of reltol times the norm and abstol. An abstol of 0.0 gives ncm_spectral_compute_chebyshev_coeffs_adaptive(); a positive one stops the refinement of a function known to be negligible.

As ncm_spectral_compute_chebyshev_coeffs_adaptive_full(), stopping at the smaller of level_cap and NcmSpectral:max-level and reporting the outcome through converged instead of failing. It uses the single-function buffers, so it may be called from inside the callback of a batch expansion.

Computes the Chebyshev coefficients of the n_comp components of $F$ on one shared grid, as ncm_spectral_compute_chebyshev_coeffs_adaptive_full(). Each node is evaluated once for all components. Every component must converge, so the one needing the highest level sets it for all. Reaching NcmSpectral:max-level without converging is an error.

As ncm_spectral_compute_chebyshev_coeffs_batch_adaptive(), stopping at the smaller of level_cap and NcmSpectral:max-level. When fatal is FALSE, not converging leaves coeffs unchanged and returns 0, and the last doubling is skipped when the level below predicts that it cannot converge. That suits a caller that splits its interval on failure.

Decreases the reference count of spectral by one.

Increases the reference count of spectral by one.

Sets NcmSpectral:max-level, reallocating the buffers when it changes.