NumCosmoMathSpectral

Converts Chebyshev $T_n$ coefficients to Gegenbauer $C^{(1)}_n$ coefficients ($\alpha=1$), where $C^{(1)}_n = U_n$. See the Spectral Methods page for the conversion relation.

Converts Chebyshev $T_n$ coefficients to Gegenbauer $C^{(2)}_k$ coefficients ($\alpha=2$) via the basis-projection formula. See the Spectral Methods page for the formula.

Evaluates the first derivative of a Chebyshev expansion at $t$ using a fused backward recurrence and Clenshaw algorithm, without explicitly forming the derivative series. See the Spectral Methods page for the derivative-coefficient relations.

Evaluates the first derivative of a Chebyshev expansion at a point x in [a_v, b]. This function converts x to t using $t = (2x - (a_v+b))/(b-a_v)$, evaluates the derivative with respect to t using ncm_spectral_chebyshev_deriv(), and then applies the chain rule: $df/dx = (df/dt) \cdot (dt/dx) = (df/dt) \cdot 2/(b-a_v)$.

Evaluates a Chebyshev expansion $f(t) = \sum_{k=0}^{N-1} a_k T_k(t)$ at t using the Clenshaw recurrence, switching to the Reinsch modification near the endpoints to avoid cancellation. The variable t should be in the interval [-1, 1]. To evaluate at a point x in [a, b], use ncm_spectral_chebyshev_eval_x() or first convert x to t using ncm_spectral_x_to_t(). See the Spectral Methods page for details.

Evaluates a Chebyshev expansion at a point x in [a_v, b]. This function converts x to t using $t = (2x - (a_v+b))/(b-a_v)$ and then calls ncm_spectral_chebyshev_eval().

If spectral is not NULL, decreases the reference count of spectral by one and sets the pointer to NULL.

Computes row k of the second derivative operator $\frac{d^2}{dx^2}$ that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Computes row k of the first derivative operator $\frac{d}{dx}$ that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Computes row k of the projection operator that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients (ultraspherical basis with $\lambda=2$). This is the identity operator expressed in different bases.

Computes row k of the $x^2 \cdot \frac{d^2}{dx^2}$ operator that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Computes row k of the multiplication by $x^2$ operator that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Computes row k of the $x \cdot \frac{d^2}{dx^2}$ operator that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Computes row k of the $x \cdot \frac{d}{dx}$ operator that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Computes row k of the multiplication by x operator that maps Chebyshev coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Evaluates a Gegenbauer $C^{(1)}_n$ expansion at t using a stable recurrence (with $C^{(1)}_n = U_n$). The variable t should be in the interval [-1, 1]. To evaluate at a point x in [a, b], use ncm_spectral_gegenbauer_alpha1_eval_x() or first convert x to t using ncm_spectral_x_to_t().

Evaluates a Gegenbauer $C^{(1)}_n$ expansion at a point x in [a, b]. This function converts x to t using $t = (2x - (a+b))/(b-a)$ and then calls ncm_spectral_gegenbauer_alpha1_eval().

Evaluates a Gegenbauer $C^{(2)}_n$ expansion at t using a stable recurrence. The variable t should be in the interval [-1, 1]. To evaluate at a point x in [a, b], use ncm_spectral_gegenbauer_alpha2_eval_x() or first convert x to t using ncm_spectral_x_to_t(). See the Spectral Methods page for the $C^{(2)}_n$ recurrence.

Evaluates a Gegenbauer $C^{(2)}_n$ expansion at a point x in [a, b]. This function converts x to t using $t = (2x - (a+b))/(b-a)$ and then calls ncm_spectral_gegenbauer_alpha2_eval().

Returns the second derivative operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $\frac{d^2f}{dx^2}$.

Returns the derivative operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $\frac{df}{dx}$.

Returns the projection (identity) operator matrix that transforms Chebyshev $T_n$ coefficients to Gegenbauer $C^{(2)}_k$ coefficients.

Returns the $x^2 \cdot \frac{d^2}{dx^2}$ operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $x^2 \cdot \frac{d^2f}{dx^2}$.

Returns the multiplication by $x^2$ operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $x^2 \cdot f(x)$.

Returns the $x \cdot \frac{d^2}{dx^2}$ operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $x \cdot \frac{d^2f}{dx^2}$.

Returns the $x \cdot \frac{d}{dx}$ operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $x \cdot \frac{df}{dx}$.

Returns the multiplication by $x$ operator matrix that transforms Chebyshev $T_n$ coefficients of $f(x)$ to Gegenbauer $C^{(2)}_k$ coefficients of $x \cdot f(x)$.

Computes order Chebyshev coefficients of $f(x)$ on $[a,b]$ at fixed resolution, sampling F at the Chebyshev–Lobatto nodes and applying FFTW DCT-I. See the Spectral Methods page for the node placement and coefficient normalization.

Computes Chebyshev coefficients of $f(x)$ on $[a,b]$ adaptively, starting at level k_min and doubling the nested Chebyshev–Lobatto grid until the coefficients converge to tol or max-order is reached. Only the new odd nodes are evaluated at each refinement. See the Spectral Methods page for the nested grids and convergence criterion.

Computes the Chebyshev coefficients of the weighted function $F(x(t))\sqrt{1-t^2}\,h$, with $h = (b-a)/2$, using the same adaptive nested grids as ncm_spectral_compute_chebyshev_coeffs_adaptive(). The weight turns the expansion into a Clenshaw–Curtis quadrature: $\int_a^b F(x)\,dx = \pi\,$ coeffs[0], and weighted inner products $\int_a^b F(x)G(x)\,dx$ follow from these coefficients and the standard coefficients of $G$. See the Spectral Methods page for the derivation.

Decreases the reference count of spectral by one. If the reference count reaches zero, the object is freed.

Gets the maximum refinement order for adaptive computations.

Increases the reference count of spectral by one.

Sets the maximum refinement order for adaptive computations.