NumCosmoMathSpectral

Converts Chebyshev $T_n$ coefficients to Gegenbauer $C^{(1)}n$ coefficients ($\alpha=1$). Uses the relationship: $T_n = \frac{1}{2}(C^{(1)}_n + C^{(1)}{n-2})$ for $n \geq 2$.

Converts Chebyshev $T_n$ coefficients to Gegenbauer $C^{(2)}_k$ coefficients ($\alpha=2$).

Evaluates the first derivative of a Chebyshev expansion at $t$.

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 Clenshaw recurrence with Reinsch modification near endpoints. 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().

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 Clenshaw recurrence. For $\alpha=1$, $C^{(1)}_n(t) = U_n(t)$ (Chebyshev polynomials of the second kind). 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 Clenshaw recurrence. For $\alpha=2$, the recurrence relation is: $(n+1) C^{(2)}{n+1}(t) = 2(n+2)t C^{(2)}n(t) - (n+3) C^{(2)}{n-1}(t)$ 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().

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 Chebyshev coefficients of f(x) on [a,b] using FFTW DCT-I. The function F is sampled at Chebyshev nodes $x_k = (a+b)/2 - (b-a)/2\cos(k\pi/(N-1))$ which correspond to the Chebyshev points $t_k = \cos(k\pi/(N-1))$ in $[-1,1]$ transformed to $[a,b]$. The Chebyshev expansion is $f(x) = f(t) = \sum_{k=0}^{N-1} a_k T_k(t)$ where $t = (2x - (a+b))/(b-a)$.

Computes Chebyshev coefficients adaptively using nested Chebyshev-Lobatto nodes. The function F is evaluated at points x in [a,b]. Starts at level k_min and refines by doubling until spectral convergence is achieved or max_order is reached. Uses nested nodes: only new odd nodes are computed at each refinement level. The Chebyshev expansion is $f(x) = f(t) = \sum_{k=0}^{N-1} a_k T_k(t)$ where $t = (2x - (a+b))/(b-a)$.

Computes Chebyshev coefficients of $F(x(t)) \sqrt{1-t^2} h$ adaptively using nested Chebyshev-Lobatto nodes, where $h = (b-a)/2$. The weight factor $\sqrt{1-t^2} \cdot h$ enables direct integral computation: the integral $\int_a^b F(x)dx = \pi \cdot$ coeffs[0].

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.