NumCosmoMathFftlog

Abstract class for implementing logarithm fast fourier transform.

This class provides the tools to compute the Fast Fourier Transform of any function, which is assumed to be a periodic sequence of logarithmically spaced points. It is inspired on the approach FFTLog developed by [Hamilton (2000)][XHamilton2000] [arXiv], which was extended as described below.

A function $G(r)$ is written as \begin{equation}\label{eq:Gr} G(r) = \int_0^\infty F(k) \ K(kr) dk, \end{equation} where $F(k)$ is defined in the fundamental interval $[\ln k_0 - L/2, \ln k_0 + L/2]$, $L$ is the period, $\ln k_0$ is the center value and $K(kr)$ is a kernel function. Assuming that $F(k)$ can be written in terms of the $N$ lowest Fourier modes, we have

$$F(k) = \sum_{n} c_n e^{\frac{2\pi i n}{L} \ln\left(\frac{k}{k_0}\right)}.$$ Substituting $F(k)$ in Eq. \eqref{eq:Gr} and changing the variable $k \rightarrow t = kr$, thus \begin{align}\label{eq:Gr_decomp} r G(r) &= \sum_n c_n \int_0^\infty \frac{k}{k_0}^{\frac{2\pi i n}{L}} K(kr)^2 d(kr) \ &= \sum_n c_n \int_0^\infty \frac{t}{k_0 r}^{\frac{2\pi i n}{L}} K(t) dt \ &= \sum_n c_n e^{-\frac{2\pi i n}{L} \ln\left(\frac{r}{r_0}\right)} e^{-\frac{2\pi i n}{L} \ln(k_0 r_0)} Y_n, \end{align} where $$Y_n = \int_0^\infty t^{\frac{2\pi i n}{L}} K(t) dt,$$ and the Fourier coefficients are $$c_n = \frac{1}{N} \sum_m F(k_m) e^{- \frac{2\pi i nm}{N}}.$$ The total number of points $N$ corresponds to the number of knots in the fundamental interval, which is equally spaced.

The variables discretization is different depending whether $N$ is even or odd, in general $$ k_n = k_0 \mathrm{e}^{n L / N}, \qquad r_m = r_0 \mathrm{e}^{m L / N}. $$ If $N$ is odd, $n$ and $m$ runs from $[-\lfloor N/2\rfloor, \lfloor N/2\rfloor]$, where $\lfloor N/2\rfloor$ is the round-down (largest integer smaller than $N/2$) of $N/2$. In this case \begin{align} \ln\left(k_{-\lfloor N/2\rfloor}\right) = \ln(k_0) - \frac{(N-1)}{N} \frac{L}{2}, \ \ln\left(k_{+\lfloor N/2\rfloor}\right) = \ln(k_0) + \frac{(N-1)}{N} \frac{L}{2}. \end{align} This means that for odd $N$ the values of $k_n$ (and $r_n$) never touches the borders $\ln(k_0) \pm L/2$ and $\ln(r_0) \pm L/2$. On the other hand if $N$ is even ($\lfloor N/2\rfloor = N/2$) \begin{align} \ln\left(k_{-\lfloor N/2\rfloor}\right) = \ln(k_0) - \frac{L}{2}, \ \ln\left(k_{+\lfloor N/2\rfloor}\right) = \ln(k_0) + \frac{L}{2}. \end{align} However, since we are assuming that these functions are periodic with period $L$ these two points refer to the same value of the functions. Thus, we do not need to include both points and in the case of even $N$ we include the point $\ln(k_0) - \frac{L}{2}$ only. In the original [FFTLog][XHamilton2000] paper they include both points but give them a $1/2$ weight, here we avoid this complication by using the lower end point only.

The user must provide the following input values: $\ln k_0$ - ncm_fftlog_set_lnk0(), $\ln r_0$ - ncm_fftlog_set_lnr0(), $L$ - ncm_fftlog_set_length(), padding percentage - ncm_fftlog_set_padding(), $N$ - ncm_fftlog_set_size(), $F(k)$ (or $F(k_m)$ — see description below).

• Since the algorithm assumes that the function to be decomposed is periodic, it is worth extending the interval in $\ln k$ such that $F(k) \equiv 0$ in the intervals $\left[\ln k_0 -\frac{L_T}{2}, \ln k_0 - \frac{L}{2} \right)$ and $ \left(\ln k_0 + \frac{L}{2}, \ln k_0 + \frac{L_T}{2}\right]$, where the total period $L_T$ is defined by the final number of knots, i.e., $N_f = N (1 + \mathrm{padding})$.
• $N$ knots are equally distributed in the fundamental interval and $N \times \mathrm{padding}$ knots are distributed in the two symmetric intervals as mentioned above.
• For the sake of optimization, the final number of points $N_f$ is substituted by the smallest number $N_f^\prime$ (bigger than $N_f$) which can be decomposed as $N_f \leq N_f^\prime = N^\prime (1 + \mathrm{padding}) = 2^a 3^b 5^c 7^d$, where $a$, $b$, $c$ and $d$ are positive integers.
• The function $F(k)$ can be provided as:
• a gsl_function - ncm_fftlog_eval_by_gsl_function() - whose values are computed at the knots within the fundamental interval, and set to zero within the padding intervals.

• as a vector - ncm_fftlog_eval_by_vector() - first one must get the vector of $\ln k$ knots, ncm_fftlog_get_lnk_vector(), and then pass a vector containing the values of the function computed at each knot.

• Regarding $Y_n$, see the different implementations of NcmFftlog, e.g., NcmFftlogTophatwin2 and NcmFftlogGausswin2.