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| double | Lk | Read / Write / Construct Only |
| guint | N | Read / Write |
| double | eval-r-max | Read / Write / Construct |
| double | eval-r-min | Read / Write / Construct |
| double | lnk0 | Read / Write / Construct Only |
| double | lnr0 | Read / Write / Construct Only |
| char * | name | Read |
| guint | nderivs | Read / Write / Construct |
| gboolean | no-ringing | Read / Write / Construct |
| double | padding | Read / Write / Construct |
| double | smooth-padding-scale | Read / Write / Construct |
| gboolean | use-eval-int | Read / Write / Construct |
| gboolean | use-smooth-padding | Read / Write / Construct |
GObject ╰── NcmFftlog ├── NcmFftlogGausswin2 ├── NcmFftlogSBesselJ ├── NcmFftlogSBesselJLJM ╰── NcmFftlogTophatwin2
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) [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 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 simetric 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.
NcmFftlog *
ncm_fftlog_ref (NcmFftlog *fftlog);
Increases the reference count of fftlog
by one.
void
ncm_fftlog_free (NcmFftlog *fftlog);
Decreases the reference count of fftlog
by one.
void
ncm_fftlog_clear (NcmFftlog **fftlog);
If fftlog
is different from NULL, decreases the reference count of
fftlog
by one and sets fftlog
to NULL.
const gchar *
ncm_fftlog_peek_name (NcmFftlog *fftlog);
This function peeks the fftlog
's associated name.
void
ncm_fftlog_reset (NcmFftlog *fftlog);
Reset the evaluation and internal coefficients forcing their recomputation.
void ncm_fftlog_set_nderivs (NcmFftlog *fftlog,guint nderivs);
Sets nderivs
as the number of derivatives to calculate.
guint
ncm_fftlog_get_nderivs (NcmFftlog *fftlog);
Gets the number of derivatives the object is currently calculating.
void ncm_fftlog_set_lnr0 (NcmFftlog *fftlog,const gdouble lnr0);
Sets the center of the transform output $\ln(r_0)$.
gdouble
ncm_fftlog_get_lnr0 (NcmFftlog *fftlog);
Gets the center of the transform output.
void ncm_fftlog_set_lnk0 (NcmFftlog *fftlog,const gdouble lnk0);
Sets the center of the transform input $\ln(k_0)$.
gdouble
ncm_fftlog_get_lnk0 (NcmFftlog *fftlog);
Gets the center of the transform input $\ln(k_0)$.
void ncm_fftlog_set_size (NcmFftlog *fftlog,guint n);
Sets the number of knots $N_f^\prime$ where the integrated function is evaluated,
given the input number of knots n
, plus padding.
void ncm_fftlog_set_padding (NcmFftlog *fftlog,gdouble pad_p);
Sets the size of the padding in percetange of the interval.
gdouble
ncm_fftlog_get_padding (NcmFftlog *fftlog);
Gets the padding percentage.
void ncm_fftlog_set_noring (NcmFftlog *fftlog,gboolean active);
Sets whether to use the no-ringing adjustment of $\ln(r_0)$.
void ncm_fftlog_set_length (NcmFftlog *fftlog,gdouble Lk);
Sets the length of the period Lk
, where the function is periodic in logarithmic space $\ln k$.
void ncm_fftlog_use_eval_interval (NcmFftlog *fftlog,gboolean use_eval_interal);
Sets whether to use a restricted evaluation interval $[r_\mathrm{min}, r_\mathrm{max}]$.
See ncm_fftlog_set_eval_r_min() and ncm_fftlog_set_eval_r_max().
void ncm_fftlog_use_smooth_padding (NcmFftlog *fftlog,gboolean use_smooth_padding);
Sets whether to use pad the fft using a power-law continuation of the
input function which is continuous at the border and drops to a scale
determined by ncm_fftlog_set_smooth_padding_scale().
void ncm_fftlog_set_smooth_padding_scale (NcmFftlog *fftlog,gdouble log10sc);
Sets the value of the smoothing scale $s$ which is used if ncm_fftlog_use_smooth_padding()
is turned on.
gdouble
ncm_fftlog_get_smooth_padding_scale (NcmFftlog *fftlog);
Gets the log10 of the current value of the smoothing scale $s$.
void ncm_fftlog_set_eval_r_min (NcmFftlog *fftlog,const gdouble eval_r_min);
Sets $r_\mathrm{min}$ to r_min
.
void ncm_fftlog_set_eval_r_max (NcmFftlog *fftlog,const gdouble eval_r_max);
Sets $r_\mathrm{max}$ to r_max
.
gdouble * ncm_fftlog_get_Ym (NcmFftlog *fftlog,guint *size);
Computes the $Y_m$ vector.
void ncm_fftlog_get_lnk_vector (NcmFftlog *fftlog,NcmVector *lnk);
Computes the $\ln k$ vector.
void ncm_fftlog_eval_by_vector (NcmFftlog *fftlog,NcmVector *Fk);
Fk
is a vector which contains the values of the function at each knot $\ln k_m$.
void ncm_fftlog_eval_by_function (NcmFftlog *fftlog,NcmFftlogFunc Fk,gpointer user_data);
Evaluates the function Fk
at each knot $\ln k_m$.
void ncm_fftlog_eval_by_gsl_function (NcmFftlog *fftlog,gsl_function *Fk);
Evaluates the function Fk
at each knot $\ln k_m$.
[skip]
void
ncm_fftlog_prepare_splines (NcmFftlog *fftlog);
Prepares the set of splines respective to the function $G(r)$ and, if required, its n-order derivatives.
NcmVector *
ncm_fftlog_get_vector_lnr (NcmFftlog *fftlog);
Gets the vector of the $\ln r$ knots.
NcmVector * ncm_fftlog_get_vector_Gr (NcmFftlog *fftlog,guint nderiv);
Gets the vector of the transformed function $G(r)$, nderiv
= 0, or
its nderiv
-th derivative with respect to $\ln r$.
NcmSpline * ncm_fftlog_peek_spline_Gr (NcmFftlog *fftlog,guint nderiv);
Peeks the spline of $G(r)$, nderiv
= 0,
or the spline of the nderiv
-th derivative of $G(r)$ with
respect to $\ln r$.
gdouble ncm_fftlog_eval_output (NcmFftlog *fftlog,guint nderiv,const gdouble lnr);
Evaluates the function $G(r)$, or the nderiv
-th derivative,
at the point lnr
.
void ncm_fftlog_calibrate_size (NcmFftlog *fftlog,NcmFftlogFunc Fk,gpointer user_data,const gdouble reltol);
Increases the original (input) number of knots until the $G(r)$ splines reach
the required precision reltol
.
void ncm_fftlog_calibrate_size_gsl (NcmFftlog *fftlog,gsl_function *Fk,const gdouble reltol);
Increases the original (input) number of knots until the $G(r)$ splines reach
the required precision reltol
.
[skip]
guint
ncm_fftlog_get_size (NcmFftlog *fftlog);
Gets the number of knots $N^\prime$ where the integrated function is evaluated.
gint
ncm_fftlog_get_full_size (NcmFftlog *fftlog);
Gets the number of knots $N_f^\prime$ where the integrated function is evaluated plus padding.
gdouble
ncm_fftlog_get_norma (NcmFftlog *fftlog);
Gets the number of knots $N_f^\prime$ where the integrated function is evaluated plus padding.
gdouble
ncm_fftlog_get_length (NcmFftlog *fftlog);
Gets the value of the ``physical'' period, i.e., period of the fundamental interval.
gdouble
ncm_fftlog_get_full_length (NcmFftlog *fftlog);
Gets the value of the total period, i.e., period defined by the fundamental interval plus the padding size.
gint ncm_fftlog_get_mode_index (NcmFftlog *fftlog,gint i);
Gets the index of the mode i
of the Fourier decomposition. This index corresponds
to the label $n$ in Eq. \eqref{eq:Gr_decomp}.
gint ncm_fftlog_get_array_index (NcmFftlog *fftlog,gint phys_i);
Gets the array index i
of the Fourier decomposition. This index corresponds the position
in the fft array of the element $n$ in Eq. \eqref{eq:Gr_decomp}.
“Lk” property “Lk” double
The function $F(k)$'s period in natural logarithm base.
Owner: NcmFftlog
Flags: Read / Write / Construct Only
Default value: 1
“N” property“N” guint
The number of knots in the fundamental interval.
Owner: NcmFftlog
Flags: Read / Write
Default value: 10
“eval-r-max” property “eval-r-max” double
Evaluation r_max.
Owner: NcmFftlog
Flags: Read / Write / Construct
Allowed values: >= 0
Default value: 1.79769e+308
“eval-r-min” property “eval-r-min” double
Evaluation r_min.
Owner: NcmFftlog
Flags: Read / Write / Construct
Allowed values: >= 0
Default value: 0
“lnk0” property “lnk0” double
The Center value for $\ln(k)$.
Owner: NcmFftlog
Flags: Read / Write / Construct Only
Default value: 0
“lnr0” property “lnr0” double
The Center value for $\ln(r)$.
Owner: NcmFftlog
Flags: Read / Write / Construct Only
Default value: 0
“name” property “name” char *
FFTW Plan wisdown's name to perform the transformation.
Owner: NcmFftlog
Flags: Read
Default value: "fftlog_default_wisdown"
“nderivs” property“nderivs” guint
The number of derivatives to be estimated.
Owner: NcmFftlog
Flags: Read / Write / Construct
Default value: 0
“no-ringing” property“no-ringing” gboolean
True to use the no-ringing adjustment of $\ln(r_0)$ and False otherwise.
Owner: NcmFftlog
Flags: Read / Write / Construct
Default value: TRUE
“padding” property “padding” double
The padding percentage of the number of knots $N$.
Owner: NcmFftlog
Flags: Read / Write / Construct
Allowed values: >= 0
Default value: 1
“smooth-padding-scale” property “smooth-padding-scale” double
Log10 of the smoothing scale.
Owner: NcmFftlog
Flags: Read / Write / Construct
Default value: -200
“use-eval-int” property“use-eval-int” gboolean
Whether to use evaluation interval
Owner: NcmFftlog
Flags: Read / Write / Construct
Default value: FALSE
“use-smooth-padding” property“use-smooth-padding” gboolean
Whether to use a smooth padding.
Owner: NcmFftlog
Flags: Read / Write / Construct
Default value: FALSE