# Multiplicatively Convolutional Fast Integral Transforms

## Features

- Compute integral transforms
- Inverse transform without analytic inversion
- Integral kernels as derivatives
- Transform input array along any axis
- Output the matrix form
- 1-to-n transform for multiple kernels (TODO)
- Easily extensible for other kernels

## Installation

`pip install mcfit`

## Algorithm

mcfit computes integral transforms of the form

G(y) = \int_0^\infty F(x) K(xy) \frac{dx}x

where `F(x)` is the input function, `G(y)` is the output function,
and `K(xy)` is the integral kernel.
One is free to scale all three functions by a power law

g(y) = \int_0^\infty f(x) k(xy) \frac{dx}x

where `f(x)=x^{-q}F(x)`, `g(y)=y^q G(y)`, and `k(t)=t^q K(t)`.
And `q` is a tilt parameter serving to shift power of `x` between
the input function and the kernel.

mcfit implements the FFTLog algorithm.
The idea is to take advantage of the convolution theorem in `\ln x` and
`\ln y`.
It approximates the input function with truncated Fourier series over one
period of a periodic approximant, and use the exact Fourier transform of the
kernel.
One can calculate the latter analytically as a Mellin transform.
This algorithm is optimal when the input function is smooth in `\ln x`,
and is ideal for oscillatory kernels with input spanning a wide range in
`\ln x`.

## Examples

One can perform the following pair of Hankel transforms

e^{-y} &= \int_0^\infty (1+x^2)^{-\frac32} J_0(xy) x dx (1+y^2)^{-\frac32} &= \int_0^\infty e^{-x} J_0(xy) x dx

easily as follows

```
def F_fun(x): return 1 / (1 + x*x)**1.5
def G_fun(y): return numpy.exp(-y)
from mcfit import Hankel
x = numpy.logspace(-3, 3, num=60, endpoint=False)
F = F_fun(x)
H = Hankel(x, lowring=True)
y, G = H(F, extrap=True)
numpy.allclose(G, G_fun(y), rtol=1e-8, atol=1e-8)
y = numpy.logspace(-4, 2, num=60, endpoint=False)
G = G_fun(y)
H_inv = Hankel(y, lowring=True)
x, F = H_inv(G, extrap=True)
numpy.allclose(F, F_fun(x), rtol=1e-10, atol=1e-10)
```

Cosmologists often need to transform a power spectrum to its correlation function

```
from mcfit import P2xi
k, P = numpy.loadtxt('P.txt', unpack=True)
r, xi = P2xi(k)(P)
```

and the other way around

```
from mcfit import xi2P
r, xi = numpy.loadtxt('xi.txt', unpack=True)
k, P = xi2P(r)(xi)
```

Similarly for the quadrupoles

```
k, P2 = numpy.loadtxt('P2.txt', unpack=True)
r, xi2 = P2xi(k, l=2)(P2)
```

Also useful to the cosmologists is the tool below that computes the variance of
the overdensity field as a function of radius, from which `\sigma_8` can
be interpolated.

```
from mcfit import TophatVar
R, var = TophatVar(k, lowring=True)(P, extrap=True)
from scipy.interpolate import CubicSpline
varR = CubicSpline(R, var)
sigma8 = numpy.sqrt(varR(8))
```