Python functions implementing numerical integration of functions in log-space.

# lintegrate

A numerical integration library for when you want/need to work with the natural logarithm of the function requiring integration.

This library provides three numerical integration functions, heavily based on GSL functions, to integrate a function when only its natural logarithm is given, and return the natural logarithm of that integral. The three functions:

• `lintegrate_qag`
• `lintegrate_qng`
• `lintegrate_cquad`

are equivalents of the GSL functions:

respectively. These can be useful when, e.g., you can calculate the natural logarithm of a Gaussian likelihood function (in cases where the exponentiation of the Gaussian function would lead to zeros or infinities) and you want to numerically find the integral of the Gaussian function itself.

The functions `lintegrate_qag`, `lintegrate_qng`, and `lintegrate_cquad`, all have wrappers functions (with `_split` appended to their names) that allow the user to specify a set of intervals that the integrals will be split into when performing the calculation. The intervals could, for example, be spaced evenly in log-space, for cases where the integral function has a very pronounced peak as it approaches zero.

## Example

An example of the use the functions is:

```/* example using lintegrate functionality */

#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <gsl/gsl_math.h>
#include <gsl/gsl_integration.h>

#include <lintegrate.h>

/* create function for integration */
double lintegrand(double x, void *params);

struct intparams {
double mu;
double sig;
};

double lintegrand(double x, void *params){
struct intparams * p = (struct intparams *)params;
double mu = p->mu;
double sig = p->sig;

return -0.5*(mu-x)*(mu-x)/(sig*sig);
}

double integrand(double x, void *params){
struct intparams * p = (struct intparams *)params;
double mu = p->mu;
double sig = p->sig;

return exp(-0.5*(mu-x)*(mu-x)/(sig*sig));
}

int main( int argv, char **argc ){
gsl_function F;
struct intparams params;
gsl_integration_workspace *w = gsl_integration_workspace_alloc (100);
double abserr = 0.;
size_t neval = 0;

double minlim = -6.; /* minimum for integration range */
double maxlim = 6.;  /* maximum for integration range */

double abstol = 1e-10; /* absolute tolerance */
double reltol = 1e-10; /* relative tolerance */

params.mu = 0.;
params.sig = 1.;

F.function = &lintegrand;
F.params = &params;

/* integrate log of function using QAG */
lintegration_qag(&F, minlim, maxlim, abstol, reltol, 100, GSL_INTEG_GAUSS31, w, &qaganswer, &abserr);

/* integrate log of function using QNG */
lintegration_qng(&F, minlim, maxlim, abstol, reltol, &qnganswer, &abserr, &neval);

/* integrate log of function using CQUAD */

/* integrate function using GSL QAG */
F.function = &integrand;
gsl_integration_qag(&F, minlim, maxlim, abstol, reltol, 100, GSL_INTEG_GAUSS31, w, &answer, &abserr);

gsl_integration_workspace_free(w);

fprintf(stdout, "Analytical answer = %.8lf\n", log(sqrt(2.*M_PI)));

return 0;
}
```

## Requirements

• GSL - on Debian/Ubuntu (16.04) install with e.g. `sudo apt-get install libgsl-dev`

## Installation

The library can be built using scons by just typing `sudo scons` in the base directory. To install the library system-wide (in `/usr/local/lib` by default) run:

``````sudo scons
sudo scons install
``````

A Python module containing wrappers to the functions can be built by running, e.g.:

``````sudo python setup.py install
``````

for a system-wide install (add `--user` and remove `sudo` if just wanting to install for a single user, and using `--prefix=INSTALLPATH` if wanting to specify this install location).

The Python module can also be installed from PyPI using pip with:

```pip install lintegrate
```

or in a Conda environment with:

```conda install -c conda-forge lintegrate
```

## Python

If the Python module has been installed it has the following functions:

• `lqng` - a wrapper to `lintegration_qng`
• `lqag` - a wrapper to `lintegration_qag`
• `lcquad` - a wrapper to `lintegration_cquad`
• `logtrapz` - using the trapezium rule for integration on a grid of values

The `lqng`, `lqag`, and `lcquad` functions are used in a similar way to the scipy `quad` function.

An example of their use would be:

```from lintegrate import lqag, lqng, lcquad, logtrapz
import numpy as np

# define the log of the function to be integrated
def integrand(x, args):
mu, sig = args # unpack extra arguments
return -0.5*((x-mu)/sig)**2

# set integration limits
xmin = -6.
xmax = 6.

mu = 0.
sig = 1.

resqag = lqag(integrand, xmin, xmax, args=(mu, sig))
resqng = lqng(integrand, xmin, xmax, args=(mu, sig))
restrapz = logtrapz(integrand, np.linspace(xmin, xmax, 100), args=(mu, sig))
```

## R

In R one can use the reticulate package to call the functions in `lintegrate`. The above example would be:

```library(reticulate)
lint <- import("lintegrate", convert = FALSE)
integrand <- function(x, args){
mu = args
sig = args
return(-.5 * ((x-mu)/sig)^2 )
}
integrand <- Vectorize(integrand)
mu <- 0
sig <- 1
mmin <- -10
mmax <- 10
lint\$lqag(py_func(integrand), r_to_py(mmin), r_to_py(mmax), c(mu, sig))
```

## Project details

This version 0.1.3 0.1.2 0.1.1 0.1.0 0.0.12 0.0.11 0.0.10 0.0.9 0.0.8 0.0.7 0.0.6 0.0.5 0.0.4