# pyroots

## Abstract

A Python 3 library implementing various root finding methods for single-variable functions.

Currently the following methods have been implemented:

• The bisect method.
• The ridder method.
• The brent method (there are 2 implementations, one following the Wikipedia article and one following the scipy implementation. The latter seems to converge faster).

## Rationale

The functionality of pyroots is already implemented in scipy, so the natural question is why rediscover the wheel?

Well, the main reason is that scipy is a huge dependency. Pyroots on the other hand is just a single package that is easily installed and that you can easily bundle with py2exe or similar projects. It doesn't even need to get installed, just throw the pyroots folder in your project and you are ready to go.

Apart from that, the API used by scipy's functions is not very user-friendly. For example you can't use keyword arguments for your functions. Finally, in scipy there is no reliable way to define how many digits of accuracy you want in the obtained root. For example, you may ask for 6 digits, but scipy may calculate up to 14 (or 12 or whatever) digits. The main implication of this "glitch" is that scipy's method may evaluate the function more times than those really needed. If the function calculates something trivial like the functions in the following examples, then these extra function calls are no big deal, but if your functions take significant time to evaluate (e.g. more than seconds), then this can quickly become annoying, or even, simply unacceptable (e.g. the function takes some minutes to return a value).

## Usage

All the functions share the same API, so you can easily switch between the, methods, although, it should be noted that the method you probably want to use is brent.

Anyway, the function whose root you are searching must take at least a single argument and return a single number. This first argument is also the dependent variable and, apart from that, the function can also take any number of positional/keyword arguments. For example the following functions are totally valid ones:

def f(x, a):
return x ** 2 - a + 1

def g(x, a, b, c=3):
# calculations
return x ** 2 + a ** b - c


In order to find the root of f you must first define an interval that contains the root. Let's say that this interval is defined as [xa, xb]. In that case you call the methods of pyroots like this:

result = brent(f, xa, xb, a)


All the methods return a Result object that has the following attributes:

result.x0               # the root
result.fx0              # the value of f(x0)
result.convergence      # True/False
result.iterations       # the number of iterations
result.func_calls       # the number of function evaluations.
result.msg              # a descriptive message regarding the convergence (or the failure of convergence)


If, for some reason, convergence cannot be achieved, then a ConvergenceError is raised. If you don't want that to happen, then you have to pass False as the value of raise_on_fail argument:

def f(x):
return x ** 2 - 1

result = brent(f, xa=-10, xb=-5, raise_on_fail=False):
print(result)    # Outputs the following line
# iter=  0, func_calls=  2, convergence=False, x0=None, f(x0)=None, msg=Root is not bracketed.


## API

The definition of each root-finding function is the following one:

brent(f, xa, xb, *args, ftol=1e-6, xtol=EPS, max_iter=500, raise_on_fail=True, **kwargs)


We have already covered f, xa and xb. At first, let's examine the arguments that regard the function that we want to solve:

• f is the function whose root we are searching.
• xa is the lower bracket of the interval of the solution we search.
• xb is the upper bracket of the interval of the solution we search.
• *args are passed as positional arguments when f is evaluated.
• **kwargs are passed as keyword arguments when f is evaluated.

The remaining arguments are "keyword only arguments" (a feature unique in Python 3) and they are the following ones:

• ftol is the required precision of the solution, i.e. a solution is achieved when |f(x0)| is smaller than ftol.
• xtol is
• max_iter is the maximum allowed number of iterations.
• raise_on_fail is a boolean flag indicating whether or not an exception should be raised if convergence fails. It defaults to True

## Example

A simple example:

from pyroots import brent

# define a function f(x) that returns a single variable.
def f(x, y, z):
return x**y + z

# solve using the brent method
result = brent(f, -10, 10, y=23, z=0.3, ftol=1e-5)


## Installation

with pip:

pip install pyroots


or from source:

python setup.py install
`

## Documentation

For the time being documentation is not yet ready, but when it is prepared, it will be uploaded at: