pynverse 0.1.4.4
A library for calculating the numerical inverse of a function
License: MIT
Keywords: inverse,function,numerical

pynverse |PyPI version|
=======================

A module specialized on calculating the numerical inverse of any
invertible continuous function.

Requirements
------------

|Scipy| |Numpy|

Install
-------

In order to install this tool you'll need ``pip``:

::

    pip install pynverse

Usage
-----

Pynverse provides a main function ``inversefunc`` that calculates the
numerical inverse of a function ``f`` passed as the first argument in
the form of a callable.

.. code:: python

        >>> from pynverse import inversefunc

It can be used to calculate the inverse function at certain ``y_values``
points:

.. code:: python

        >>> cube = (lambda x: x**3)
        >>> invcube = inversefunc(cube, y_values=3)
        array(3.0000000063797567)

Or to obtain a callable that will calculate the inverse values at any
other points if ``y_values`` is not provided:

.. code:: python

        >>> invcube = inversefunc(cube)
        >>> invcube(27)
        array(3.0000000063797567)

It requires the function to be continuous and strictly monotonic (i.e.
purely decreasing or purely increasing) within the domain of the
function. By default, the domain includes all real numbers, but it can
be restricted to an inverval using the ``domain`` argument:

.. code:: python

        >>> import numpy as np
        >>> inversefunc(np.cos, y_values=[1, 0, -1], # Should give [0, pi / 2, pi]
        ...             domain=[0, np.pi])
        array([ 0.        ,  1.57079632,  3.14159265])

Additionally, the argument ``open_domain`` can be used to specify the
open/closed character of each of the ends of the domain interval:

.. code:: python

        >>> inversefunc(np.log10, y_values=-2, # Should give 0.01
        ...             domain=0, open_domain=[True, False])
        array(0.0099999999882423)

Or on both ends simultaneously:

.. code:: python

        >>> invtan = inversefunc(np.tan,
        ...                      domain=[-np.pi / 2, np.pi / 2],
        ...                      open_domain=True)
        >>> invtan([1, 0, -1]) # Should give [pi / 4, 0, -pi / 4]
        array([  7.85398163e-01,   1.29246971e-26,  -7.85398163e-01])

Additional parameters may be passed to the function for easier
reusability of callables using the ``args`` argument:

.. code:: python

        >>> invsquare = inversefunc(np.power, args=(2), domain=0)
        >>> invsquare([4, 16, 64])
        array([ 2.,  4.,  8.])

The image of the function in the interval may be also provided for cases
where the function is non continuous right at the end of an open
interval with the ``image`` argument:

.. code:: python

        >>> invmod = inversefunc(np.mod, args=(1), domain=[5,6], 
        ...                      open_domain=[False,True], image=[0,1])
        >>> invmod([0.,0.3,0.5])
        array([ 5. ,  5.3,  5.5])

Additionally an argument can be used to check for the number of digits
of accuracy in the results, giving a warning in case this is not meet:

.. code:: python

        >>> inversefunc(np.log10, y_values=-8, # Should give 0.01
        ...             domain=0, open_domain=True, accuracy=6)
        pynverse\inverse.py:195: RuntimeWarning: Results obtained with less than 6 decimal digits of accuracy
        array(9.999514710830838e-09)

As it is compatible with arrays, it can very easily used to obtain the
inverse for broad ranges. These are some examples of using the returned
numerical inverse callables with arrays to make plots, and compare them
to the analytical inverse, each of them calculated as simply as:

.. code:: python

    log = lambda x: np.log10(x)
    invlog = scipy.misc.inversefunc(log, domain=0, open_domain=True)
    x1=np.linspace(0.00001,10,100)
    x2=np.linspace(-5,1,100)
    ax1.plot(x1,log(y1),'b-')
    ax2.plot(x2,invlog(x2),'b-')

    invlog_a = lambda x: 10**x
    ax2.plot(x2,invlog_a(x2),'r--')

.. figure:: https://cloud.githubusercontent.com/assets/12649253/19738042/cf22460a-9bad-11e6-9c17-6fdd6cda0991.png
   :alt: 

In particular, for the definition of piecewise functions, there is a
``piecewise`` utility function provided that solves the issues of
np.piecewise when working with both scalars and arrays. For example, the
inverse for the last plot was obtained as:

.. code:: python

    from pynverse import inversefunc, piecewise

    pw=lambda x: piecewise(x,[x<1,(x>=1)*(x<3),x>=3],[lambda x: x, lambda x: x**2, lambda x: x+6])
    invpw =inversefunc(pw) 

Disclaimer
----------

The problem of calculating the numerical inverse of an arbitrary funtion
in unlimited or open intervals is still an open question in applied
mathematics. The main purpose of this package is not to be fast, or as
accurate as it could be if the inverse was calculated specifically for a
known function, using more specialised techniques. The current
implementation essentially uses the existing tools in scipy to solve the
particular problem of finding the inverse of a function meeting the
continuity and monotonicity conditions, but while it performs really
well it may fail under certain conditions. For example when inverting a
``log10`` it is known to start giving inccacurate values when being
asked to invert -10, which should correspond to 0.0000000001 (1e-10),
but gives instead 0.0000000000978 (0.978e-10).

The advantage about estimating the inverse function is that the accuracy
can always be verified by checking if f(finv(x))==x..

Details about the implementation
--------------------------------

The summarized internal strategy is the following:

1. Figure out if the function is increasing or decreasing. For this two
   reference points ref1 and ref2 are needed:

   -  In case of a finite interval, the points ref points are 1/4 and
      3/4 through the interval.
   -  In an infinite interval any two values work really.
   -  If f(ref1)<f(ref2), the function is increasing, otherwise is
      decreasing.

2. Figure out the image of the function in the interval.

   -  If values are provided, then those are used.
   -  In a closed interval just calculate f(a) and f(b), where a and b
      are the ends of the interval.
   -  In an open interval try to calculate f(a) and f(b), if this works
      those are used, otherwise it will be assume to be (-Inf, Inf).

3. Built a bounded function with the following conditions:

   -  bounded\_f(x):

      -  return -Inf if x below interval, and f is increasing.
      -  return +Inf if x below interval, and f is decreasing.
      -  return +Inf if x above interval, and f is increasing.
      -  return -Inf if x above interval, and f is decreasing.
      -  return f(x) otherwise

4. If the required number y0 for the inverse is outside the image, raise
   an exception.
5. Find roots for bounded\_f(x)-y0, by minimizing
   (bounded\_f(x)-y0)\*\*2, using the ``Brent`` method, making sure that
   the algorithm for minimising starts in a point inside the original
   interval by setting ref1, ref2 as brackets. As soon as if goes
   outside the allowed intervals, bounded\_f returns infinite, forcing
   the algorithm to go back to search inside the interval.
6. Check that the solutions are accurate and they meet f(x0)=y0 to some
   desired precision, raising a warning otherwise.

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