Units
=====  

Casting rules
-------------
In Brian 1, a distinction is made between scalars and numpy arrays (including
scalar arrays): Scalars could be multiplied with a unit, resulting in a Quantity
object whereas the multiplication of an array with a unit resulted in a
(unitless) array. Accordingly, scalars were considered as dimensionless
quantities for the purpose of unit checking (e.g.. 1 + 1 * mV raised an error)
whereas arrays were not (e.g. array(1) + 1 * mV resulted in 1.001 without any
errors). Brian 2 no longer makes this distinction and treats both scalars and
arrays as dimensionless for unit checking and make all operations involving
quantities return a quantity.::

    >>> 1 + 1*second   # doctest: +ELLIPSIS
    Traceback (most recent call last):
    ...
    DimensionMismatchError: Cannot calculate 1. s + 1, units do not match (units are second and 1).

    >>> np.array([1]) + 1*second   # doctest: +ELLIPSIS
    Traceback (most recent call last):
    ...
    DimensionMismatchError: Cannot calculate 1. s + [1], units do not match (units are second and 1).

    >>> 1*second + 1*second
    2. * second
    >>> np.array([1])*second + 1*second
    array([ 2.]) * second

As one exception from this rule, a scalar or array ``0`` is considered as having
"any unit", i.e. ``0 + 1 * second`` will result in ``1 * second`` without a
dimension mismatch error and ``0 == 0 * mV`` will evaluate to ``True``. This
seems reasonable from a mathematical viewpoint and makes some sources of error
disappear. For example, the Python builtin ``sum`` (not numpy's version) adds
the value of the optional argument ``start``, which defaults to 0, to its
main argument. Without this exception, ``sum([1 * mV, 2 * mV])`` would therefore
raise an error.

The above rules also apply to all comparisons (e.g. ``==`` or ``<``) with one
further exception: ``inf`` and ``-inf`` also have "any unit", therefore an
expression like ``v <= inf`` will never raise an exception (and always return
``True``).

Functions and units
-------------------

ndarray methods
~~~~~~~~~~~~~~~
All methods that make sense on quantities should work, i.e. they check for the
correct units of their arguments and return quantities with units were
appropriate. Most of the methods are overwritten using thin function wrappers:

``wrap_function_keep_dimension``:
	Strips away the units before giving the array to the method of ``ndarray``,
	then reattaches the unit to the result (examples: sum, mean, max)

``wrap_function_change_dimension``:
	Changes the dimensions in a simple way that is independent of function
	arguments, the shape of the array, etc. (examples: sqrt, var, power)

``wrap_function_dimensionless``:
	Raises an error if the method is called on a quantity with dimensions (i.e.
	it works on dimensionless quantities). 

**List of methods**

``all``, ``any``, ``argmax``, ``argsort``, ``clip``, ``compress``, ``conj``, ``conjugate``, 
``copy``, ``cumsum``, ``diagonal``, ``dot``, ``dump``, ``dumps``, ``fill``, ``flatten``, ``getfield``, 
``item``, ``itemset``, ``max``, ``mean``, ``min``, ``newbyteorder``, ``nonzero``, ``prod``, ``ptp``, 
``put``, ``ravel``, ``repeat``, ``reshape``, ``round``, ``searchsorted``, ``setasflat``, ``setfield``, 
``setflags``, ``sort``, ``squeeze``, ``std``, ``sum``, ``take``, ``tolist``, ``trace``, ``transpose``, 
``var``, ``view``

**Notes**

* Methods directly working on the internal data buffer (``setfield``,
  ``getfield``, ``newbyteorder``) ignore the dimensions of the quantity.
* The type of a quantity cannot be int, therefore ``astype`` does not quite
  work when trying to convert the array into integers.
* ``choose`` is only defined for integer arrays and therefore does not work
* ``tostring`` and ``tofile`` only return/save the pure array data without the
  unit (but you can use ``dump`` or ``dumps`` to pickle a quantity array)
* ``resize`` does not work: ``ValueError: cannot resize this array: it does not
  own its data``
* ``cumprod`` would result in different dimensions for different elements and is
  therefore forbidden
* ``item`` returns a pure Python float by definition
* ``itemset`` does not check for units

Numpy ufuncs
~~~~~~~~~~~~

All of the standard `numpy ufuncs`_ (functions that operate element-wise on numpy
arrays) are supported, meaning that they check for correct units and return
appropriate arrays. These functions are often called implicitly, for example
when using operators like ``<`` or ``**``.

*Math operations:*
	``add``, ``subtract``, ``multiply``, ``divide``, ``logaddexp``, ``logaddexp2``, 
        ``true_divide``, ``floor_divide``, ``negative``, ``power``, ``remainder``, ``mod``, 
        ``fmod``, ``absolute``, ``rint``, ``sign``, ``conj``, ``conjugate``, ``exp``, ``exp2``, 
        ``log``, ``log2``, ``log10``, ``expm1``, ``log1p``, ``sqrt``, ``square``, ``reciprocal``, 
        ``ones_like``
	
*Trigonometric functions:*
	``sin``, ``cos``, ``tan``, ``arcsin``, ``arccos``, ``arctan``, ``arctan2``, ``hypot``, 
        ``sinh``, ``cosh``, ``tanh``, ``arcsinh``, ``arccosh``, ``arctanh``, ``deg2rad``, ``rad2deg``

*Bitwise functions:*
	``bitwise_and``, ``bitwise_or``, ``bitwise_xor``, ``invert``, ``left_shift``, ``right_shift``

*Comparison functions:* 
	``greater``, ``greater_equal``, ``less``, ``less_equal``, ``not_equal``,
	``equal``, ``logical_and``, ``logical_or``, ``logical_xor``, ``logical_not``, ``maximum``, 
        ``minimum``
	
*Floating functions:*
	``isreal``, ``iscomplex``, ``isfinite``, ``isinf``, ``isnan``, ``floor``, ``ceil``, ``trunc``, 
        ``fmod``

Not taken care of yet: ``signbit``, ``copysign``, ``nextafter``, ``modf``, ``ldexp``, ``frexp``

**Notes**

* Everything involving ``log`` or ``exp``, as well as trigonometric functions
  only works on dimensionless array (for ``arctan2`` and ``hypot`` this is
  questionable, though)
* Unit arrays can only be raised to a scalar power, not to an array of
  exponents as this would lead to differing dimensions across entries. For
  simplicity, this is enforced even for dimensionless quantities.
* Bitwise functions never works on quantities (numpy will by itself throw a 
  ``TypeError`` because they are floats not integers).
* All comparisons only work for matching dimensions (with the exception of
  always allowing comparisons to 0) and return a pure boolean array.
* All logical functions treat quantities as boolean values in the same
  way as floats are treated as boolean: Any non-zero value is True.

.. _numpy ufuncs: http://docs.scipy.org/doc/numpy/reference/ufuncs.html

Numpy functions
~~~~~~~~~~~~~~~
Many numpy functions are functional versions of ndarray methods (e.g. ``mean``,
``sum``, ``clip``). They therefore work automatically when called on quantities,
as numpy propagates the call to the respective method.

There are some functions in numpy that do not propagate their call to the
corresponding method (because they use np.asarray instead of np.asanyarray,
which might actually be a bug in numpy): ``trace``, ``diagonal``, ``ravel``,
``dot``. For these, wrapped functions in ``unitsafefunctions.py`` are provided.

**Wrapped numpy functions in unitsafefunctions.py**

These functions are thin wrappers around the numpy functions to correctly check
for units and return quantities when appropriate:

``log``, ``exp``, ``sin``, ``cos``, ``tan``, ``arcsin``, ``arccos``, ``arctan``, ``sinh``, 
``cosh``, ``tanh``, ``arcsinh``, ``arccosh``, ``arctanh``, ``diagonal``, ``ravel``, ``trace``, 
``dot``
         
**numpy functions that work unchanged**

This includes all functional counterparts of the methods mentioned above (with
the exceptions mentioned above). Some other functions also work correctly, as
they are only using functions/methods that work with quantities:

* ``linspace``, ``diff``, ``digitize`` [1]_
* ``trim_zeros``, ``fliplr``, ``flipud``, ``roll``, ``rot90``, ``shuffle``
* ``corrcoeff`` [1]_

.. [1] But does not care about the units of its input.

**numpy functions that return a pure numpy array instead of quantities**

* ``arange``
* ``cov``
* ``random.permutation``
* ``histogram``, ``histogram2d``
* ``cross``, ``inner``, ``outer``
* ``where``

**numpy functions that do something wrong**

* ``insert``, ``delete`` (return a quantity array but without units)
* ``correlate`` (returns a quantity with wrong units)
* ``histogramdd`` (raises a ``DimensionMismatchError``)

User-defined functions and units
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For performance and simplicity reasons, code within the Brian core does not use
Quantity objects but unitless numpy arrays instead. See :doc:`functions` for
details on how to make use user-defined functions with Brian's unit system.

