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Generalized Universal Function API
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There is a general need for looping over not only functions on scalars
but also over functions on vectors (or arrays), as explained on
http://scipy.org/scipy/numpy/wiki/GeneralLoopingFunctions. We propose
to realize this concept by generalizing the universal functions
(ufuncs), and provide a C implementation that adds ~500 lines
to the numpy code base. In current (specialized) ufuncs, the elementary
function is limited to element-by-element operations, whereas the
generalized version supports "sub-array" by "sub-array" operations.
The Perl vector library PDL provides a similar functionality and its
terms are re-used in the following.
Each generalized ufunc has information associated with it that states
what the "core" dimensionality of the inputs is, as well as the
corresponding dimensionality of the outputs (the element-wise ufuncs
have zero core dimensions). The list of the core dimensions for all
arguments is called the "signature" of a ufunc. For example, the
ufunc numpy.add has signature ``(),()->()`` defining two scalar inputs
and one scalar output.
Another example is (see the GeneralLoopingFunctions page) the function
``inner1d(a,b)`` with a signature of ``(i),(i)->()``. This applies the
inner product along the last axis of each input, but keeps the
remaining indices intact. For example, where ``a`` is of shape ``(3,5,N)``
and ``b`` is of shape ``(5,N)``, this will return an output of shape ``(3,5)``.
The underlying elementary function is called 3*5 times. In the
signature, we specify one core dimension ``(i)`` for each input and zero core
dimensions ``()`` for the output, since it takes two 1-d arrays and
returns a scalar. By using the same name ``i``, we specify that the two
corresponding dimensions should be of the same size (or one of them is
of size 1 and will be broadcasted).
The dimensions beyond the core dimensions are called "loop" dimensions. In
the above example, this corresponds to ``(3,5)``.
The usual numpy "broadcasting" rules apply, where the signature
determines how the dimensions of each input/output object are split
into core and loop dimensions:
#. While an input array has a smaller dimensionality than the corresponding
number of core dimensions, 1's are pre-pended to its shape.
#. The core dimensions are removed from all inputs and the remaining
dimensions are broadcasted; defining the loop dimensions.
#. The output is given by the loop dimensions plus the output core dimensions.
Definitions
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Elementary Function
Each ufunc consists of an elementary function that performs the
most basic operation on the smallest portion of array arguments
(e.g. adding two numbers is the most basic operation in adding two
arrays). The ufunc applies the elementary function multiple times
on different parts of the arrays. The input/output of elementary
functions can be vectors; e.g., the elementary function of inner1d
takes two vectors as input.
Signature
A signature is a string describing the input/output dimensions of
the elementary function of a ufunc. See section below for more
details.
Core Dimension
The dimensionality of each input/output of an elementary function
is defined by its core dimensions (zero core dimensions correspond
to a scalar input/output). The core dimensions are mapped to the
last dimensions of the input/output arrays.
Dimension Name
A dimension name represents a core dimension in the signature.
Different dimensions may share a name, indicating that they are of
the same size (or are broadcastable).
Dimension Index
A dimension index is an integer representing a dimension name. It
enumerates the dimension names according to the order of the first
occurrence of each name in the signature.
Details of Signature
--------------------
The signature defines "core" dimensionality of input and output
variables, and thereby also defines the contraction of the
dimensions. The signature is represented by a string of the
following format:
* Core dimensions of each input or output array are represented by a
list of dimension names in parentheses, ``(i_1,...,i_N)``; a scalar
input/output is denoted by ``()``. Instead of ``i_1``, ``i_2``,
etc, one can use any valid Python variable name.
* Dimension lists for different arguments are separated by ``","``.
Input/output arguments are separated by ``"->"``.
* If one uses the same dimension name in multiple locations, this
enforces the same size (or broadcastable size) of the corresponding
dimensions.
The formal syntax of signatures is as follows::
::= "->"