### Table Of Contents

- Generalized Universal Function API
  - Definitions
  - Details of Signature
  - C-API for implementing Elementary Functions

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# Generalized Universal Function API

There is a general need for looping over not only functions on scalars but also over functions on vectors (or arrays). This concept is realized in Numpy by generalizing the universal functions (ufuncs). In regular ufuncs, the elementary function is limited to element-by-element operations, whereas the generalized version (gufuncs) 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 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:

1. While an input array has a smaller dimensionality than the corresponding number of core dimensions, 1’s are pre-pended to its shape.
2. The core dimensions are removed from all inputs and the remaining dimensions are broadcasted; defining the loop dimensions.
3. The output is given by the loop dimensions plus the output core dimensions.

## Definitions

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:

    <Signature>            ::= <Input arguments> "->" <Output arguments>
    <Input arguments>      ::= <Argument list>
    <Output arguments>     ::= <Argument list>
    <Argument list>        ::= nil | <Argument> | <Argument> "," <Argument list>
    <Argument>             ::= "(" <Core dimension list> ")"
    <Core dimension list>  ::= nil | <Dimension name> |
                               <Dimension name> "," <Core dimension list>
    <Dimension name>       ::= valid Python variable name

Notes:

1. All quotes are for clarity.
2. Core dimensions that share the same name must be broadcastable, as the two `i` in our example above. Each dimension name typically corresponding to one level of looping in the elementary function’s implementation.
3. White spaces are ignored.

Here are some examples of signatures:

add | `(),()->()` |
inner1d | `(i),(i)->()` |
sum1d | `(i)->()` |
dot2d | `(m,n),(n,p)->(m,p)` | matrix multiplication
outer_inner | `(i,t),(j,t)->(i,j)` | inner over the last dimension, outer over the second to last, and loop/broadcast over the rest.

## C-API for implementing Elementary Functions

The current interface remains unchanged, and `PyUFunc_FromFuncAndData` can still be used to implement (specialized) ufuncs, consisting of scalar elementary functions.

One can use `PyUFunc_FromFuncAndDataAndSignature` to declare a more general ufunc. The argument list is the same as `PyUFunc_FromFuncAndData`, with an additional argument specifying the signature as C string.

Furthermore, the callback function is of the same type as before, `void (*foo)(char **args, intp *dimensions, intp *steps, void *func)`. When invoked, `args` is a list of length `nargs` containing the data of all input/output arguments. For a scalar elementary function, `steps` is also of length `nargs`, denoting the strides used for the arguments. `dimensions` is a pointer to a single integer defining the size of the axis to be looped over.

For a non-trivial signature, `dimensions` will also contain the sizes of the core dimensions as well, starting at the second entry. Only one size is provided for each unique dimension name and the sizes are given according to the first occurrence of a dimension name in the signature.

The first `nargs` elements of `steps` remain the same as for scalar ufuncs. The following elements contain the strides of all core dimensions for all arguments in order.

For example, consider a ufunc with signature `(i,j),(i)->()`. In this case, `args` will contain three pointers to the data of the input/output arrays `a`, `b`, `c`. Furthermore, `dimensions` will be `[N, I, J]` to define the size of `N` of the loop and the sizes `I` and `J` for the core dimensions `i` and `j`. Finally, `steps` will be `[a_N, b_N, c_N, a_i, a_j, b_i]`, containing all necessary strides.