Universal functions (ufunc) basics#
See also
A universal function (or ufunc for short) is a function that
operates on ndarrays in an element-by-element fashion,
supporting array broadcasting, type
casting, and several other standard features. That
is, a ufunc is a “vectorized” wrapper for a function
that takes a fixed number of specific inputs and produces a fixed number of
specific outputs.
In NumPy, universal functions are instances of the
numpy.ufunc class. Many of the built-in functions are
implemented in compiled C code. The basic ufuncs operate on scalars, but
there is also a generalized kind for which the basic elements are sub-arrays
(vectors, matrices, etc.), and broadcasting is done over other dimensions.
The simplest example is the addition operator:
>>> np.array([0,2,3,4]) + np.array([1,1,-1,2])
array([1, 3, 2, 6])
One can also produce custom numpy.ufunc instances using the
numpy.frompyfunc factory function.
Ufunc methods#
All ufuncs have four methods. They can be found at
Methods. However, these methods only make sense on scalar
ufuncs that take two input arguments and return one output argument.
Attempting to call these methods on other ufuncs will cause a
ValueError.
The reduce-like methods all take an axis keyword, a dtype
keyword, and an out keyword, and the arrays must all have dimension >= 1.
The axis keyword specifies the axis of the array over which the reduction
will take place (with negative values counting backwards). Generally, it is an
integer, though for numpy.ufunc.reduce, it can also be a tuple of
int to reduce over several axes at once, or None, to reduce over all
axes. For example:
>>> x = np.arange(9).reshape(3,3)
>>> x
array([[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> np.add.reduce(x, 1)
array([ 3, 12, 21])
>>> np.add.reduce(x, (0, 1))
36
The dtype keyword allows you to manage a very common problem that arises
when naively using ufunc.reduce. Sometimes you may
have an array of a certain data type and wish to add up all of its
elements, but the result does not fit into the data type of the
array. This commonly happens if you have an array of single-byte
integers. The dtype keyword allows you to alter the data type over which
the reduction takes place (and therefore the type of the output). Thus,
you can ensure that the output is a data type with precision large enough
to handle your output. The responsibility of altering the reduce type is
mostly up to you. There is one exception: if no dtype is given for a
reduction on the “add” or “multiply” operations, then if the input type is
an integer (or Boolean) data-type and smaller than the size of the
numpy.int_ data type, it will be internally upcast to the int_
(or numpy.uint) data-type. In the previous example:
>>> x.dtype
dtype('int64')
>>> np.multiply.reduce(x, dtype=float)
array([ 0., 28., 80.])
Finally, the out keyword allows you to
provide an output array (for single-output ufuncs, which are currently the only
ones supported; for future extension, however, a tuple with a single argument
can be passed in). If out is given, the dtype argument is ignored.
Considering x from the previous example:
>>> y = np.zeros(3, dtype=int)
>>> y
array([0, 0, 0])
>>> np.multiply.reduce(x, dtype=float, out=y)
array([ 0, 28, 80])
Ufuncs also have a fifth method, numpy.ufunc.at, that allows in place
operations to be performed using advanced indexing. No
buffering is used on the dimensions where
advanced indexing is used, so the advanced index can
list an item more than once and the operation will be performed on the result
of the previous operation for that item.
Output type determination#
The output of the ufunc (and its methods) is not necessarily an
ndarray, if all input arguments are not
ndarrays. Indeed, if any input defines an
__array_ufunc__ method,
control will be passed completely to that function, i.e., the ufunc is
overridden.
If none of the inputs overrides the ufunc, then
all output arrays will be passed to the
__array_prepare__ and
__array_wrap__ methods of the input (besides
ndarrays, and scalars) that defines it and has
the highest __array_priority__
of any other input to the universal function. The default
__array_priority__ of the
ndarray is 0.0, and the default __array_priority__ of a subtype
is 0.0. Matrices have __array_priority__ equal to 10.0.
All ufuncs can also take output arguments. If necessary, output will
be cast to the data-type(s) of the provided output array(s). If a class
with an __array__ method is used for the output,
results will be written to the object returned by __array__.
Then, if the class also has an __array_prepare__ method, it is
called so metadata may be determined based on the context of the ufunc (the
context consisting of the ufunc itself, the arguments passed to the ufunc, and
the ufunc domain.) The array object returned by
__array_prepare__ is passed to the ufunc for computation.
Finally, if the class also has an __array_wrap__ method, the
returned ndarray result will be passed to that method just before
passing control back to the caller.
Broadcasting#
See also
Each universal function takes array inputs and produces array outputs by performing the core function element-wise on the inputs (where an element is generally a scalar, but can be a vector or higher-order sub-array for generalized ufuncs). Standard broadcasting rules are applied so that inputs not sharing exactly the same shapes can still be usefully operated on.
By these rules, if an input has a dimension size of 1 in its shape, the first data entry in that dimension will be used for all calculations along that dimension. In other words, the stepping machinery of the ufunc will simply not step along that dimension (the stride will be 0 for that dimension).
Type casting rules#
Note
In NumPy 1.6.0, a type promotion API was created to encapsulate the
mechanism for determining output types. See the functions
numpy.result_type, numpy.promote_types, and
numpy.min_scalar_type for more details.
At the core of every ufunc is a one-dimensional strided loop that
implements the actual function for a specific type combination. When a
ufunc is created, it is given a static list of inner loops and a
corresponding list of type signatures over which the ufunc operates.
The ufunc machinery uses this list to determine which inner loop to
use for a particular case. You can inspect the .types attribute for a particular ufunc to see which type
combinations have a defined inner loop and which output type they
produce (character codes are used
in said output for brevity).
Casting must be done on one or more of the inputs whenever the ufunc does not have a core loop implementation for the input types provided. If an implementation for the input types cannot be found, then the algorithm searches for an implementation with a type signature to which all of the inputs can be cast “safely.” The first one it finds in its internal list of loops is selected and performed, after all necessary type casting. Recall that internal copies during ufuncs (even for casting) are limited to the size of an internal buffer (which is user settable).
Note
Universal functions in NumPy are flexible enough to have mixed type
signatures. Thus, for example, a universal function could be defined
that works with floating-point and integer values. See
numpy.ldexp for an example.
By the above description, the casting rules are essentially
implemented by the question of when a data type can be cast “safely”
to another data type. The answer to this question can be determined in
Python with a function call: can_cast(fromtype, totype). The example below shows the results of this call for
the 24 internally supported types on the author’s 64-bit system. You
can generate this table for your system with the code given in the example.
Example
Code segment showing the “can cast safely” table for a 64-bit system. Generally the output depends on the system; your system might result in a different table.
>>> mark = {False: ' -', True: ' Y'}
>>> def print_table(ntypes):
... print('X ' + ' '.join(ntypes))
... for row in ntypes:
... print(row, end='')
... for col in ntypes:
... print(mark[np.can_cast(row, col)], end='')
... print()
...
>>> print_table(np.typecodes['All'])
X ? b h i l q p B H I L Q P e f d g F D G S U V O M m
? Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y - Y
b - Y Y Y Y Y Y - - - - - - Y Y Y Y Y Y Y Y Y Y Y - Y
h - - Y Y Y Y Y - - - - - - - Y Y Y Y Y Y Y Y Y Y - Y
i - - - Y Y Y Y - - - - - - - - Y Y - Y Y Y Y Y Y - Y
l - - - - Y Y Y - - - - - - - - Y Y - Y Y Y Y Y Y - Y
q - - - - Y Y Y - - - - - - - - Y Y - Y Y Y Y Y Y - Y
p - - - - Y Y Y - - - - - - - - Y Y - Y Y Y Y Y Y - Y
B - - Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y - Y
H - - - Y Y Y Y - Y Y Y Y Y - Y Y Y Y Y Y Y Y Y Y - Y
I - - - - Y Y Y - - Y Y Y Y - - Y Y - Y Y Y Y Y Y - Y
L - - - - - - - - - - Y Y Y - - Y Y - Y Y Y Y Y Y - -
Q - - - - - - - - - - Y Y Y - - Y Y - Y Y Y Y Y Y - -
P - - - - - - - - - - Y Y Y - - Y Y - Y Y Y Y Y Y - -
e - - - - - - - - - - - - - Y Y Y Y Y Y Y Y Y Y Y - -
f - - - - - - - - - - - - - - Y Y Y Y Y Y Y Y Y Y - -
d - - - - - - - - - - - - - - - Y Y - Y Y Y Y Y Y - -
g - - - - - - - - - - - - - - - - Y - - Y Y Y Y Y - -
F - - - - - - - - - - - - - - - - - Y Y Y Y Y Y Y - -
D - - - - - - - - - - - - - - - - - - Y Y Y Y Y Y - -
G - - - - - - - - - - - - - - - - - - - Y Y Y Y Y - -
S - - - - - - - - - - - - - - - - - - - - Y Y Y Y - -
U - - - - - - - - - - - - - - - - - - - - - Y Y Y - -
V - - - - - - - - - - - - - - - - - - - - - - Y Y - -
O - - - - - - - - - - - - - - - - - - - - - - - Y - -
M - - - - - - - - - - - - - - - - - - - - - - Y Y Y -
m - - - - - - - - - - - - - - - - - - - - - - Y Y - Y
You should note that, while included in the table for completeness, the ‘S’, ‘U’, and ‘V’ types cannot be operated on by ufuncs. Also, note that on a 32-bit system the integer types may have different sizes, resulting in a slightly altered table.
Mixed scalar-array operations use a different set of casting rules that ensure that a scalar cannot “upcast” an array unless the scalar is of a fundamentally different kind of data (i.e., under a different hierarchy in the data-type hierarchy) than the array. This rule enables you to use scalar constants in your code (which, as Python types, are interpreted accordingly in ufuncs) without worrying about whether the precision of the scalar constant will cause upcasting on your large (small precision) array.
Use of internal buffers#
Internally, buffers are used for misaligned data, swapped data, and
data that has to be converted from one data type to another. The size
of internal buffers is settable on a per-thread basis. There can
be up to \(2 (n_{\mathrm{inputs}} + n_{\mathrm{outputs}})\)
buffers of the specified size created to handle the data from all the
inputs and outputs of a ufunc. The default size of a buffer is
10,000 elements. Whenever buffer-based calculation would be needed,
but all input arrays are smaller than the buffer size, those
misbehaved or incorrectly-typed arrays will be copied before the
calculation proceeds. Adjusting the size of the buffer may therefore
alter the speed at which ufunc calculations of various sorts are
completed. A simple interface for setting this variable is accessible
using the function numpy.setbufsize.
Error handling#
Universal functions can trip special floating-point status registers
in your hardware (such as divide-by-zero). If available on your
platform, these registers will be regularly checked during
calculation. Error handling is controlled on a per-thread basis,
and can be configured using the functions numpy.seterr and
numpy.seterrcall.
Overriding ufunc behavior#
Classes (including ndarray subclasses) can override how ufuncs act on them by defining certain special methods. For details, see Standard array subclasses.