Universal functions (ufunc)#
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. For detailed information on universal functions, see
Universal functions (ufunc) basics.
ufunc#
Functions that operate element by element on whole arrays. |
Optional keyword arguments#
All ufuncs take optional keyword arguments. Most of these represent advanced usage and will not typically be used.
out
The first output can be provided as either a positional or a keyword parameter. Keyword ‘out’ arguments are incompatible with positional ones.
The ‘out’ keyword argument is expected to be a tuple with one entry per output (which can be None for arrays to be allocated by the ufunc). For ufuncs with a single output, passing a single array (instead of a tuple holding a single array) is also valid.
If ‘out’ is None (the default), a uninitialized output array is created,
which will be filled in the ufunc. At the end, this array is returned
unless it is zero-dimensional, in which case it is converted to a scalar;
this conversion can be avoided by passing in out=.... This can also be
spelled out=Ellipsis if you think that is clearer.
Note that the output is filled only in the places that the broadcast ‘where’ is True. If ‘where’ is the scalar True (the default), then this corresponds to all elements of the output, but in other cases, the elements not explicitly filled are left with their uninitialized values.
Operations where ufunc input and output operands have memory overlap are
defined to be the same as for equivalent operations where there
is no memory overlap. Operations affected make temporary copies
as needed to eliminate data dependency. As detecting these cases
is computationally expensive, a heuristic is used, which may in rare
cases result in needless temporary copies. For operations where the
data dependency is simple enough for the heuristic to analyze,
temporary copies will not be made even if the arrays overlap, if it
can be deduced copies are not necessary. As an example,
np.add(a, b, out=a) will not involve copies.
where
Accepts a boolean array which is broadcast together with the operands. Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone. This argument cannot be used for generalized ufuncs as those take non-scalar input.
Note that if an uninitialized return array is created, values of False will leave those values uninitialized.
axes
A list of tuples with indices of axes a generalized ufunc should operate
on. For instance, for a signature of (i,j),(j,k)->(i,k) appropriate
for matrix multiplication, the base elements are two-dimensional matrices
and these are taken to be stored in the two last axes of each argument.
The corresponding axes keyword would be [(-2, -1), (-2, -1), (-2, -1)].
For simplicity, for generalized ufuncs that operate on 1-dimensional arrays
(vectors), a single integer is accepted instead of a single-element tuple,
and for generalized ufuncs for which all outputs are scalars, the output
tuples can be omitted.
axis
A single axis over which a generalized ufunc should operate. This is a
short-cut for ufuncs that operate over a single, shared core dimension,
equivalent to passing in axes with entries of (axis,) for each
single-core-dimension argument and () for all others. For instance,
for a signature (i),(i)->(), it is equivalent to passing in
axes=[(axis,), (axis,), ()].
keepdims
If this is set to True, axes which are reduced over will be left in the
result as a dimension with size one, so that the result will broadcast
correctly against the inputs. This option can only be used for generalized
ufuncs that operate on inputs that all have the same number of core
dimensions and with outputs that have no core dimensions, i.e., with
signatures like (i),(i)->() or (m,m)->(). If used, the location of
the dimensions in the output can be controlled with axes and axis.
casting
May be ‘no’, ‘equiv’, ‘safe’, ‘same_kind’, or ‘unsafe’.
See can_cast for explanations of the parameter values.
Provides a policy for what kind of casting is permitted. For compatibility with previous versions of NumPy, this defaults to ‘unsafe’ for numpy < 1.7. In numpy 1.7 a transition to ‘same_kind’ was begun where ufuncs produce a DeprecationWarning for calls which are allowed under the ‘unsafe’ rules, but not under the ‘same_kind’ rules. From numpy 1.10 and onwards, the default is ‘same_kind’.
order
Specifies the calculation iteration order/memory layout of the output array. Defaults to ‘K’. ‘C’ means the output should be C-contiguous, ‘F’ means F-contiguous, ‘A’ means F-contiguous if the inputs are F-contiguous and not also not C-contiguous, C-contiguous otherwise, and ‘K’ means to match the element ordering of the inputs as closely as possible.
dtype
Overrides the DType of the output arrays the same way as the signature. This should ensure a matching precision of the calculation. The exact calculation DTypes chosen may depend on the ufunc and the inputs may be cast to this DType to perform the calculation.
subok
Defaults to true. If set to false, the output will always be a strict array, not a subtype.
signature
Either a Dtype, a tuple of DTypes, or a special signature string indicating the input and output types of a ufunc.
This argument allows the user to specify exact DTypes to be used for the
calculation. Casting will be used as necessary. The actual DType of the
input arrays is not considered unless signature is None for
that array.
When all DTypes are fixed, a specific loop is chosen or an error raised
if no matching loop exists.
If some DTypes are not specified and left None, the behaviour may
depend on the ufunc.
At this time, a list of available signatures is provided by the types
attribute of the ufunc. (This list may be missing DTypes not defined
by NumPy.)
The signature only specifies the DType class/type. For example, it
can specify that the operation should be datetime64 or float64
operation. It does not specify the datetime64 time-unit or the
float64 byte-order.
For backwards compatibility this argument can also be provided as sig, although the long form is preferred. Note that this should not be confused with the generalized ufunc signature that is stored in the signature attribute of the of the ufunc object.
Attributes#
There are some informational attributes that universal functions possess. None of the attributes can be set.
__doc__ |
A docstring for each ufunc. The first part of the docstring is dynamically generated from the number of outputs, the name, and the number of inputs. The second part of the docstring is provided at creation time and stored with the ufunc. |
__name__ |
The name of the ufunc. |
__signature__ |
The call signature of the ufunc, as an |
The number of inputs. |
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The number of outputs. |
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The number of arguments. |
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The number of types. |
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Returns a list with types grouped input->output. |
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The identity value. |
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Definition of the core elements a generalized ufunc operates on. |
Methods#
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Reduces |
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Accumulate the result of applying the operator to all elements. |
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Performs a (local) reduce with specified slices over a single axis. |
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Apply the ufunc op to all pairs (a, b) with a in A and b in B. |
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Performs unbuffered in place operation on operand 'a' for elements specified by 'indices'. |
Warning
A reduce-like operation on an array with a data-type that has a
range “too small” to handle the result will silently wrap. One
should use dtype to increase the size of the data-type over which
reduction takes place.
Available ufuncs#
There are currently more than 60 universal functions defined in
numpy on one or more types, covering a wide variety of
operations. Some of these ufuncs are called automatically on arrays
when the relevant infix notation is used (e.g., add(a, b)
is called internally when a + b is written and a or b is an
ndarray). Nevertheless, you may still want to use the ufunc
call in order to use the optional output argument(s) to place the
output(s) in an object (or objects) of your choice.
Recall that each ufunc operates element-by-element. Therefore, each scalar ufunc will be described as if acting on a set of scalar inputs to return a set of scalar outputs.
Note
The ufunc still returns its output(s) even if you use the optional output argument(s).
Math operations#
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Tip
The optional output arguments can be used to help you save memory
for large calculations. If your arrays are large, complicated
expressions can take longer than absolutely necessary due to the
creation and (later) destruction of temporary calculation
spaces. For example, the expression G = A * B + C is equivalent to
T1 = A * B; G = T1 + C; del T1. It will be more quickly executed
as G = A * B; add(G, C, G) which is the same as
G = A * B; G += C.
Trigonometric functions#
All trigonometric functions use radians when an angle is called for. The ratio of degrees to radians is \(180^{\circ}/\pi.\)
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Bit-twiddling functions#
These function all require integer arguments and they manipulate the bit-pattern of those arguments.
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Comparison functions#
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Warning
Do not use the Python keywords and and or to combine
logical array expressions. These keywords will test the truth
value of the entire array (not element-by-element as you might
expect). Use the bitwise operators & and | instead.
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Warning
The bit-wise operators & and | are the proper way to perform
element-by-element array comparisons. Be sure you understand the
operator precedence: (a > 2) & (a < 5) is the proper syntax because
a > 2 & a < 5 will result in an error due to the fact that 2 & a
is evaluated first.
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Tip
The Python function max() will find the maximum over a one-dimensional
array, but it will do so using a slower sequence interface. The reduce
method of the maximum ufunc is much faster. Also, the max() method
will not give answers you might expect for arrays with greater than
one dimension. The reduce method of minimum also allows you to compute
a total minimum over an array.
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Warning
the behavior of maximum(a, b) is different than that of max(a, b).
As a ufunc, maximum(a, b) performs an element-by-element comparison
of a and b and chooses each element of the result according to which
element in the two arrays is larger. In contrast, max(a, b) treats
the objects a and b as a whole, looks at the (total) truth value of
a > b and uses it to return either a or b (as a whole). A similar
difference exists between minimum(a, b) and min(a, b).
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Floating functions#
Recall that all of these functions work element-by-element over an array, returning an array output. The description details only a single operation.
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