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 scalar inputs and produces a fixed number of
scalar outputs.

In Numpy, universal functions are instances of the
`numpy.ufunc` class. Many of the built-in functions are
implemented in compiled C code, but `ufunc` instances can also
be produced using the `frompyfunc` factory function.

Each universal function takes array inputs and produces array outputs by performing the core function element-wise on the inputs. Standard broadcasting rules are applied so that inputs not sharing exactly the same shapes can still be usefully operated on. Broadcasting can be understood by four rules:

- All input arrays with
`ndim`smaller than the input array of largest`ndim`, have 1’s prepended to their shapes. - The size in each dimension of the output shape is the maximum of all the input sizes in that dimension.
- An input can be used in the calculation if its size in a particular dimension either matches the output size in that dimension, or has value exactly 1.
- 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).

Broadcasting is used throughout NumPy to decide how to handle
disparately shaped arrays; for example, all arithmetic operations (`+`,
`-`, `*`, ...) between `ndarrays` broadcast the
arrays before operation.

A set of arrays is called “*broadcastable*” to the same shape if
the above rules produce a valid result, *i.e.*, one of the following
is true:

- The arrays all have exactly the same shape.
- The arrays all have the same number of dimensions and the length of each dimensions is either a common length or 1.
- The arrays that have too few dimensions can have their shapes prepended with a dimension of length 1 to satisfy property 2.

Example

If `a.shape` is (5,1), `b.shape` is (1,6), `c.shape` is (6,)
and `d.shape` is () so that *d* is a scalar, then *a*, *b*, *c*,
and *d* are all broadcastable to dimension (5,6); and

*a*acts like a (5,6) array where`a[:,0]`is broadcast to the other columns,*b*acts like a (5,6) array where`b[0,:]`is broadcast to the other rows,*c*acts like a (1,6) array and therefore like a (5,6) array where`c[:]`is broadcast to every row, and finally,*d*acts like a (5,6) array where the single value is repeated.

The output of the ufunc (and its methods) is not necessarily an
`ndarray`, if all input arguments are not `ndarrays`.

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 1.0. 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.

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 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

setbufsize(size) |
Set the size of the buffer used in ufuncs. |

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

seterr([all, divide, over, under, invalid]) |
Set how floating-point errors are handled. |

seterrcall(func) |
Set the floating-point error callback function or log object. |

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 `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 Figure below shows the results of this call for
the 21 internally supported types on the author’s 32-bit system. You
can generate this table for your system with the code given in the Figure.

Figure

Code segment showing the “can cast safely” table for a 32-bit system.

```
>>> def print_table(ntypes):
... print 'X',
... for char in ntypes: print char,
... print
... for row in ntypes:
... print row,
... for col in ntypes:
... print int(np.can_cast(row, col)),
... print
>>> print_table(np.typecodes['All'])
X ? b h i l q p B H I L Q P f d g F D G S U V O
? 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
b 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1
h 0 0 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1
i 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1
l 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1
q 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1
p 0 0 0 1 1 1 1 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1
B 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
H 0 0 0 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
I 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1
L 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1
Q 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 1 1 1 1
P 0 0 0 0 0 1 0 0 0 1 1 1 1 0 1 1 0 1 1 1 1 1 1
f 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1
d 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 1 1 1
g 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 1 1
F 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1
D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1
G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1
S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1
U 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1
V 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
O 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
```

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 64-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.

All ufuncs take optional keyword arguments. These represent rather advanced usage and will not typically be used by most Numpy users.

*sig*

Either a data-type, a tuple of data-types, or a special signature string indicating the input and output types of a ufunc. This argument allows you to provide a specific signature for the 1-d loop to use in the underlying calculation. If the loop specified does not exist for the ufunc, then a TypeError is raised. Normally, a suitable loop is found automatically by comparing the input types with what is available and searching for a loop with data-types to which all inputs can be cast safely. This keyword argument lets you bypass that search and choose a particular loop. A list of available signatures is provided by thetypesattribute of the ufunc object.

*extobj*

a list of length 1, 2, or 3 specifying the ufunc buffer-size, the error mode integer, and the error call-back function. Normally, these values are looked up in a thread-specific dictionary. Passing them here circumvents that look up and uses the low-level specification provided for the error mode. This may be useful, for example, as an optimization for calculations requiring many ufunc calls on small arrays in a loop.

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. |

ufunc.nin |
The number of inputs. |

ufunc.nout |
The number of outputs. |

ufunc.nargs |
The number of arguments. |

ufunc.ntypes |
The number of types. |

ufunc.types |
Returns a list with types grouped input->output. |

ufunc.identity |
The identity value. |

All ufuncs have four methods. However, these methods only make sense on
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
and a *dtype* 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 and may be negative, but must be an integer. The
*dtype* keyword allows you to manage a very common problem that arises
when naively using *{op}.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
`int_` data type, it will be internally upcast to the `int_`
(or `uint`) data-type.

ufunc.reduce(a[, axis, dtype, out]) |
Reduces a‘s dimension by one, by applying ufunc along one axis. |

ufunc.accumulate(array[, axis, dtype, out]) |
Accumulate the result of applying the operator to all elements. |

ufunc.reduceat(a, indices[, axis, dtype, out]) |
Performs a (local) reduce with specified slices over a single axis. |

ufunc.outer(A, B) |
Apply the ufunc op to all pairs (a, b) with a in A and b in B. |

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.

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 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).

add(x1) |
Add arguments element-wise. |

subtract(x1) |
Subtract arguments, element-wise. |

multiply(x1) |
Multiply arguments element-wise. |

divide(x1) |
Divide arguments element-wise. |

logaddexp(x1) |
Logarithm of the sum of exponentiations of the inputs. |

logaddexp2(x1) |
Logarithm of the sum of exponentiations of the inputs in base-2. |

true_divide(x1) |
Returns a true division of the inputs, element-wise. |

floor_divide(x1) |
Return the largest integer smaller or equal to the division of the inputs. |

negative() |
Returns an array with the negative of each element of the original array. |

power(x1) |
Returns element-wise base array raised to power from second array. |

remainder(x1) |
Return element-wise remainder of division. |

mod(x1) |
Return element-wise remainder of division. |

fmod(x1) |
Return the element-wise remainder of division. |

absolute() |
Calculate the absolute value element-wise. |

rint() |
Round elements of the array to the nearest integer. |

sign() |
Returns an element-wise indication of the sign of a number. |

conj() |
Return the complex conjugate, element-wise. |

exp() |
Calculate the exponential of all elements in the input array. |

exp2() |
Calculate 2**p for all p in the input array. |

log() |
Natural logarithm, element-wise. |

log2(x[, y]) |
Return the base 2 logarithm of the input array, element-wise. |

log10() |
Return the base 10 logarithm of the input array, element-wise. |

expm1() |
Calculate exp(x) - 1 for all elements in the array. |

log1p() |
Return the natural logarithm of one plus the input array, element-wise. |

sqrt() |
Return the positive square-root of an array, element-wise. |

square() |
Return the element-wise square of the input. |

reciprocal() |
Return the reciprocal of the argument, element-wise. |

ones_like() |
Returns an array of ones with the same shape and type as a given array. |

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`.

All trigonometric functions use radians when an angle is called for. The ratio of degrees to radians is

sin() |
Trigonometric sine, element-wise. |

cos() |
Cosine elementwise. |

tan() |
Compute tangent element-wise. |

arcsin() |
Inverse sine elementwise. |

arccos() |
Trigonometric inverse cosine, element-wise. |

arctan() |
Trigonometric inverse tangent, element-wise. |

arctan2(x1) |
Elementwise arc tangent of x1/x2 choosing the quadrant correctly. |

hypot(x1) |
Given the “legs” of a right triangle, return its hypotenuse. |

sinh() |
Hyperbolic sine, element-wise. |

cosh() |
Hyperbolic cosine, element-wise. |

tanh() |
Compute hyperbolic tangent element-wise. |

arcsinh() |
Inverse hyperbolic sine elementwise. |

arccosh() |
Inverse hyperbolic cosine, elementwise. |

arctanh() |
Inverse hyperbolic tangent elementwise. |

deg2rad() |
Convert angles from degrees to radians. |

rad2deg() |
Convert angles from radians to degrees. |

These function all require integer arguments and they manipulate the bit-pattern of those arguments.

bitwise_and(x1) |
Compute the bit-wise AND of two arrays element-wise. |

bitwise_or(x1) |
Compute the bit-wise OR of two arrays element-wise. |

bitwise_xor(x1) |
Compute the bit-wise XOR of two arrays element-wise. |

invert() |
Compute bit-wise inversion, or bit-wise NOT, element-wise. |

left_shift(x1) |
Shift the bits of an integer to the left. |

right_shift(x1) |
Shift the bits of an integer to the right. |

greater(x1) |
Return (x1 > x2) element-wise. |

greater_equal(x1) |
Return (x1 >= x2) element-wise. |

less(x1) |
Return (x1 < x2) element-wise. |

less_equal(x1) |
Return (x1 <= x2) element-wise. |

not_equal(x1) |
Return (x1 != x2) element-wise. |

equal(x1) |
Return (x1 == x2) element-wise. |

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.

logical_and(x1) |
Compute the truth value of x1 AND x2 elementwise. |

logical_or(x1) |
Compute the truth value of x1 OR x2 elementwise. |

logical_xor(x1) |
Compute the truth value of x1 XOR x2 elementwise. |

logical_not() |
Compute the truth value of NOT x elementwise. |

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.

maximum(x1) |
Element-wise maximum of array elements. |

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.

minimum(x1) |
Element-wise minimum of array elements. |

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)`.

Recall that all of these functions work element-by-element over an array, returning an array output. The description details only a single operation.

isreal(x) |
Returns a bool array, where True if input element is real. |

iscomplex(x) |
Returns a bool array, where True if input element is complex. |

isfinite() |
Test element-wise for finite-ness (not infinity or not Not a Number). |

isinf() |
Test element-wise for positive or negative infinity, return result as bool array. |

isnan() |
Test element-wise for Not a Number (NaN), return result as a bool array. |

signbit() |
Returns element-wise True where signbit is set (less than zero). |

copysign(x1) |
Change the sign of x1 to that of x2, element-wise. |

nextafter(x1) |
Return the next representable floating-point value after x1 in the direction of x2 element-wise. |

modf(out1) |
Return the fractional and integral parts of an array, element-wise. |

ldexp(x1) |
Compute y = x1 * 2**x2. |

frexp(out1) |
Split the number, x, into a normalized fraction (y1) and exponent (y2) |

fmod(x1) |
Return the element-wise remainder of division. |

floor() |
Return the floor of the input, element-wise. |

ceil() |
Return the ceiling of the input, element-wise. |

trunc() |
Return the truncated value of the input, element-wise. |