15.7 Universal Functions (ufuncs)

Numeric supplies named functions with the same semantics as Python's arithmetic, comparison, and bitwise operators. Similar semantics (element-wise operation, broadcasting, coercion) are also available with other mathematical functions, both binary and unary, that Numeric supplies. For example, Numeric supplies typical mathematical functions similar to those supplied by built-in module math, such as sin, cos, log, and exp.

These functions are objects of type ufunc (which stands for universal function) and share several traits in addition to those they have in common with array operators. Every ufunc instance u is callable, is applicable to sequences as well as to arrays, and lets you specify an optional output argument. If u is binary (i.e., if u accepts two operand arguments), u also has four callable attributes, named u.accumulate, u.outer, u.reduce, and u.reduceat. The ufunc objects supplied by Numeric apply only to arrays with numeric type codes (i.e., not to arrays with type code 'O' or 'c').

Any ufunc u applies to sequences, not just to arrays. When you start with a list L, it's faster to call u directly on L rather than to convert L to an array. u's return value is an array a; you can perform further computation, if any, on a, and then, if you need a list result, you can convert the resulting array to a list by calling its method tolist. For example, say you must compute the logarithm of each item of a list and return another list. On my system, with N set to 2222 and using python -O, a list comprehension such as:

def logsupto(N):
    return [math.log(x) for x in range(2,N)]

takes about 5.6 milliseconds. Using Python's built-in map:

def logsupto(N):
    return map(math.log, range(2,N))

takes around half the time, 2.8 milliseconds. Using Numeric's ufunc named log:

def logsupto(N):
    return Numeric.log(range(2,N)).tolist(  )

reduces the time to about 2.0 milliseconds. Taking some care to exploit the output argument to the log ufunc:

def logsupto(N):
    temp = Numeric.arange(2, N, typecode=Numeric.Float)
    Numeric.log(temp, temp)
    return temp.tolist(  )

further reduces the time, down to just 0.9 milliseconds. The ability to accelerate such simple but massive computations (here by about 6 times) with so little effort is a good part of the attraction of Numeric, and particularly of Numeric's ufunc objects.

15.7.1 The Optional output Argument

Any ufunc u accepts an optional last argument output that specifies an output array. If supplied, output must be an array or array slice of the right shape and type for u's results (i.e., no coercion, no broadcasting). u stores results in output and does not create a new array. output can be the same as an input array argument a of u. Indeed, output is normally specified in order to substitute common idioms such as a=u(a,b) with faster equivalents such as u(a,b,a). However, output cannot share data with a without being a (i.e., output can't be a different view of some or all of a's data). If you pass such a disallowed output argument, Numeric is normally unable to diagnose your error and raise an exception, so instead you get wrong results.

Whether you pass the optional output argument or not, a ufunc u returns its results as the function's return value. When you do not pass output, u stores the results it returns in a new array object, so you normally bind u's return value to some reference in order to be able to access u's results later. When you pass the output argument, u stores the results in output, so you need not bind u's return value. You can later access u's results as the new contents of the array object passed as output.

15.7.2 Callable Attributes

Every binary ufunc u supplies four attributes that are also callable objects.

u.accumulate(a,axis=0)

Returns an array r with the same shape and type code as a. Each element of r is the accumulation of elements of a along the given axis with the function or operator underlying u. For example:

print add.accumulate(range(10))      
# prints: [0 1 3 6 10 15 21 28 36 45]

Since add's underlying operator is +, and a is sequence 0,1,2,...,9, r is 0,0+1,0+1+2,...,0+1+...+8+9. In other words, r[0] is a[0], r[1] is r[0] + a[1], r[2] is r[1] + a[2], and so on (i.e., each r[i] is r[i-1] + a[i]).

u.outer(a,b)

Returns an array r whose shape tuple is a.shape+b.shape. For each tuple ta indexing a and tb indexing b, a[ta], operated (with the function or operator underlying u) with b[tb], is put in r[ta+tb] (the + here indicates tuple concatenation). The overall operation is known in mathematics as the outer product when u is multiply. For example:

a = Numeric.arange(3, 5)
b = Numeric.arange(1, 6)
c = Numeric.multiply.outer(a, b)
print a.shape, b.shape, c.shape # prints: (2,) (5,) (2,5)
print c                         # prints: [[3 6 9 12 15]
                                #          [4 8 12 16 20]]

c.shape is (2,5), the concatenation of the shape tuples of operands a and b. Each i row of c is the whole of b multiplied by the corresponding i element of a.

u.reduce(a,axis=0)

Returns an array r with the same type code as a and rank one less than a's rank. Each element of r is the reduction of the elements of a, along the given axis, with the function or operator underlying u. The functionality of u.reduce is therefore close to that of Python's built-in reduce function, covered in Chapter 8. For example, since 0+1+2+...+9 is 45, add.reduce(range(10)) is 45. This is just like, when using built-in reduce and import operator, reduce(operator.add,range(10)) is also 45.

u.reduceat(a,indices)

Returns an array r with the same type code as a and the same shape as indices. Each element of r is the reduction, with the function or operator underlying u, of elements of a starting from the corresponding item of indices up to the next one excluded (up to the end, for the last one). For example:

print add.reduceat(range(10),(2,6,8)) # prints: [14 13 17]

Here, r's elements are the partial sums 2+3+4+5, 6+7, and 8+9.

15.7.3 ufunc Objects Supplied by Numeric

Numeric supplies several ufunc objects, as listed in Table 15-4.

absolute

add

arccos

arccosh

arcsin

arcsinh

arctan

arctanh

bitwise_and

bitwise_not

bitwise_or

bitwise_xor

ceil

conjugate

cos

cosh

divide

equal

exp

fabs

floor

fmod

greater

greater_equal

less

less_equal

log

log10

logical_and

logical_not

logical_or

logical_xor

maximum

minimum

multiply

not_equal

power

remainder

sin

sinh

sqrt

subtract

tan

tanh

Here's how you might use the maximum ufunc to get a numeric ramp that goes down and then back up again:

print Numeric.maximum(range(1,20),range(20,1,-1))
# prints: [20 19 18 17 16 15 14 13 12 11 11 12 13 14 15 16 17 18 19]

15.7.4 Shorthand for Commonly Used ufunc Methods

Numeric defines function synonyms for some commonly used methods of ufunc objects, as listed in Table 15-5.

alltrue

logical_and.reduce

cumproduct

multiply.accumulate

cumsum

add.accumulate

product

multiply.reduce

sometrue

logical_or.reduce

sum

add.reduce