(octave.info)Broadcasting
19.2 Broadcasting
=================
Broadcasting refers to how Octave binary operators and functions behave
when their matrix or array operands or arguments differ in size. Since
version 3.6.0, Octave now automatically broadcasts vectors, matrices,
and arrays when using elementwise binary operators and functions.
Broadly speaking, smaller arrays are “broadcast” across the larger one,
until they have a compatible shape. The rule is that corresponding
array dimensions must either
1. be equal, or
2. one of them must be 1.
In case all dimensions are equal, no broadcasting occurs and ordinary
element-by-element arithmetic takes place. For arrays of higher
dimensions, if the number of dimensions isn’t the same, then missing
trailing dimensions are treated as 1. When one of the dimensions is 1,
the array with that singleton dimension gets copied along that dimension
until it matches the dimension of the other array. For example,
consider
x = [1 2 3;
4 5 6;
7 8 9];
y = [10 20 30];
x + y
Without broadcasting, ‘x + y’ would be an error because the dimensions
do not agree. However, with broadcasting it is as if the following
operation were performed:
x = [1 2 3
4 5 6
7 8 9];
y = [10 20 30
10 20 30
10 20 30];
x + y
⇒ 11 22 33
14 25 36
17 28 39
That is, the smaller array of size ‘[1 3]’ gets copied along the
singleton dimension (the number of rows) until it is ‘[3 3]’. No actual
copying takes place, however. The internal implementation reuses
elements along the necessary dimension in order to achieve the desired
effect without copying in memory.
Both arrays can be broadcast across each other, for example, all
pairwise differences of the elements of a vector with itself:
y - y'
⇒ 0 10 20
-10 0 10
-20 -10 0
Here the vectors of size ‘[1 3]’ and ‘[3 1]’ both get broadcast into
matrices of size ‘[3 3]’ before ordinary matrix subtraction takes place.
A special case of broadcasting that may be familiar is when all
dimensions of the array being broadcast are 1, i.e., the array is a
scalar. Thus for example, operations like ‘x - 42’ and ‘max (x, 2)’ are
basic examples of broadcasting.
For a higher-dimensional example, suppose ‘img’ is an RGB image of
size ‘[m n 3]’ and we wish to multiply each color by a different scalar.
The following code accomplishes this with broadcasting,
img .*= permute ([0.8, 0.9, 1.2], [1, 3, 2]);
Note the usage of permute to match the dimensions of the ‘[0.8, 0.9,
1.2]’ vector with ‘img’.
For functions that are not written with broadcasting semantics,
‘bsxfun’ can be useful for coercing them to broadcast.
-- : bsxfun (F, A, B)
Apply a binary function F element-by-element to two array arguments
A and B, expanding singleton dimensions in either input argument as
necessary.
F is a function handle, inline function, or string containing the
name of the function to evaluate. The function F must be capable
of accepting two column-vector arguments of equal length, or one
column vector argument and a scalar.
The dimensions of A and B must be equal or singleton. The
singleton dimensions of the arrays will be expanded to the same
dimensionality as the other array.
See also: Note: arrayfun, Note: cellfun.
Broadcasting is only applied if either of the two broadcasting
conditions hold. As usual, however, broadcasting does not apply when
two dimensions differ and neither is 1:
x = [1 2 3
4 5 6];
y = [10 20
30 40];
x + y
This will produce an error about nonconformant arguments.
Besides common arithmetic operations, several functions of two
arguments also broadcast. The full list of functions and operators that
broadcast is
plus + .+
minus - .-
times .*
rdivide ./
ldivide .\
power .^ .**
lt <
le <=
eq ==
gt >
ge >=
ne != ~=
and &
or |
atan2
hypot
max
min
mod
rem
xor
+= -= .+= .-= .*= ./= .\= .^= .**= &= |=
Beware of resorting to broadcasting if a simpler operation will
suffice. For matrices A and B, consider the following:
C = sum (permute (A, [1, 3, 2]) .* permute (B, [3, 2, 1]), 3);
This operation broadcasts the two matrices with permuted dimensions
across each other during elementwise multiplication in order to obtain a
larger 3-D array, and this array is then summed along the third
dimension. A moment of thought will prove that this operation is simply
the much faster ordinary matrix multiplication, ‘C = A*B;’.
A note on terminology: “broadcasting” is the term popularized by the
Numpy numerical environment in the Python programming language. In
other programming languages and environments, broadcasting may also be
known as _binary singleton expansion_ (BSX, in MATLAB, and the origin of
the name of the ‘bsxfun’ function), _recycling_ (R programming
language), _single-instruction multiple data_ (SIMD), or _replication_.
19.2.1 Broadcasting and Legacy Code
-----------------------------------
The new broadcasting semantics almost never affect code that worked in
previous versions of Octave. Consequently, all code inherited from
MATLAB that worked in previous versions of Octave should still work
without change in Octave. The only exception is code such as
try
c = a.*b;
catch
c = a.*a;
end_try_catch
that may have relied on matrices of different size producing an error.
Because such operation is now valid Octave syntax, this will no longer
produce an error. Instead, the following code should be used:
if (isequal (size (a), size (b)))
c = a .* b;
else
c = a .* a;
endif
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