# A simple Octave tip

Matlab and Octave are both great for quickly trying out numeric code, especially using linear algebra. Many times code can be written to use the built in matrix operators instead of using loops. This makes the code clearer and faster. The following examples all use Octave.

First, consider this formula

\begin{equation*} S = \sum_{i=1}^n x_i y_i \end{equation*}

x and y are one dimensional vectors in Octave, so we can find $S$ with this code:

S = 0;
N = length(x);
for i = 1:N
S += x(i) * y(i);
end


That code is straight forward. Octave can do more, though. The =.*= operator does an element by element multiplication over matrices, and =sum= adds up the elements in a matrix. Since our matrices are one-dimensional, we can compose these functions to get our result, at the cost of creating a temporary matrix.

S = sum(x .* y)


Finally, Octave already has a function that does what we want. The formula we are looking at is the dot product of two vectors, so in Octave, we can do

S = dot(x, y)


The first attempt is the clearest when you do not know what .*, sum, and dot do. But, once you know what sum and dot do, they are much clearer. Also, they are much faster in Octave than the naive loop.

The next listing shows a test driver to compare the three methods.

fprintf("\n||||\n| method | size| result | time |\n")
fmt = "| %s | %d | %f | %f |\n";

for SIZE = [100, 1000,10000,100000,1000000]
x = rand(SIZE, 1);
y = rand(SIZE, 1);

fprintf("|---\n")
tic
S = 0;
for i = 1:SIZE
S += x(i) * y(i);
end
t1 = toc;
fprintf(fmt, "for", SIZE, S, t1);

tic
S = sum(x .* y);
t1 = toc;
fprintf(fmt, "sum", SIZE, S, t1);

tic
S = dot(x, y);
t1 = toc;
fprintf(fmt, "dot", SIZE, S, t1);
end


Running this code gives us the following results.

method

size

result

time

for

100

24.266291

0.000744

sum

100

24.266291

0.000055

dot

100

24.266291

0.000018

for

1000

247.065638

0.007198

sum

1000

247.065638

0.000016

dot

1000

247.065638

0.000017

for

10000

2507.387623

0.074163

sum

10000

2507.387623

0.000083

dot

10000

2507.387623

0.000029

for

100000

25090.247927

0.736105

sum

100000

25090.247927

0.000656

dot

100000

25090.247927

0.000095

for

1000000

249863.296971

7.334975

sum

1000000

249863.296971

0.006773

dot

1000000

249863.296971

0.001246

As a general rule, we should always try to use the language and tools, instead of doing things ourselves. Often, this is mostly a matter of learning what is available.

Sometimes, though we want to learn how something works. Then, of course, it makes sense to do it ourselves instead of just making a call to existing code.