: Notice that indices start at 0, as in Perl, C and Java (and unlike
MATLAB and IDL).
But with PDL, we can just write:
pdl> $b = $a + 3
pdl> p $b
[
[ 3 4 5 6 7 8 9 10 11 12 13]
[ 14 15 16 17 18 19 20 21 22 23 24]
[ 25 26 27 28 29 30 31 32 33 34 35]
[ 36 37 38 39 40 41 42 43 44 45 46]
[ 47 48 49 50 51 52 53 54 55 56 57]
[ 58 59 60 61 62 63 64 65 66 67 68]
[ 69 70 71 72 73 74 75 76 77 78 79]
[ 80 81 82 83 84 85 86 87 88 89 90]
[ 91 92 93 94 95 96 97 98 99 100 101]
]
This is the simplest example of threading, and it is something that
all numerical software tools do. The C<+ 3> operation was automatically
applied along two dimensions. Now suppose you want to to subtract a
line from every row in C<$a>:
pdl> $line = sequence(11)
pdl> p $line
[0 1 2 3 4 5 6 7 8 9 10]
pdl> $c = $a - $line
pdl> p $c
[
[ 0 0 0 0 0 0 0 0 0 0 0]
[11 11 11 11 11 11 11 11 11 11 11]
[22 22 22 22 22 22 22 22 22 22 22]
[33 33 33 33 33 33 33 33 33 33 33]
[44 44 44 44 44 44 44 44 44 44 44]
[55 55 55 55 55 55 55 55 55 55 55]
[66 66 66 66 66 66 66 66 66 66 66]
[77 77 77 77 77 77 77 77 77 77 77]
[88 88 88 88 88 88 88 88 88 88 88]
]
Two things to note here: First, the value of C<$a> is still the same. Try
C to check. Second, PDL automatically subtracted C<$line> from each
row in C<$a>. Why did it do that? Let's look at the dimensions of C<$a>,
C<$line> and C<$c>:
pdl> p $line->info => PDL: Double D [11]
pdl> p $a->info => PDL: Double D [11,9]
pdl> p $c->info => PDL: Double D [11,9]
So, both C<$a> and C<$line> have the same number of elements in the 0th
dimension! What PDL then did was thread over the higher dimensions in C<$a>
and repeated the same operation 9 times to all the rows on C<$a>. This is
PDL threading in action.
What if you want to subtract C<$line> from the first line in C<$a> only?
You can do that by specifying the line explicitly:
pdl> $a(:,0) -= $line
pdl> p $a
[
[ 0 0 0 0 0 0 0 0 0 0 0]
[11 12 13 14 15 16 17 18 19 20 21]
[22 23 24 25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40 41 42 43]
[44 45 46 47 48 49 50 51 52 53 54]
[55 56 57 58 59 60 61 62 63 64 65]
[66 67 68 69 70 71 72 73 74 75 76]
[77 78 79 80 81 82 83 84 85 86 87]
[88 89 90 91 92 93 94 95 96 97 98]
]
See L and L to
learn more about specifying subsets from piddles.
The true power of threading comes when you realise that the piddle can
have any number of dimensions! Let's make a 4 dimensional piddle:
pdl> $piddle_4D = sequence(11,3,7,2)
pdl> $c = $piddle_4D - $line
Now C<$c> is a piddle of the same dimension as C<$piddle_4D>.
pdl> p $piddle_4D->info => PDL: Double D [11,3,7,2]
pdl> p $c->info => PDL: Double D [11,3,7,2]
This time PDL has threaded over three higher dimensions automatically,
subtracting C<$line> all the way.
But, maybe you don't want to subtract from the rows (dimension 0), but from
the columns (dimension 1). How do I subtract a column of numbers from each
column in C<$a>?
pdl> $cols = sequence(9)
pdl> p $a->info => PDL: Double D [11,9]
pdl> p $cols->info => PDL: Double D [9]
Naturally, we can't just type C<$a - $cols>. The dimensions don't match:
pdl> p $a - $cols
PDL: PDL::Ops::minus(a,b,c): Parameter 'b'
PDL: Mismatched implicit thread dimension 0: should be 11, is 9
How do we tell PDL that we want to subtract from dimension 1 instead?
=head1 MANIPULATING DIMENSIONS
There are many PDL functions that let you rearrange the dimensions of PDL
arrays. They are mostly covered in L. The three
most common ones are:
xchg
mv
reorder
=head2 Method: C
The C method "B" two dimensions in a piddle:
pdl> $a = sequence(6,7,8,9)
pdl> $a_xchg = $a->xchg(0,3)
pdl> p $a->info => PDL: Double D [6,7,8,9]
pdl> p $a_xchg->info => PDL: Double D [9,7,8,6]
| |
V V
(dim 0) (dim 3)
Notice that dimensions 0 and 3 were exchanged without affecting the other
dimensions. Notice also that C does not alter C<$a>. The original
variable C<$a> remains untouched.
=head2 Method: C
The C method "B" one dimension, in a piddle, shifting other
dimensions as necessary.
pdl> $a = sequence(6,7,8,9) (dim 0)
pdl> $a_mv = $a->mv(0,3) |
pdl> V _____
pdl> p $a->info => PDL: Double D [6,7,8,9]
pdl> p $a_mv->info => PDL: Double D [7,8,9,6]
----- |
V
(dim 3)
Notice that when dimension 0 was moved to position 3, all the other dimensions
had to be shifted as well. Notice also that C does not alter C<$a>. The
original variable C<$a> remains untouched.
=head2 Method: C
The C method is a generalization of the C and C methods.
It "B" the dimensions in any way you specify:
pdl> $a = sequence(6,7,8,9)
pdl> $a_reorder = $a->reorder(3,0,2,1)
pdl>
pdl> p $a->info => PDL: Double D [6,7,8,9]
pdl> p $a_reorder->info => PDL: Double D [9,6,8,7]
| | | |
V V v V
dimensions: 0 1 2 3
Notice what happened. When we wrote C we instructed PDL to:
* Put dimension 3 first.
* Put dimension 0 next.
* Put dimension 2 next.
* Put dimension 1 next.
When you use the C method, all the dimensions are shuffled. Notice that
C does not alter C<$a>. The original variable C<$a> remains untouched.
=head1 GOTCHA: LINKING VS ASSIGNMENT
=head2 Linking
By default, piddles are B so that changes on one will go
back and affect the original B.
pdl> $a = sequence(4,5)
pdl> $a_xchg = $a->xchg(1,0)
Here, C<$a_xchg> B. It is merely a different way
of looking at C<$a>. Any change in C<$a_xchg> will appear in C<$a> as well.
pdl> p $a
[
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
]
pdl> $a_xchg += 3
pdl> p $a
[
[ 3 4 5 6]
[ 7 8 9 10]
[11 12 13 14]
[15 16 17 18]
[19 20 21 22]
]
=head2 Assignment
Some times, linking is not the behaviour you want. If you want to make the
piddles independent, use the C method:
pdl> $a = sequence(4,5)
pdl> $a_xchg = $a->copy->xchg(1,0)
Now C<$a> and C<$a_xchg> are completely separate objects:
pdl> p $a
[
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
]
pdl> $a_xchg += 3
pdl> p $a
[
[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]
[12 13 14 15]
[16 17 18 19]
]
pdl> $a_xchg
[
[ 3 7 11 15 19]
[ 4 8 12 16 20]
[ 5 9 13 17 21]
[ 6 10 14 18 22]
]
=head1 PUTTING IT ALL TOGETHER
Now we are ready to solve the problem that motivated this whole discussion:
pdl> $a = sequence(11,9)
pdl> $cols = sequence(9)
pdl>
pdl> p $a->info => PDL: Double D [11,9]
pdl> p $cols->info => PDL: Double D [9]
How do we tell PDL to subtract C<$cols> along dimension 1 instead of dimension 0?
The simplest way is to use the C method and rely on PDL linking:
pdl> p $a
[
[ 0 1 2 3 4 5 6 7 8 9 10]
[11 12 13 14 15 16 17 18 19 20 21]
[22 23 24 25 26 27 28 29 30 31 32]
[33 34 35 36 37 38 39 40 41 42 43]
[44 45 46 47 48 49 50 51 52 53 54]
[55 56 57 58 59 60 61 62 63 64 65]
[66 67 68 69 70 71 72 73 74 75 76]
[77 78 79 80 81 82 83 84 85 86 87]
[88 89 90 91 92 93 94 95 96 97 98]
]
pdl> $a->xchg(1,0) -= $cols
pdl> p $a
[
[ 0 1 2 3 4 5 6 7 8 9 10]
[10 11 12 13 14 15 16 17 18 19 20]
[20 21 22 23 24 25 26 27 28 29 30]
[30 31 32 33 34 35 36 37 38 39 40]
[40 41 42 43 44 45 46 47 48 49 50]
[50 51 52 53 54 55 56 57 58 59 60]
[60 61 62 63 64 65 66 67 68 69 70]
[70 71 72 73 74 75 76 77 78 79 80]
[80 81 82 83 84 85 86 87 88 89 90]
]
=over 5
=item General Strategy:
Move the dimensions you want to operate on to the start of your piddle's
dimension list. Then let PDL thread over the higher dimensions.
=back
=head1 EXAMPLE: CONWAY'S GAME OF LIFE
Okay, enough theory. Let's do something a bit more interesting: We'll write
B in PDL and see how powerful PDL can be!
The B is a simulation run on a big two dimensional grid. Each
cell in the grid can either be alive or dead (represented by 1 or 0). The
next generation of cells in the grid is calculated with simple rules according
to the number of living cells in it's immediate neighbourhood:
1) If an empty cell has exactly three neighbours, a living cell is generated.
2) If a living cell has less than two neighbours, it dies of overfeeding.
3) If a living cell has 4 or more neighbours, it dies from starvation.
Only the first generation of cells is determined by the programmer. After that,
the simulation runs completely according to these rules. To calculate the next
generation, you need to look at each cell in the 2D field (requiring two loops),
calculate the number of live cells adjacent to this cell (requiring another two
loops) and then fill the next generation grid.
=head2 Classical implementation
Here's a classic way of writing this program in Perl. We only use PDL for
addressing individual cells.
#!/usr/bin/perl -w
use PDL;
use PDL::NiceSlice;
# Make a board for the game of life.
my $nx = 20;
my $ny = 20;
# Current generation.
my $a = zeroes($nx, $ny);
# Next generation.
my $n = zeroes($nx, $ny);
# Put in a simple glider.
$a(1:3,1:3) .= pdl ( [1,1,1],
[0,0,1],
[0,1,0] );
for (my $i = 0; $i < 100; $i++) {
$n = zeroes($nx, $ny);
$new_a = $a->copy;
for ($x = 0; $x < $nx; $x++) {
for ($y = 0; $y < $ny; $y++) {
# For each cell, look at the surrounding neighbours.
for ($dx = -1; $dx <= 1; $dx++) {
for ($dy = -1; $dy <= 1; $dy++) {
$px = $x + $dx;
$py = $y + $dy;
# Wrap around at the edges.
if ($px < 0) {$px = $nx-1};
if ($py < 0) {$py = $ny-1};
if ($px >= $nx) {$px = 0};
if ($py >= $ny) {$py = 0};
$n($x,$y) .= $n($x,$y) + $a($px,$py);
}
}
# Do not count the central cell itself.
$n($x,$y) -= $a($x,$y);
# Work out if cell lives or dies:
# Dead cell lives if n = 3
# Live cell dies if n is not 2 or 3
if ($a($x,$y) == 1) {
if ($n($x,$y) < 2) {$new_a($x,$y) .= 0};
if ($n($x,$y) > 3) {$new_a($x,$y) .= 0};
} else {
if ($n($x,$y) == 3) {$new_a($x,$y) .= 1}
}
}
}
print $a;
$a = $new_a;
}
If you run this, you will see a small glider crawl diagonally across the grid
of zeroes. On my machine, it prints out a couple of generations per second.
=head2 Threaded PDL implementation
And here's the threaded version in PDL. Just four lines of PDL code, and
one of those is printing out the latest generation!
#!/usr/bin/perl -w
use PDL;
use PDL::NiceSlice;
my $a = zeroes(20,20);
# Put in a simple glider.
$a(1:3,1:3) .= pdl ( [1,1,1],
[0,0,1],
[0,1,0] );
my $n;
for (my $i = 0; $i < 100; $i++) {
# Calculate the number of neighbours per cell.
$n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1);
$n = $n->sumover->sumover - $a;
# Calculate the next generation.
$a = ((($n == 2) + ($n == 3))* $a) + (($n==3) * !$a);
print $a;
}
The threaded PDL version is much faster:
Classical => 32.79 seconds.
Threaded => 0.41 seconds.
=head2 Explanation
How does the threaded version work?
There are many PDL functions designed to help you carry out PDL threading.
In this example, the key functions are:
=head3 Method: C
At the simplest level, the C method is a different way to select a
portion of a piddle. Instead of using the C<$a(2,3)> notation, we use
another piddle.
pdl> $a = sequence(6,7)
pdl> p $a
[
[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]
[12 13 14 15 16 17]
[18 19 20 21 22 23]
[24 25 26 27 28 29]
[30 31 32 33 34 35]
[36 37 38 39 40 41]
]
pdl> p $a->range( pdl [1,2] )
13
pdl> p $a(1,2)
[
[13]
]
At this point, the C method looks very similar to a regular PDL slice.
But the C method is more general. For example, you can select several
components at once:
pdl> $index = pdl [ [1,2],[2,3],[3,4],[4,5] ]
pdl> p $a->range( $index )
[13 20 27 34]
Additionally, C takes a second parameter which determines the size
of the chunk to return:
pdl> $size = 3
pdl> p $a->range( pdl([1,2]) , $size )
[
[13 14 15]
[19 20 21]
[25 26 27]
]
We can use this to select one or more 3x3 boxes.
Finally, C can take a third parameter called the "boundary" condition.
It tells PDL what to do if the size box you request goes beyond the edge of
the piddle. We won't go over all the options. We'll just say that the option
C means that the piddle "wraps around". For example:
pdl> p $a
[
[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]
[12 13 14 15 16 17]
[18 19 20 21 22 23]
[24 25 26 27 28 29]
[30 31 32 33 34 35]
[36 37 38 39 40 41]
]
pdl> $size = 3
pdl> p $a->range( pdl([4,2]) , $size , "periodic" )
[
[16 17 12]
[22 23 18]
[28 29 24]
]
pdl> p $a->range( pdl([5,2]) , $size , "periodic" )
[
[17 12 13]
[23 18 19]
[29 24 25]
]
Notice how the box wraps around the boundary of the piddle.
=head3 Method: C
The C method is a convenience method that returns an enumerated
list of coordinates suitable for use with the C method.
pdl> p $piddle = sequence(3,3)
[
[0 1 2]
[3 4 5]
[6 7 8]
]
pdl> p ndcoords($piddle)
[
[
[0 0]
[1 0]
[2 0]
]
[
[0 1]
[1 1]
[2 1]
]
[
[0 2]
[1 2]
[2 2]
]
]
This can be a little hard to read. Basically it's saying that the coordinates
for every element in C<$piddle> is given by:
(0,0) (1,0) (2,0)
(1,0) (1,1) (2,1)
(2,0) (2,1) (2,2)
=head3 Combining C and C
What really matters is that C is designed to work together with
C, with no C<$size> parameter, you get the same piddle back.
pdl> p $piddle
[
[0 1 2]
[3 4 5]
[6 7 8]
]
pdl> p $piddle->range( ndcoords($piddle) )
[
[0 1 2]
[3 4 5]
[6 7 8]
]
Why would this be useful? Because now we can ask for a series of "boxes" for
the entire piddle. For example, 2x2 boxes:
pdl> p $piddle->range( ndcoords($piddle) , 2 , "periodic" )
The output of this function is difficult to read because the "boxes" along
the last two dimension. We can make the result more readable by rearranging
the dimensions:
pdl> p $piddle->range( ndcoords($piddle) , 2 , "periodic" )->reorder(2,3,0,1)
[
[
[
[0 1]
[3 4]
]
[
[1 2]
[4 5]
]
...
]
Here you can see more clearly that
[0 1]
[3 4]
Is the 2x2 box starting with the (0,0) element of C<$piddle>.
We are not done yet. For the game of life, we want 3x3 boxes from C<$a>:
pdl> p $a
[
[ 0 1 2 3 4 5]
[ 6 7 8 9 10 11]
[12 13 14 15 16 17]
[18 19 20 21 22 23]
[24 25 26 27 28 29]
[30 31 32 33 34 35]
[36 37 38 39 40 41]
]
pdl> p $a->range( ndcoords($a) , 3 , "periodic" )->reorder(2,3,0,1)
[
[
[
[ 0 1 2]
[ 6 7 8]
[12 13 14]
]
...
]
We can confirm that this is the 3x3 box starting with the (0,0) element of C<$a>.
But there is one problem. We actually want the 3x3 box to be B on
(0,0). That's not a problem. Just subtract 1 from all the coordinates in
C. Remember that the "periodic" option takes care of making
everything wrap around.
pdl> p $a->range( ndcoords($a) - 1 , 3 , "periodic" )->reorder(2,3,0,1)
[
[
[
[41 36 37]
[ 5 0 1]
[11 6 7]
]
[
[36 37 38]
[ 0 1 2]
[ 6 7 8]
]
...
Now we see a 3x3 box with the (0,0) element in the centre of the box.
=head3 Method: C
The C method adds along only the first dimension. If we apply it
twice, we will be adding all the elements of each 3x3 box.
pdl> $n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1)
pdl> p $n
[
[
[
[41 36 37]
[ 5 0 1]
[11 6 7]
]
[
[36 37 38]
[ 0 1 2]
[ 6 7 8]
]
...
pdl> p $n->sumover->sumover
[
[144 135 144 153 162 153]
[ 72 63 72 81 90 81]
[126 117 126 135 144 135]
[180 171 180 189 198 189]
[234 225 234 243 252 243]
[288 279 288 297 306 297]
[216 207 216 225 234 225]
]
Use a calculator to confirm that 144 is the sum of all the elements in the
first 3x3 box and 135 is the sum of all the elements in the second 3x3 box.
=head3 Counting neighbours
We are almost there!
Adding up all the elements in a 3x3 box is not B what we want. We
don't want to count the center box. Fortunately, this is an easy fix:
pdl> p $n->sumover->sumover - $a
[
[144 134 142 150 158 148]
[ 66 56 64 72 80 70]
[114 104 112 120 128 118]
[162 152 160 168 176 166]
[210 200 208 216 224 214]
[258 248 256 264 272 262]
[180 170 178 186 194 184]
]
When applied to Conway's Game of Life, this will tell us how many living
neighbours each cell has:
pdl> $a = zeroes(10,10)
pdl> $a(1:3,1:3) .= pdl ( [1,1,1],
..( > [0,0,1],
..( > [0,1,0] )
pdl> p $a
[
[0 0 0 0 0 0 0 0 0 0]
[0 1 1 1 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
pdl> $n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1)
pdl> $n = $n->sumover->sumover - $a
pdl> p $n
[
[1 2 3 2 1 0 0 0 0 0]
[1 1 3 2 2 0 0 0 0 0]
[1 3 5 3 2 0 0 0 0 0]
[0 1 1 2 1 0 0 0 0 0]
[0 1 1 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
For example, this tells us that cell (0,0) has 1 living neighbour, while
cell (2,2) has 5 living neighbours.
=head3 Calculating the next generation
At this point, the variable C<$n> has the number of living neighbours for
every cell. Now we apply the rules of the game of life to calculate the next
generation.
=over 5
=item If an empty cell has exactly three neighbours, a living cell is generated.
Get a list of cells that have exactly three neighbours:
pdl> p ($n == 3)
[
[0 0 1 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
Get a list of B cells that have exactly three neighbours:
pdl> p ($n == 3) * !$a
=item If a living cell has less than 2 or more than 3 neighbours, it dies.
Get a list of cells that have exactly 2 or 3 neighbours:
pdl> p (($n == 2) + ($n == 3))
[
[0 1 1 1 0 0 0 0 0 0]
[0 0 1 1 1 0 0 0 0 0]
[0 1 0 1 1 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
Get a list of B cells that have exactly 2 or 3 neighbours:
pdl> p (($n == 2) + ($n == 3)) * $a
=back
Putting it all together, the next generation is:
pdl> $a = ((($n == 2) + ($n == 3)) * $a) + (($n == 3) * !$a)
pdl> p $a
[
[0 0 1 0 0 0 0 0 0 0]
[0 0 1 1 0 0 0 0 0 0]
[0 1 0 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0]
]
=head2 Bonus feature: Graphics!
If you have L installed, you can
make a graphical version of the program by just changing three lines:
#!/usr/bin/perl
use PDL;
use PDL::NiceSlice;
use PDL::Graphics::TriD;
my $a = zeroes(20,20);
# Put in a simple glider.
$a(1:3,1:3) .= pdl ( [1,1,1],
[0,0,1],
[0,1,0] );
my $n;
for (my $i = 0; $i < 100; $i++) {
# Calculate the number of neighbours per cell.
$n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1);
$n = $n->sumover->sumover - $a;
# Calculate the next generation.
$a = ((($n == 2) + ($n == 3))* $a) + (($n==3) * !$a);
# Display.
nokeeptwiddling3d();
imagrgb [$a];
}
But if we really want to see something interesting, we should make a few more
changes:
1) Start with a random collection of 1's and 0's.
2) Make the grid larger.
3) Add a small timeout so we can see the game evolve better.
4) Use a while loop so that the program can run as long as it needs to.
#!/usr/bin/perl
use PDL;
use PDL::NiceSlice;
use PDL::Graphics::TriD;
use Time::HiRes qw(usleep);
my $a = random(100,100);
$a = ($a < 0.5);
my $n;
while (1) {
# Calculate the number of neighbours per cell.
$n = $a->range(ndcoords($a)-1,3,"periodic")->reorder(2,3,0,1);
$n = $n->sumover->sumover - $a;
# Calculate the next generation.
$a = ((($n == 2) + ($n == 3))* $a) + (($n==3) * !$a);
# Display.
nokeeptwiddling3d();
imagrgb [$a];
# Sleep for 0.1 seconds.
usleep(100000);
}
=head1 CONCLUSION: GENERAL STRATEGY
The general strategy is: I
Threading is a powerful tool that helps eliminate for-loops and can make your
code more concise. Hopefully this tutorial has shown why it is worth getting
to grips with threading in PDL.
=head1 COPYRIGHT
Copyright 2010 Matthew Kenworthy (kenworthy@strw.leidenuniv.nl) and
Daniel Carrera (dcarrera@gmail.com). You can distribute and/or modify this
document under the same terms as the current Perl license.
See: http://dev.perl.org/licenses/