=head1 NAME

PDL::MATLAB - A guide for MATLAB users.

=head1 INTRODUCTION

If you are a MATLAB user, this page is for you. It explains the key
differences between MATLAB and PDL to help you get going as quickly
as possible.

B<This document is not a tutorial>. For that, go to L<PDL::QuickStart|
PDL::QuickStart>. This document B<complements> the Quick Start guide, as
it highlights the key differences between MATLAB and PDL.

=head1 Perl

The key differences between MATLAB and PDL are broadcasting, and B<Perl>.

Broadcasting means you can get a reference to just a part of your data,
and operate on it in a way that makes sense for your application. Those
operations will be reflected in the original data.

Perl is a general purpose programming language with thousands of modules
freely available on the web. PDL is an extension of Perl. This gives PDL
programs access to more features than most numerical tools can dream of.
At the same time, most syntax differences between MATLAB and PDL are a
result of its Perl foundation.

B<You do not have to learn much Perl to be effective with PDL>. But
if you wish to learn Perl, there is excellent documentation available
on-line (L<http://perldoc.perl.org>) or through the command C<perldoc perl>.
There is also a beginner's portal (L<http://perl-begin.org>).

Perl's module repository is called CPAN (L<http://www.cpan.org>) and it
has a vast array of modules. Run C<perldoc cpan> for more information.

=head1 TERMINOLOGY: NDARRAY

MATLAB typically refers to vectors, matrices, and arrays. Perl already
has arrays, and the terms "vector" and "matrix" typically refer to one-
and two-dimensional collections of data. Having no good term to describe
their object, PDL developers coined the term "I<ndarray>" to give a name to
their data type.

An I<ndarray> consists of a series of numbers organized as an N-dimensional
data set. ndarrays provide efficient storage and fast computation of large
N-dimensional matrices. They are highly optimized for numerical work.

For more information, see "B<ndarrays vs Perl Arrays>" later in this document.

=head1 COMMAND WINDOW AND IDE

Unlike MATLAB, PDL does not come with a dedicated IDE. It does however
come with an interactive shell and you can use a Perl IDE to develop
PDL programs.

=head2 PDL interactive shell

To start the interactive shell, open a terminal and run C<perldl>.
As in MATLAB, the interactive shell is the best way to learn the
language. To exit the shell, type C<exit>, just like MATLAB.

=head2 Writing PDL programs

One popular IDE for Perl is called Padre (L<http://padre.perlide.org>).
It is cross platform and easy to use.

Whenever you write a stand-alone PDL program (i.e. outside the
C<perldl> shell) you must start the program with C<use PDL;>.
This command imports the PDL module into Perl. Here is a sample
PDL program:

  use PDL;             # Import main PDL module.
  use PDL::NiceSlice;  # Import additional PDL module.
  use PDL::AutoLoader; # Import additional PDL module.

  $y = pdl [2,3,4];              # Statements end in semicolon.
  $A = pdl [ [1,2,3],[4,5,6] ];  # 2-dimensional matrix.

  print $A x $y->transpose;

Save this file as C<myprogram.pl> and run it with:

  perl myprogram.pl

=head2 New: Flexible syntax

In current versions of PDL (version 2.4.7 or later) there is
a flexible matrix syntax that can look extremely similar to MATLAB:

1) Use spaces to separate elements:

  $y = pdl q[ 2 3 4 ];

2) Use a ';' to delimit rows:

  $A = pdl q[ 1,2,3 ; 4,5,6 ];

Basically, as long as you put a C<q> in front of the opening bracket,
PDL should "do what you mean". So you can write in a syntax that is
more comfortable for you.

=head1 MODULES FOR MATLAB USERS

There are two modules that MATLAB users will want to use:

=over 5

=item L<PDL::NiceSlice>

Gives PDL a syntax for slices (sub-matrices) that is shorter and
more familiar to MATLAB users.

  % MATLAB
  b(1:5)            -->  Selects the first 5 elements from b.

  # PDL without NiceSlice
  $y->slice("0:4")  -->  Selects the first 5 elements from $y.

  # PDL with NiceSlice
  $y(0:4)           -->  Selects the first 5 elements from $y.

=item L<PDL::AutoLoader>

Provides a MATLAB-style autoloader for PDL. If an unknown function
C<foo()> is called, PDL looks for a file called C<foo.pdl>. If it
finds one, it reads it.

=back

=head1 BASIC FEATURES

This section explains how PDL's syntax differs from MATLAB. Most
MATLAB users will want to start here.

=head2 General "gotchas"

=over 5

=item Indices

In PDL, indices start at '0' (like C and Java), not 1 (like MATLAB or FORTRAN).
For example, if C<$y> is an array with 5 elements, the elements would be
numbered from 0 to 4. This is different, but less difficult as soon as
you need to do calculations based on offsets.

=item Displaying an object

MATLAB normally displays object contents automatically unless you
end with a C<;>. In the PDL shells you need to
display objects explicitly with the C<print> command or the shortcut C<p>:

MATLAB:

 >> a = 12
 a =  12
 >> b = 23;       % Suppress output.
 >>

PDL Shell (perldl):

 pdl> $x = 12    # No output.
 pdl> print $x   # Print object.
 12
 pdl> p $x       # "p" is a shorthand for "print" in the shell.
 12
 pdl>

In L<perldl> there is the C<do_print> command that can turn on the "print"
mode, which defaults to off. In "print" mode, expressions you enter on
the command line will have their values printed. Just like MATLAB,
if you end the command with C<;>, nothing will be printed.

=back

=head2 Creating ndarrays

=over 5

=item Variables in PDL

Variables always start with the '$' sign.

 MATLAB:    value  = 42
 PerlDL:    $value = 42

=item Basic syntax

Use the "pdl" constructor to create a new I<ndarray>.

 MATLAB:    v  = [1,2,3,4]
 PerlDL:    $v = pdl [1,2,3,4]

 MATLAB:    A  =      [ 1,2,3  ;  3,4,5 ]
 PerlDL:    $A = pdl [ [1,2,3] , [3,4,5] ]

=item Simple matrices

                      MATLAB       PDL
                      ------       ------
  Matrix of ones      ones(5)      ones 5,5
  Matrix of zeros     zeros(5)     zeros 5,5
  Random matrix       rand(5)      random 5,5
  Linear vector       1:5          sequence 5

Notice that in PDL the parenthesis in a function call are often optional.
It is important to keep an eye out for possible ambiguities. For example:

  pdl> p zeros 2, 2 + 2

Should this be interpreted as C<zeros(2,2) + 2> or as C<zeros 2, (2+2)>?
Both are valid statements:

  pdl> p zeros(2,2) + 2
  [
   [2 2]
   [2 2]
  ]
  pdl> p zeros 2, (2+2)
  [
   [0 0]
   [0 0]
   [0 0]
   [0 0]
  ]

Rather than trying to memorize Perl's order of precedence, it is best
to use parentheses to make your code unambiguous. Remember you may need
to come back to your code, and parentheses make your own (as well as
others') comprehension easier.

=item Linearly spaced sequences

  MATLAB:   >> linspace(2,10,5)
            ans = 2 4 6 8 10

  PerlDL:   pdl> p zeroes(5)->xlinvals(2,10)
            [2 4 6 8 10]

B<Explanation>: Start with a 1-dimensional ndarray of 5 elements and give
it equally spaced values from 2 to 10.

MATLAB has a single function call for this. On the other hand, PDL's
method is more flexible:

  pdl> p zeros(5,5)->xlinvals(2,10)
  [
   [ 2  4  6  8 10]
   [ 2  4  6  8 10]
   [ 2  4  6  8 10]
   [ 2  4  6  8 10]
   [ 2  4  6  8 10]
  ]
  pdl> p zeros(5,5)->ylinvals(2,10)
  [
   [ 2  2  2  2  2]
   [ 4  4  4  4  4]
   [ 6  6  6  6  6]
   [ 8  8  8  8  8]
   [10 10 10 10 10]
  ]
  pdl> p zeros(3,3,3)->zlinvals(2,6)
  [
   [
    [2 2 2]
    [2 2 2]
    [2 2 2]
   ]
   [
    [4 4 4]
    [4 4 4]
    [4 4 4]
   ]
   [
    [6 6 6]
    [6 6 6]
    [6 6 6]
   ]
  ]

=item Slicing and indices

Extracting a subset from a collection of data is known as I<slicing>.
PDL and MATLAB have a similar syntax for slicing, but there are two
important differences:

1) PDL indices start at 0, as in C and Java. MATLAB starts indices at 1.

2) In MATLAB you think "rows and columns". In PDL, think "x and y".

  MATLAB                         PerlDL
  ------                         ------
  >> A                           pdl> p $A
  A =                            [
       1   2   3                  [1 2 3]
       4   5   6                  [4 5 6]
       7   8   9                  [7 8 9]
                                 ]
  -------------------------------------------------------
  (row = 2, col = 1)             (x = 0, y = 1)
  >> A(2,1)                      pdl> p $A(0,1)
  ans =                          [
         4                        [4]
                                 ]
  -------------------------------------------------------
  (row = 2 to 3, col = 1 to 2)   (x = 0 to 1, y = 1 to 2)
  >> A(2:3,1:2)                  pdl> p $A(0:1,1:2)
  ans =                          [
         4   5                    [4 5]
         7   8                    [7 8]
                                 ]

=over 5

=item B<Warning>

When you write a stand-alone PDL program, if you want the "nice slice" syntax, you have
to include the L<PDL::NiceSlice> module. See the
previous section "B<MODULES FOR MATLAB USERS>" for more information.

  use PDL;             # Import main PDL module.
  use PDL::NiceSlice;  # Nice syntax for slicing.
  use PDL::AutoLoader; # MATLAB-like autoloader.

  $A = random 4,4;
  print $A(0,1);

=back

=back

=head2 Matrix Operations

=over 10

=item Matrix multiplication

 MATLAB:    A * B
 PerlDL:    $A x $B

=item Element-wise multiplication

 MATLAB:    A .* B
 PerlDL:    $A * $B

=item Transpose

 MATLAB:    A'
 PerlDL:    $A->transpose

=back


=head2 Functions that aggregate data

Some functions (like C<sum>, C<max> and C<min>) aggregate data
for an N-dimensional data set. This is a place where MATLAB and
PDL take a different approach:

=over 10

=item In MATLAB, these functions all work along one dimension.

  >> A = [ 1,5,4  ;  4,2,1 ]
  A = 1  5  4
      4  2  1
  >> max(A)
  ans = 4  5  4
  >> max(A')
  ans = 5  4

If you want the maximum for the entire data set, you can use the special
C<A(:)> notation which basically turns the entire data set into a single
1-dimensional vector.

  >> max(A(:))
  ans =  5
  >> A = ones(2,2,2,2)
  >> max(A(:))
  ans = 1

=item PDL offers two functions for each feature.

  sum   vs   sumover
  avg   vs   average
  max   vs   maximum
  min   vs   minimum

The B<long name> works over a dimension, while the B<short name>
works over the entire ndarray.

  pdl> p $A = pdl [ [1,5,4] , [4,2,1] ]
  [
   [1 5 4]
   [4 2 1]
  ]
  pdl> p $A->maximum
  [5 4]
  pdl> p $A->transpose->maximum
  [4 5 4]
  pdl> p $A->max
  5
  pdl> p ones(2,2,2)->max
  1
  pdl> p ones(2,2,2,2)->max
  1

=back

=over 5

=item B<Note>

Notice that PDL aggregates horizontally while MATLAB aggregates
vertically. In other words:

  MATLAB              PerlDL
  max(A)       ==     $A->transpose->maximum
  max(A')      ==     $A->maximum

B<TIP>: In MATLAB you think "rows and columns". In PDL, think "x and y".

=back

=head2 Higher dimensional data sets

A related issue is how MATLAB and PDL understand data sets of higher
dimension. MATLAB was designed for 1D vectors and 2D matrices. Higher
dimensional objects ("N-D arrays") were added on top. In contrast, PDL
was designed for N-dimensional ndarrays from the start. This leads to
a few surprises in MATLAB that don't occur in PDL:

=over 5

=item MATLAB sees a vector as a 2D matrix.

  MATLAB                       PerlDL
  ------                       ------
  >> vector = [1,2,3,4];       pdl> $vector = pdl [1,2,3,4]
  >> size(vector)              pdl> p $vector->dims
  ans = 1 4                    4

MATLAB sees C<[1,2,3,4]> as a 2D matrix (1x4 matrix). PDL sees it
as a 1D vector: A single dimension of size 4.

=item But MATLAB ignores the last dimension of a 4x1x1 matrix.

  MATLAB                       PerlDL
  ------                       ------
  >> A = ones(4,1,1);          pdl> $A = ones 4,1,1
  >> size(A)                   pdl> p $A->dims
  ans = 4 1                    4 1 1

=item And MATLAB treats a 4x1x1 matrix differently from a 1x1x4 matrix.

  MATLAB                       PerlDL
  ------                       ------
  >> A = ones(1,1,4);          pdl> $A = ones 1,1,4
  >> size(A)                   pdl> p $A->dims
  ans = 1 1 4                  1 1 4

=item MATLAB has no direct syntax for N-D arrays.

  pdl> $A = pdl [ [[1,2,3],[4,5,6]], [[2,3,4],[5,6,7]] ]
  pdl> p $A->dims
  3 2 2

=item Feature support.

In MATLAB, several features such as sparse matrix support are not
available for N-D arrays. In PDL, just about any feature supported by
1D and 2D ndarrays, is equally supported by N-dimensional ndarrays.
There is usually no distinction.

=back

=head2 Loop Structures

Perl has many loop structures, but we will only show the one that
is most familiar to MATLAB users:

  MATLAB              PerlDL
  ------              ------
  for i = 1:10        for $i (1..10) {
      disp(i)             print $i
  endfor              }

=over 5

=item B<Note>

Never use for-loops for numerical work. Perl's for-loops are faster
than MATLAB's, but they both pale against a "vectorized" operation.
PDL has many tools that facilitate writing vectorized programs.
These are beyond the scope of this guide. To learn more, see:
L<PDL::Indexing>, L<PDL::Broadcasting>,
and L<PDL::PP>.

Likewise, never use C<1..10> for numerical work, even outside a for-loop.
C<1..10> is a Perl array. Perl arrays are designed for flexibility, not
speed. Use I<ndarrays> instead. To learn more, see the next section.

=back

=head2 ndarrays vs Perl Arrays

It is important to note the difference between an I<ndarray> and a Perl
array. Perl has a general-purpose array object that can hold any
type of element:

  @perl_array = 1..10;
  @perl_array = ( 12, "Hello" );
  @perl_array = ( 1, 2, 3, \@another_perl_array, sequence(5) );

Perl arrays allow you to create powerful data structures (see
B<Data structures> below), B<but they are not designed for numerical work>.
For that, use I<ndarrays>:

  $pdl = pdl [ 1, 2, 3, 4 ];
  $pdl = sequence 10_000_000;
  $pdl = ones 600, 600;

For example:

  $points =  pdl  1..10_000_000    # 4.7 seconds
  $points = sequence 10_000_000    # milliseconds

B<TIP>: You can use underscores in numbers (C<10_000_000> reads better
than C<10000000>).

=head2 Conditionals

Perl has many conditionals, but we will only show the one that is
most familiar to MATLAB users:

  MATLAB                          PerlDL
  ------                          ------
  if value > MAX                  if ($value > $MAX) {
      disp("Too large")               print "Too large\n";
  elseif value < MIN              } elsif ($value < $MIN) {
      disp("Too small")               print "Too small\n";
  else                            } else {
      disp("Perfect!")                print "Perfect!\n";
  end                             }

=over 5

=item B<Note>

Here is a "gotcha":

  MATLAB:  elseif
  PerlDL:  elsif

If your conditional gives a syntax error, check that you wrote
your C<elsif>'s correctly.

=back

=head2 TIMTOWDI (There Is More Than One Way To Do It)

One of the most interesting differences between PDL and other tools
is the expressiveness of the Perl language. TIMTOWDI, or "There Is
More Than One Way To Do It", is Perl's motto.

Perl was written by a linguist, and one of its defining properties
is that statements can be formulated in different ways to give the
language a more natural feel. For example, you are unlikely to say
to a friend:

 "While I am not finished, I will keep working."

Human language is more flexible than that. Instead, you are more
likely to say:

 "I will keep working until I am finished."

Owing to its linguistic roots, Perl is the only programming language
with this sort of flexibility. For example, Perl has traditional
while-loops and if-statements:

  while ( ! finished() ) {
      keep_working();
  }

  if ( ! wife_angry() ) {
      kiss_wife();
  }

But it also offers the alternative B<until> and B<unless> statements:

  until ( finished() ) {
      keep_working();
  }

  unless ( wife_angry() ) {
      kiss_wife();
  }

And Perl allows you to write loops and conditionals in "postfix" form:

  keep_working() until finished();

  kiss_wife() unless wife_angry();

In this way, Perl often allows you to write more natural, easy to
understand code than is possible in more restrictive programming
languages.

=head2 Functions

PDL's syntax for declaring functions differs significantly from MATLAB's.

  MATLAB                          PerlDL
  ------                          ------
  function retval = foo(x,y)      sub foo {
      retval = x.**2 + x.*y           my ($x, $y) = @_;
  endfunction                         return $x**2 + $x*$y;
                                  }

Don't be intimidated by all the new syntax. Here is a quick run through
a function declaration in PDL:

1) "B<sub>" stands for "subroutine".

2) "B<my>" declares variables to be local to the function. This helps
you not accidentally use undeclared variables, which is enforced if you
C<use strict>. See L<strict> for more.

3) "B<@_>" is a special Perl array that holds all the function parameters.
This might seem like a strange way to do functions, but it allows you
to make functions that take a variable number of parameters. For example,
the following function takes any number of parameters and adds them
together:

  sub mysum {
      my ($i, $total) = (0, 0);
      for $i (@_) {
          $total += $i;
      }
      return $total;
  }

In more recent versions of Perl, you can C<use signatures> for a different
syntax for declaring function parameters. See L<signatures> for more.

4) You can assign values to several variables at once using the syntax:

  ($x, $y, $z) = (1, 2, 3);

So, in the previous examples:

  # This declares two local variables and initializes them to 0.
  my ($i, $total) = (0, 0);

  # This takes the first two elements of @_ and puts them in $x and $y.
  my ($x, $y) = @_;

5) The "B<return>" statement gives the return value of the function, if any.

=head1 ADDITIONAL FEATURES

=head2 ASCII File IO

To read data files containing whitespace separated columns of
numbers (as would be read using the MATLAB I<load> command)
one uses the PDL I<rcols> in L<PDL::IO::Misc>.  For a general
review of the IO functionality available in PDL, see the
documentation for L<PDL::IO>, e.g., C<help PDL::IO> in the I<perldl>
shell or C< pdldoc PDL::IO > from the shell command line.

=head2 Data structures

To create complex data structures, MATLAB uses "I<cell arrays>" and
"I<structure arrays>". Perl's arrays and hashes offer similar functionality
but are more powerful and flexible. This section is only a quick overview
of what Perl has to offer. To learn more about this, please go to
L<http://perldoc.perl.org/perldata.html> or run the command C<perldoc perldata>.

=over 5

=item Arrays

Perl arrays are similar to MATLAB's cell arrays, but more flexible. For
example, in MATLAB, a cell array is still fundamentally a matrix. It is
made of rows, and rows must have the same length.

  MATLAB
  ------
  array = {1, 12, 'hello'; rand(3, 2), ones(3), 'junk'}
  => OK
  array = {1, 12, 'hello'; rand(3, 2), ones(3) }
  => ERROR

A Perl array is a general purpose, sequential data structure. It can contain
any data type.

  PerlDL
  ------
  @array = ( [1, 12, 'hello'] , [ random(3,2), ones(3,3), 'junk' ] )
  => OK
  @array = ( [1, 12, 'hello'] , [ random(3,2), ones(3,3) ] )
  => OK
  @array = ( 5 , {'name' => 'Mike'} , [1, 12, 'hello'] )
  => OK

Notice that Perl array's start with the "@" prefix instead of the "$" used by
ndarrays.

I<To learn about Perl arrays, please go to L<http://perldoc.perl.org/perldata.html>
or run the command C<perldoc perldata>.>

=item Hashes

Perl hashes are similar to MATLAB's structure arrays:

  MATLAB
  ------
  >> drink = struct('type', 'coke', 'size', 'large', 'myarray', {1,2,3})
  >> drink.type = 'sprite'
  >> drink.price = 12          % Add new field to structure array.

  PerlDL
  ------
  pdl> %drink = ( type => 'coke' , size => 'large', myndarray => ones(3,3,3) )
  pdl> $drink{type} = 'sprite'
  pdl> $drink{price} = 12   # Add new field to hash.

Notice that Perl hashes start with the "%" prefix instead of the "@" for
arrays and "$" used by ndarrays.

I<To learn about Perl hashes, please go to L<http://perldoc.perl.org/perldata.html>
or run the command C<perldoc perldata>.>

=back

=head2 Performance

PDL has powerful performance features, some of which are not normally
available in numerical computation tools. The following pages will guide
you through these features:

=over 5

=item L<PDL::Indexing>

B<Level>: Beginner

This beginner tutorial covers the standard "vectorization" feature that
you already know from MATLAB. Use this page to learn how to avoid for-loops
to make your program more efficient.

=item L<PDL::Broadcasting>

B<Level>: Intermediate

PDL's "vectorization" feature goes beyond what most numerical software
can do. In this tutorial you'll learn how to "broadcast" over higher dimensions,
allowing you to vectorize your program further than is possible in MATLAB.

=item Benchmarks

B<Level>: Intermediate

Perl comes with an easy to use benchmarks module to help you find how
long it takes to execute different parts of your code. It is a great
tool to help you focus your optimization efforts. You can read about it
online (L<http://perldoc.perl.org/Benchmark.html>) or through the
command C<perldoc Benchmark>.

=item L<PDL::PP>

B<Level>: Advanced

PDL's Pre-Processor is one of PDL's most powerful features. You
write a function definition in special markup and the pre-processor
generates real C code which can be compiled. With PDL:PP
you get the full speed of native C code without having to deal with
the full complexity of the C language.

=back

=head2 Plotting

PDL has full-featured plotting abilities. Unlike MATLAB, PDL relies more on
third-party libraries (pgplot and PLplot) for its 2D plotting features.
Its 3D plotting and graphics uses OpenGL for performance and portability.
PDL has three main plotting modules:

=over 5

=item L<PDL::Graphics::Simple>

B<Best for>: Plotting 2D functions and data sets.

Provides a uniform interface to PGPLOT, PLplot, Gnuplot, and Prima.

=item L<PDL::Graphics::Gnuplot>

B<Best for>: Plotting 2D functions as well as 2D and 3D data sets.

This is an interface to Gnuplot,
a modern, open source program for making scientific plots.
It supports plots of both 2D and 3D data sets. It is
supported for Unix/Linux/MacOS/Win32 platforms, with an active
developers community.

=item L<PDL::Graphics::TriD>

B<Best for>: Plotting 3D functions.

The native PDL 3D graphics library using OpenGL as a backend
for 3D plots and data visualization. With OpenGL, it is easy
to manipulate the resulting 3D objects with the mouse in real
time.

=back

=head2 Writing GUIs

Through Perl, PDL has access to all the major toolkits for creating
a cross platform graphical user interface. One popular option is
wxPerl (L<http://wxperl.sourceforge.net>). These are the Perl bindings
for wxWidgets, a powerful GUI toolkit for writing cross-platform
applications.

wxWidgets is designed to make your application look and feel like
a native application in every platform. For example, the Perl
IDE B<Padre> is written with wxPerl.

=head2 Simulink

Simulink is a graphical dynamical system modeler and simulator. It
can be purchased separately as an add-on to MATLAB.
PDL and Perl do not have a direct equivalent to MATLAB's Simulink.
If this feature is important to you, then take a look at B<Scilab>:

L<http://www.scilab.org>

Scilab is another numerical analysis software. Like PDL, it is free
and open source. It doesn't have PDL's unique features, but it is
very similar to MATLAB. Scilab comes with B<Xcos> (previously Scicos),
a graphical system modeler and simulator similar to Simulink.

=head1 COMPARISON: REPEATED COPY OF MATRIX

In MATLAB, the C<repmat> function works like so:

  > A = reshape(0:5, 3, 2)' # similar to PDL::sequence(3, 2)
  ans =
     0   1   2
     3   4   5
  > repmat(A, 2, 3) # double rows, triple columns
  ans =
     0   1   2   0   1   2   0   1   2
     3   4   5   3   4   5   3   4   5
     0   1   2   0   1   2   0   1   2
     3   4   5   3   4   5   3   4   5

This works (at least in Octave) at least up to 3 dimensions.

The PDL analog:

  sub repmat {
    my $f=shift;
    my @n=@_; #number of repetitions along dimension
    my $sl = join ',', map ":,*$_", @n; # insert right-size dummy after each real
    my $r = $f->slice($sl); #result
    $r = $r->clump($_, $_+1) for 0..$#n;
    $r;
  }
  > p $x = sequence(3,2)
  [
   [0 1 2]
   [3 4 5]
  ]
  > p repmat($x, 3, 2) # triple columns, double rows
  [
   [0 1 2 0 1 2 0 1 2]
   [3 4 5 3 4 5 3 4 5]
   [0 1 2 0 1 2 0 1 2]
   [3 4 5 3 4 5 3 4 5]
  ]

=head1 COMPARISON: FLOYD-WARSHALL ALGORITHM

In L<graph theory|https://en.wikipedia.org/wiki/Graph_theory>, an
apparently-simple but difficult problem is the "shortest path" problem,
of finding the shortest path between any two nodes. A famous solution
to this, albeit expensive (it is C<O(V^3)> where C<V> is the number of
vertices) is the Floyd-Warshall algorithm, which iterates through all
the possible paths.

Both the MATLAB solution and the PDL solution use vectorisation, so
hopefully this is a useful comparison. The MATLAB version started with
the code in
L<code by Giorgos Dim|https://uk.mathworks.com/matlabcentral/fileexchange/67503-floyd-warshall-vectorized>,
but modified as that code produces an incorrect path matrix.

Sample data (reflected on both the Wikipedia page, and the Rosetta Code
website) for the weighted-edges matrix is, in PDL format:

  my $we = pdl q[
   [Inf Inf  -2 Inf]
   [  4 Inf   3 Inf]
   [Inf Inf Inf   2]
   [Inf  -1 Inf Inf]
  ];

and in MATLAB format:

  A = [0 Inf -2 Inf; 4 0 3 Inf; Inf Inf 0 2; Inf -1 Inf 0]

=head2 PDL version

To solve for only distances without capturing the shortest actual paths:

  $we .= $we->hclip($we->mslice('X', $_) + $we->mslice($_, 'X'))
    for 0..($we->dim(0)-1);

This loops over each possible intermediate point (C<k> in the
other literature), setting it to C<$_> (a Perl idiom). It uses
L<PDL::Primitive/hclip> for vectorised calculation of the distance
between the intermediate point's predecessors and successors. Those are
the two components of the addition expression, using "slices" alluded
to above. The C<.=> is the PDL syntax for updating an ndarray.

To capture the shortest-path "next vertex" matrix as well:

  use PDL::Lite;
  my $d = $we->copy->inplace;
  $d->diagonal(0, 1) .= 0;
  my $suc = $we->copy->inplace;
  my $adjacent_coords = PDL::whichND($we->isfinite);
  $suc->indexND($adjacent_coords) .= $adjacent_coords->slice(0)->flat;
  $suc->diagonal(0, 1) .= PDL::Basic::sequence($d->dim(0));
  for (my $k = $d->dim(0)-1; $k >= 0; $k--) {
    my $d_copy = $d->copy;
    $d .= $d->hclip($d->mslice('X', $k) + $d->mslice($k, 'X'));
    my $coords = PDL::whichND($d < $d_copy);
    my $from_coords = $coords->copy->inplace;
    $from_coords->slice(0) .= $k;
    $suc->indexND($coords) .= $suc->indexND($from_coords);
  }

The C<diagonal> and C<slice> expressions show how to update data via a
query syntax.

=head2 MATLAB version

Path-lengths only:

  function D = FloydWarshall(D)
    for k = 1:length(D)
      D = min(D,D(:,k) + D(k,:));
    end
  end

The path vertices-capturing as well:

  function [D,P] = FloydWarshall(D)
    P = D;
    n = length(D);
    coords = find(isfinite(P));
    P(coords) = floor((coords-1) / n)+1; % the col in 1-based
    for v = 1:n; P(v, v) = v; end
    for k = 1:n
      prevD = D;
      D = min(D,D(:,k) + D(k,:));
      coords = find(D<prevD);
      from_coords = n * (k-1) + mod(coords-1, n) + 1; % change col to k in 1-based
      P(coords) = P(from_coords);
    end
  end

By comparison, the lack of "broadcasting" means that to update the diagonal
requires a for-loop, which in the sphere of vectorised calculations is
a bad thing. The calculations of coordinates are complicated by the
1-based counting.

=head1 COPYRIGHT

Copyright 2010 Daniel Carrera (dcarrera@gmail.com). You can distribute and/or
modify this document under the same terms as the current Perl license.

See: L<http://dev.perl.org/licenses/>

=head1 ACKNOWLEDGEMENTS

I'd like to thank David Mertens, Chris Marshall and Sigrid Carrera for
their immense help reviewing earlier drafts of this guide. Without their
hours of work, this document would not be remotely as useful to MATLAB
users as it is today.
