=head1 NAME

PDL::Indexing - Introduction to indexing and slicing ndarrays.

=head1 OVERVIEW

This man page should serve as a first tutorial on the indexing and
broadcasting features of I<PDL>.

Like all vectorized languages, PDL automates looping over multi-dimensional data structures ("ndarrays") using
a variant of mathematical vector notation.  The automatic looping is called
"broadcasting", similar to NumPy and Julia. PDL also automatically runs
broadcasting computation in parallel - see L<PDL::ParallelCPU>.

A lot of the flexibility and power of PDL relies on the indexing and broadcasting
features of the Perl extension.  Indexing allows access to the data of an ndarray
in a very flexible way.  Broadcasting provides efficient vectorization of simple
operations.

The values of an ndarray are stored compactly as typed values in a single block of memory,
not (as in a normal Perl list-of-lists) as individual Perl scalars.

In the sections that follow many "methods" are called out -- these are Perl operators
that apply to ndarrays.  From the L<perldl> shell, you
can find out more about each method by typing "?" followed by the method name.

=head2 Dimension lists

A ndarray (PDL variable), in general, is an N-dimensional array where N can be
0 (for a scalar), 1 (e.g. for a sound sample), or higher values for images
and more complex structures.  Each dimension of the ndarray has a positive
integer size.  The C<perl> interpreter treats each ndarray as a special type of
Perl scalar (a blessed Perl object, actually -- but you don't have to know that
to use them) that can be used anywhere you can put a normal scalar.

You can access the dimensions of an ndarray as a Perl list and otherwise determine
the size of an ndarray with several methods.  The important ones are:

=over 3

=item nelem - the total number of elements in an ndarray

=item ndims - returns the number of dimensions in an ndarray

=item dims - returns the dimension list of an ndarray as a Perl list

=item dim - returns the size of a particular dimension of an ndarray

=back

=head2 Indexing and Dataflow

PDL maintains a notion of "dataflow" between an ndarray and indexed subfields of
that ndarray.  When you produce an indexed subfield or single element of a parent
ndarray, the child and parent remain attached until you manually disconnect them.
This lets you represent the same data different ways within your code -- for example,
you can consider an RGB image simultaneously as a collection of (R,G,B) values
in a 3 x 1000 x 1000 image, and as three separate 1000 x 1000 color planes stored in
different variables.  Modifying any of the variables changes the underlying memory, and
the changes are reflected in all representations of the data.

There are two important methods that let you control dataflow connections between a
child and parent ndarray:

=over 3

=item copy - forces an explicit copy of an ndarray

=item sever - breaks the dataflow connection between an ndarray and its parents (if any)

=back

=head2 Broadcasting and Dimension Order

Most PDL operations act on the first few dimensions of their ndarray arguments.  For
example, C<sumover> sums all elements along the first dimension in the list (dimension 0).
If you feed in a three-dimensional ndarray, then the first dimension is considered the
"active" dimension and the later dimensions are "broadcast" dimensions because they are simply
looped over.  There are several ways to transpose or re-order the dimension list of an ndarray.
Those techniques are very fast since they don't touch the underlying data, only change the
way that PDL accesses the data.  The main dimension ordering functions are:

=over 3

=item mv - moves a particular dimension somewhere else in the dimension list

=item xchg - exchanges two dimensions in the dimension list, leaving the rest alone

=item reorder - allows wholesale mixing of the dimensions

=item clump - clumps together two or more small dimensions into one larger one

=item squeeze - eliminates any dimensions of size 1


=back

=head2 Physical and Dummy Dimensions

=over 5

=item *

document Perl level broadcasting (see L<PDL::Core/broadcast_define>)

=item *

broadcastids (see L</Explicit broadcasting>)

=item *

L<PDL::Slices/slice> is crucial, useful in niches are L<PDL::Slices/lags>
and L<PDL::Slices/splitdim>

=back

=head1 Indexing and broadcasting with PDL

A lot of the flexibility and power of PDL relies on the indexing and
looping features of the Perl extension. Indexing allows access to the
data of an ndarray in a very flexible way. Broadcasting provides
efficient implicit looping functionality (since the loops are
implemented as optimized C code).

ndarrays are Perl objects that
represent multidimensional arrays and operations on those. In contrast
to simple Perl C<@x> style lists the array data is compactly stored in
a single block of memory thus taking up a lot less memory and enabling
use of fast C code to implement operations (e.g. addition,
etc) on ndarrays.

=head2 ndarrays can have children

Central to many of the indexing capabilities of PDL are the relation of
"parent" and "child" between ndarrays. Many of the indexing commands
create a new ndarray from an existing ndarray. The new ndarray is the "child"
and the old one is the "parent". The data of the new ndarray is defined by a
transformation that specifies how to generate (compute) its data from
the parent's data.

Another way to interpret the ndarrays created by our indexing commands is
to view them as a kind of intelligent pointer that points back to some
portion or all of its parent's data. Therefore, it is not surprising
that the parent's data (or a portion of it) changes when manipulated
through this "pointer". We will further illustrate the pointer/dataflow
analogies in the context of some of the examples later on.

There are two different implementations of this ``smart pointer''
relationship: the first one, which is a little slower but works
for any transformation is simply to do the transformation forwards
and backwards as necessary. The other is to consider the child ndarray
a ``virtual'' ndarray, which only stores a pointer to the parent
and access information so that routines which use the child ndarray
actually directly access the data in the parent.
If the virtual ndarray is given to a routine which cannot use it,
PDL transparently physicalizes the virtual ndarray before letting
the routine use it.

Currently (still true as of 2.086) all transformations which are ``affine'',
i.e. the indices of the data item in the parent ndarray are determined
by a linear transformation (+ constant) from the indices of the
child ndarray result in virtual affine, or "vaffine" ndarrays. All other indexing
routines (e.g. C<< ->index(...) >>) result in physical ndarrays.
All routines compiled by PP can accept affine ndarrays, except
those routines that pass pointers to external library functions,
and those will simply be "physicalised" (memory allocated, data copied in).
There will then be dataflow back and forth, in a lazy-evaluation way.
See L<PDL::Dataflow> for more.

Note that whether something is affine or not does not affect the semantics
of what you do in any way: both

 $x->index(...) .= 5;
 $x->slice(...) .= 5;

change the data in C<$x>. The affinity does, however, have a significant
impact on memory usage and performance.

=head2 Slicing ndarrays

Probably the most important application of the concept of parent/child
ndarrays is the representation of rectangular slices of a physical ndarray by
a virtual ndarray. Having talked long enough about concepts let's get more
specific. Suppose we are working with a 2D ndarray representing a 5x5
image (it's unusually small so that we can print it without filling
several screens full of digits).

 pdl> $im = sequence(5,5)
 pdl> p $im

 [
  [ 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]
 ]

 pdl> help vars
 PDL variables in package main::

 Name         Type   Dimension       Flow  State          Mem
 ----------------------------------------------------------------
 $im          Double D [5,5]                P            0.20Kb

The interactive C<perldl> shells that come with PDL have
a C<help vars> command that provides information about ndarrays. Use
C<help help> for more.

Now suppose we want to create a 1-D ndarray that just references
one line of the image, say line 2; or an ndarray that represents all even
lines of the image (imagine we have to deal with even and odd frames
of an interlaced image due to some peculiar behaviour of our frame
grabber). As another frequent application of slices we might want to
create an ndarray that represents a rectangular region of the image with
top and bottom reversed. All these effects (and many more) can be
easily achieved with the powerful slice function:

 pdl> $line = $im->slice(':,(2)')
 pdl> $even = $im->slice(':,1:-1:2')
 pdl> $area = $im->slice('3:4,3:1')
 pdl> help vars  # or just PDL->vars
 PDL variables in package main::

 Name         Type   Dimension       Flow  State          Mem
 ----------------------------------------------------------------
 $even        Double D [5,2]                -C           0.00Kb
 $im          Double D [5,5]                P            0.20Kb
 $line        Double D [5]                  -C           0.00Kb
 $area        Double D [2,3]                -C           0.00Kb

All three "child" ndarrays are children of C<$im> or in the other (largely
equivalent) interpretation pointers to data of C<$im>.  Operations on
those virtual ndarrays access only those portions of the data as specified
by the argument to slice. So we can just print line 2:

 pdl> p $line
 [10 11 12 13 14]

Also note the difference in the "Flow State" of C<$area> above
and below:

 pdl> p $area
 pdl> help $area
 This variable is Double D [2,3]                VC           0.00Kb

The following demonstrates that C<$im> and C<$line> really behave as you
would expect from a pointer-like object (or in the dataflow picture:
the changes in C<$line>'s data are propagated back to C<$im>):

 pdl> $im++
 pdl> p $line
 [11 12 13 14 15]
 pdl> $line += 2
 pdl> p $im

 [
  [ 1  2  3  4  5]
  [ 6  7  8  9 10]
  [13 14 15 16 17]
  [16 17 18 19 20]
  [21 22 23 24 25]
 ]

Note how assignment operations on the child virtual ndarrays change the
parent physical ndarray and vice versa (however, the basic "=" assignment
doesn't, use ".=" to obtain that effect. See below for the reasons).
The virtual child ndarrays are
something like "live links" to the "original" parent ndarray. As
previously said, they can be thought of to work similar to a
C-pointer. But in contrast to a C-pointer they carry a lot more
information. Firstly, they specify the structure of the data they
represent (the dimensionality of the new ndarray) and secondly, specify
how to create this structure from its parents data (the way this works
is buried in the internals of PDL and not important for you to know
anyway (unless you want to hack the core in the future or would like
to become a PDL guru in general (for a definition of this strange
creature see L<PDL::Internals>)).

The previous examples have demonstrated typical usage of the slice
function. Since the slicing functionality is so important here is an
explanation of the syntax for the string argument to slice:

 $vpdl = $x->slice('ind0,ind1...')

where C<ind0> specifies what to do with index No 0 of the ndarray C<$x>,
etc. Each element of the comma separated list can have one of the
following forms:

=over 6

=item ':'

Use the whole dimension

=item 'n'

Use only index C<n>. The dimension of this index in the
resulting virtual ndarray is 1. An example involving those first two index
formats:

 pdl> $column = $im->slice('2,:')
 pdl> $row = $im->slice(':,0')
 pdl> p $column

 [
  [ 3]
  [ 8]
  [15]
  [18]
  [23]
 ]

 pdl> p $row

 [
  [1 2 3 4 5]
 ]

 pdl> help $column
 This variable is Double D [1,5]                VC           0.00Kb

 pdl> help $row
 This variable is Double D [5,1]                VC           0.00Kb

=item '(n)'

Use only index C<n>. This dimension is removed from the
resulting ndarray (relying on the fact that a dimension of size 1 can always be
removed). The distinction between this case and the previous one
becomes important in assignments where left and right hand side have to
have appropriate dimensions.

 pdl> $line = $im->slice(':,(0)')
 pdl> help $line
 This variable is Double D [5]                  -C           0.00Kb

 pdl> p $line
 [1 2 3 4 5]

Spot the difference to the previous example?

=item 'n1:n2' or 'n1:n2:n3'

Take the range of indices from C<n1> to C<n2> or (second form)
take the range of indices from C<n1> to C<n2> with step
C<n3>. An example for the use of this format is the previous
definition of the sub-image composed of even lines.

 pdl> $even = $im->slice(':,1:-1:2')

This example also demonstrates that negative indices work like they do
for normal Perl style arrays by counting backwards from the end of the
dimension. If C<n2> is smaller than C<n1> (in the example -1 is
equivalent to index 4) the elements in the virtual ndarray are effectively
reverted with respect to its parent.

=item '*[n]'

Add a dummy dimension. The size of this dimension will be 1 by default
or equal to C<n> if the optional numerical argument is given.

Now, this is really something a bit strange on first sight. What is a
dummy dimension? A dummy dimension inserts a dimension where there
wasn't one before. How is that done ? Well, in the case of the new
dimension having size 1 it can be easily explained by the way in which
you can identify a vector (with C<m> elements) with an C<(1,m)> or C<(m,1)>
matrix. The same holds obviously for higher dimensional objects. More
interesting is the case of a dummy dimensions of size greater than one
(e.g. C<slice('*5,:')>). This works in the same way as a call to the
L<dummy|PDL::Core/dummy> function creates a new dummy dimension.
So read on and check
its explanation below.

=back

=head3 Generalised diagonals

With L<PDL::Slices/diagonal> you can make I<generalised diagonals>.
The best way to explain it is probably to
give an example, here we make an ndarray that refers to the elements along
the space diagonal of its parent ndarray (a cube):

 $cube = zeroes(5,5,5);
 $sdiag = $cube->diagonal(1..3);

The above command creates a virtual ndarray that represents the diagonal
along the parents' dimension no. 0, 1 and 2 and makes its dimension 0
(the only dimension) of it. You use the extended syntax if the
dimension sizes of the parent dimensions you want to build the
diagonal from have different sizes or you want to reverse the sequence
of elements in the diagonal, e.g.

 $rect = zeroes(12,3,5,6,2);
 $vpdl = $rect->slice('2:7,0:1,(4),5:4'); # now 6,2,2,2
 $vpdl = $vpdl->diagonal(1..3); # now 6,2

So the elements of $vpdl will then be related to those of its parent
in way we can express as:

  vpdl(i,j) = rect(i+2,j,4,5-j,j)       0<=i<6, 0<=j<2

=head2 There are different kinds of assignments in PDL

The previous examples have already shown that virtual ndarrays can be used
to operate on or access portions of data of a parent ndarray. They can
also be used as lvalues in assignments (as the use of C<++> in some of
the examples above has already demonstrated). For explicit assignments
to the data represented by a virtual ndarray you have to use the
overloaded C<.=> operator (which in this context we call I<propagated
assignment>). Why can't you use the normal assignment operator C<=>?

Well, you definitely still can use the '=' operator but it wouldn't do
what you want. In recent Perls, the '=' operator can be
overloaded in the same way as other assignment operators, but for
back-compatibility reasons PDL uses '=' for a reference copy, so
the variable representing the virtual ndarray (a reference to a
blessed thingy) would after the assignment just contain the reference
to another blessed thingy which would behave to future assignments as
a "physical" copy of the original rvalue.

E.g.

 pdl> $line = $im->slice(':,(2)')
 pdl> $line = zeroes(5);
 pdl> $line++;
 pdl> p $im

 [
  [ 1  2  3  4  5]
  [ 6  7  8  9 10]
  [13 14 15 16 17]
  [16 17 18 19 20]
  [21 22 23 24 25]
 ]

 pdl> p $line
 [1 1 1 1 1]

But using C<.=>

 pdl> $line = $im->slice(':,(2)')
 pdl> $line .= zeroes(5)
 pdl> $line++
 pdl> p $im

 [
  [ 1  2  3  4  5]
  [ 6  7  8  9 10]
  [ 1  1  1  1  1]
  [16 17 18 19 20]
  [21 22 23 24 25]
 ]

 pdl> print $line
 [1 1 1 1 1]

Also, you can substitute

 pdl> $line .= 0;

for the assignment above (the zero is converted to a scalar ndarray,
with no dimensions so it can be assigned to any ndarray).

A nice feature in recent perl versions is lvalue subroutines
(i.e., versions 5.6.x and higher including all perls currently
supported by PDL).  That allows one to use the slicing syntax
on both sides of the assignment:

 pdl> $im->slice(':,(2)') .= xvals(float, 5)

Note that there can be a problem with assignments like this when
lvalue and rvalue ndarrays refer to overlapping portions of data in the
parent ndarray:

 # revert the elements of the first line of $x
 $x->slice(':,(1)') .= $x->slice('-1:0,(1)');

Currently, the parent data on the right side of the assignments is not
copied before the (internal) assignment loop proceeds. Therefore, the
outcome of this assignment will depend on the sequence in which
elements are assigned and almost certainly I<not> do what you
wanted.  So the semantics are currently B<undefined> for now and liable
to change anytime. To obtain the desired behaviour, use

 $x->slice(':,(1)') .= $x->slice('-1:0,(1)')->copy;

which makes a physical copy of the slice or

 $x->slice(':,(1)') .= $x->slice('-1:0,(1)')->sever;

which returns the same slice but first severs the connection of the slice
to its parent.

=head2 Other functions that manipulate dimensions

Having talked extensively about the
L<slice|PDL::Slices/slice> function it should be
noted that this is not the only PDL indexing function. There
are additional indexing functions which are also useful
(especially in the context of broadcasting which we will talk about
later). Here are a list and some examples how to use them.

=over 4

=item C<dummy>

inserts a dummy dimension of the size you specify (default 1) at the
chosen location. You can't wait to hear how that is achieved?  Well,
all elements with index C<(X,x,Y)> (C<< 0<=x<size_of_dummy_dim >>) just map to
the element with index C<(X,Y)> of the parent ndarray (where C<X> and C<Y> refer to
the group of indices before and after the location where the dummy
dimension was inserted.)

This example calculates the x coordinate of the centroid of an
image (later we will learn that we didn't actually need the dummy
dimension thanks to the magic of implicit broadcasting; but using dummy
dimensions the code would also work in a broadcast-less world; though once
you have worked with PDL broadcasting you wouldn't want to live without
them again).

 # centroid
 ($xd,$yd) = $im->dims;
 $xc = sum($im*xvals($xd)->dummy(1,$yd))/sum($im);

Let's explain how that works in a little more detail. First, the
product:

 $xvs = xvals($xd);
 print $xvs->dummy(1,$yd);      # repeat the line $yd times
 $prod = $im*xvs->dummy(1,$yd); # form the pixel-wise product with
                                # the repeated line of x-values

The rest is then summing the results of the pixel-wise product together
and normalizing with the sum of all pixel values in the original image
thereby calculating the x-coordinate of the "center of mass" of the
image (interpreting pixel values as local mass) which is known as the
centroid of an image.

Next is a (from the point of view of memory consumption) very
cheap conversion from grey-scale to RGB, i.e. every pixel holds now a
triple of values instead of a scalar. The three values in the triple
are, fortunately, all the same for a grey image, so that our trick
works well in that it maps all the three members of the triple to the
same source element:

 # a cheap grey-scale to RGB conversion
 $rgb = $grey->dummy(0,3)

Unfortunately this trick cannot be used to convert your old B/W
photos to color ones in the way you'd like. :(

Note that the memory usage of ndarrays with dummy dimensions
is especially sensitive to the internal representation. If the ndarray
can be represented as a virtual affine (``vaffine'') ndarray,
only the control structures are stored. But if C<$y> in

 $x = zeroes(10000);
 $y = $x->dummy(1,10000);

is made physical by some routine, you will find that the memory usage
of your program has suddenly grown by 100Mb.

=item C<diagonal>

replaces two dimensions (which have to be of equal size) by one
dimension that references all the elements along the "diagonal" along
those two dimensions. Here, we have two examples which should appear
familiar to anyone who has ever done some linear algebra. Firstly,
make a unity matrix:

 # unity matrix
 $e = zeroes(float, 3, 3); # make everything zero
 $e->diagonal(0,1) .= 1; # set the elements along the diagonal to 1
 print $e;

Or the other diagonal:

 $e->slice(':-1:0')->diagonal(0,1) .= 2;
 print $e;

(Did you notice how we used the slice function to reverse the sequence
of lines before setting the diagonal of the new child, thereby setting
the cross diagonal of the parent ?)  Or a mapping from the space of
diagonal matrices to the field over which the matrices are defined,
the trace of a matrix:

 # trace of a matrix
 $trace = sum($mat->diagonal(0,1));  # sum all the diagonal elements

=item C<xchg> and C<mv>

L<xchg|PDL::Slices/xchg> exchanges or "transposes" the two  specified dimensions.
A straightforward example:

 # transpose a matrix (without explicitly reshuffling data and
 # making a copy)
 $prod = $x x $x->xchg(0,1);

C<$prod> should now be pretty close to the unity matrix if C<$x> is an
orthogonal matrix. Often C<xchg> will be used in the context of broadcasting
but more about that later.

L<mv|PDL::Slices/mv> works in a similar fashion. It moves a dimension (specified by
its number in the parent) to a new position in the new child ndarray:

 $y = $x->mv(4,0);  # make the 5th dimension of $x the first in the
                    # new child $y

The difference between C<xchg> and C<mv> is that C<xchg> only changes
the position of two dimensions with each other, whereas C<mv>
inserts the first dimension to the place of second, moving the other
dimensions around accordingly.

=item C<clump>

collapses several dimensions into one. Its only argument specifies how
many dimensions of the source ndarray should be collapsed (starting from
the first). An (admittedly unrealistic) example is a 3D ndarray which
holds data from a stack of image files that you have just read
in. However, the data from each image really represents a 1D time
series and has only been arranged that way because it was digitized
with a frame grabber. So to have it again as an array of time
sequences you say

 pdl> $seqs = $stack->clump(2)
 pdl> help vars
 PDL variables in package main::

 Name         Type   Dimension       Flow  State          Mem
 ----------------------------------------------------------------
 $seqs        Double D [8000,50]            -C           0.00Kb
 $stack       Double D [100,80,50]          P            3.05Mb

Unrealistic as it may seem, our confocal microscope software writes
data (sometimes) this way. But more often you use clump to achieve a
certain effect when using implicit or explicit broadcasting.

=back

=head2 Calls to indexing functions can be chained

As you might have noticed in some of the examples above calls to the
indexing functions can be nicely chained since all of these functions
return a newly created child object. However, when doing extensive
index manipulations in a chain be sure to keep track of what you are
doing, e.g.

 $x->xchg(0,1)->mv(0,4)

moves the dimension 1 of C<$x> to position 4 since when the
second command is executed the original dimension 1 has been moved
to position 0 of the new child that calls the C<mv> function. I think
you get the idea (in spite of my convoluted explanations).

=head2 Propagated assignments ('.=') and dummy dimensions

A subtlety related to indexing is the assignment to ndarrays containing dummy
dimensions of size greater than 1. These assignments (using C<.=>) are
forbidden since several elements of the lvalue ndarray point to the same
element of the parent. As a consequence the value of those parent
elements are potentially ambiguous and would depend on the sequence in
which the implementation makes the assignments to elements. Therefore,
an assignment like this:

 $x = pdl [1,2,3];
 $y = $x->dummy(1,4);
 $y .= yvals(3,4);

can produce unexpected results and the results are explicitly
B<undefined>, and will (as of 2.086) throw an error.

From the point of view of dataflow the introduction of
greater-size-than-one dummy dimensions is regarded as an irreversible
transformation (similar to the terminology in thermodynamics) which
precludes backward propagation of assignment to a parent (which you had
explicitly requested using the C<.=> assignment). A similar problem to
watch out for occurs in the context of broadcasting where sometimes
dummy dimensions are created implicitly during the broadcast loop (see below).

=head2 Reasons for the parent/child (or "pointer") concept

It is very memory efficient, with only metadata existing in the child.
It provides a very flexible and powerful way of accessing portions of
ndarray data.

=head2 How to make things physical again

A "virtual affine" (or "vaffine") ndarray will be automatically
"physicalised" (have memory allocated and its data copied in) if that
is requested with the L<PDL::Core/make_physical> method, if it is
L<sever|PDL::Core/sever>ed, or if it is passed to an operation as a
parameter that is marked as C<[phys]>. See L<PDL::PP> for more.
It will still have dataflow except in the C<sever> case.

=head1 Broadcasting

In the previous paragraph on indexing we have already mentioned the
term occasionally but now its really time to talk explicitly about
"broadcasting" with ndarrays: within the framework of PDL it could
probably be loosely defined as an implicit looping
facility. It is implicit because you don't specify anything like
enclosing for-loops but rather the loops are automatically (or
'magically') generated by PDL based on the dimensions of the ndarrays
involved. This should give you a first idea why the index/dimension
manipulating functions you have met in the previous paragraphs are
especially important and useful in the context of broadcasting.
The other ingredient for broadcasting (apart from the ndarrays involved) is
a function that is broadcasting aware (generally, these are
L<PDL::PP> compiled functions) and that the ndarrays are "broadcast" over.
So much about the terminology and now let's try to shed some light on what it
all means.

=head2 Implicit broadcasting - a first example

There are two slightly different variants of broadcasting. We start
with what we call "implicit broadcasting". Let's pick a practical example
that involves looping of a function over many elements of a
ndarray. Suppose we have an RGB image that we want to convert to
grey-scale. The RGB image is represented by a 3-dim ndarray C<im(3,x,y)> where
the first dimension contains the three color components of each pixel and C<x>
and C<y> are width and height of the image, respectively. Next we need to
specify how to convert a color-triple at a given pixel into a
grey-value (to be a realistic example it should represent the relative
intensity with which our color insensitive eye cells would detect that
color to achieve what we would call a natural conversion from color to
grey-scale). An approximation that works quite well is to compute the
grey intensity from each RGB triplet (r,g,b) as a weighted sum

 grey-value = 77/256*r + 150/256*g + 29/256*b =
     inner([77,150,29]/256, [r,g,b])

where the last form indicates that we can write this as an inner
product of the 3-vector comprising the weights for red, green and blue
components with the 3-vector containing the color
components. Traditionally, we might have written a function like the
following to process the whole image:

 my @dims=$im->dims;
 # here normally check that first dim has correct size (3), etc
 $grey=zeroes(@dims[1,2]);   # make the ndarray for the resulting grey image
 $w = pdl [77,150,29] / 256; # the vector of weights
 for ($j=0;$j<dims[2];$j++) {
    for ($i=0;$i<dims[1];$i++) {
        # compute the pixel value
        $tmp = inner($w,$im->slice(':,(i),(j)'));
	set($grey,$i,$j,$tmp); # and set it in the grey-scale image
    }
 }

Now we write the same using broadcasting (noting that C<inner> is a broadcasting
aware function defined in the L<PDL::Primitive> package)

 $grey = inner($im,pdl([77,150,29]/256));

We have ended up with a one-liner that automatically
creates the ndarray C<$grey> with the right number and size of dimensions and
performs the loops automatically (these loops are implemented as fast C code
in the internals of PDL).
Well, we
still owe you an explanation how this 'magic' is achieved.

=head2 How does the example work ?

The first thing to note is that every function that is broadcasting aware
(these are without exception functions compiled from concise
descriptions by L<PDL::PP>, later just called PP-functions) expects a
defined (minimum) number of dimensions (we call them core dimensions)
from each of its ndarray arguments. The L<inner|PDL::Primitive/inner> function
expects two one-dimensional (input) parameters from which it calculates a
zero-dimensional (output) parameter. We write that symbolically as
C<inner((n),(n),[o]())> and call it C<inner>'s I<signature>, where n
represents the size of that dimension. n being equal in the first and
second parameter means that those dimensions have to be of equal size
in any call. As a different example take the outer product which takes
two 1D vectors to generate a 2D matrix, symbolically written as
C<outer((n),(m),[o](n,m))>. The C<[o]> in both examples indicates that
this (here third) argument is an output argument. In the latter
example the dimensions of first and second argument don't have to
agree but you see how they determine the size of the two dimensions of
the output ndarray.

Here is the point when broadcasting finally enters the game. If you call
PP-functions with ndarrays that have I<more> than the required core
dimensions the first dimensions of the ndarray arguments are used as the
core dimensions and the additional extra dimensions are broadcast
over. Let us demonstrate this first with our example above

 $grey = inner($im,$w); # w is the weight vector from above

In this case $w is 1D and so supplied just the core dimension, C<$im> is
3D, more specifically C<(3,x,y)>. The first dimension (of size 3) is the
required core dimension that matches (as required by inner) the first
(and only) dimension of C<$w>. The second dimension is the first broadcast
dimension (of size C<x>) and the third is here the second broadcast
dimension (of size C<y>). The output ndarray is automatically created (as
requested by setting C<$grey> to "null" prior to invocation). The output
dimensions are obtained by appending the I<loop dimensions> (here
C<(x,y)>) to the core output dimensions (here 0D) to yield the final
dimensions of the auto-created ndarray (here C<0D+2D=2D> to yield a 2D
output of size C<(x,y)>).

So the above command calls the core functionality that computes the inner
product of two 1D vectors C<x*y> times with C<$w> and all 1D slices of the
form C<(':,(i),(j)')> of C<$im> and sets the respective elements of the
output ndarray C<$grey(i,j)> to the result of each computation. We could
write that symbolically as

 $grey(0,0) = f($w,$im(:,(0),(0)))
 $grey(1,0) = f($w,$im(:,(1),(0)))
     .
     .
     .
 $grey(x-2,y-1) = f($w,$im(:,(x-2),(y-1)))
 $grey(x-1,y-1) = f($w,$im(:,(x-1),(y-1)))

But this is done automatically by PDL without writing any explicit
Perl loops.  We see that the command really creates an output ndarray with
the right dimensions and sets the elements indeed to the result of the
computation for each pixel of the input image.

When even more ndarrays and extra dimensions are involved things get a bit
more complicated. We will first give the general rules how the broadcast
dimensions depend on the dimensions of input ndarrays enabling you to
figure out the dimensionality of an auto-created output ndarray (for any
given set of input ndarrays and core dimensions of the PP-function in
question). The general rules will most likely appear a bit confusing
on first sight so that we'll set out to illustrate the usage with a set
of further examples (which will hopefully also demonstrate that there
are indeed many practical situations where broadcasting comes in extremely
handy).

=head2 A call for coding discipline

Before we point out the other technical details of broadcasting, please
note this call for programming discipline when using broadcasting:

In order to preserve human readability, I<PLEASE> comment any nontrivial
expression in your code involving broadcasting.  Most importantly, for
any subroutine, include information at the beginning about what you
expect the dimensions to represent (or ranges of dimensions).

As a warning, look at this undocumented function and try to guess what
might be going on:

 sub lookup {
   my ($im,$palette) = @_;
   $palette->xchg(0,1)->index($im->indx->dummy(0,$palette->dim(0)));
 }

Would you agree that it might be difficult to figure out expected
dimensions, purpose of the routine, etc ?
(If you want to find out what this piece of code does, see below)

=head2 How to figure out the loop dimensions

There are a couple of rules that allow you to figure out number and
size of loop dimensions (and if the size of your input ndarrays comply
with the broadcasting rules). Dimensions of any ndarray argument are broken
down into two groups in the following: Core dimensions (as defined by
the PP-function, see B<Appendix A> for a list of PDL primitives) and
extra dimensions which comprises all remaining dimensions of that
ndarray. For example calling a function C<func> with the signature
C<func((n,m),[o](n))> with an ndarray C<$x(2,4,7,1,3)> as C<f($x,($o = null))>
results in the semantic splitting of x's dimensions into:
core dimensions C<(2,4)> and extra dimensions C<(7,1,3)>.

=over 6

=item R0

Core, or "active", dimensions are identified with the first N dimensions of the
respective ndarray argument (and are required). Any further dimensions are
extra dimensions and used to determine the loop dimensions.

=item R1

The number of (implicit) loop dimensions is equal to the maximal
number of extra dimensions taken over the set of ndarray arguments.

=item R2

The size of each of the loop dimensions is derived from the size
of the respective dimensions of the ndarray arguments. The size of a
loop dimension is given by the maximal size found in any of
the ndarrays having this extra dimension.

=item R3

If an ndarray doesn't have a loop dimension, in the broadcast loop this
ndarray is "promoted" as if having a dummy dimension of size equal to the
size of that loop dimension.

=item R4

For all ndarrays that have a given extra dimension the size must be
equal to the size of the loop dimension (as determined by the
previous rule) or 1; otherwise you raise a runtime exception. If the
size of the extra dimension in an ndarray is 1 it is implicitly treated
as a dummy dimension of size equal to ("inflated" to) that loop dim size when
performing the broadcast loop.

=item R5

If output auto-creation is used (by setting the relevant ndarray to
C<< PDL->null >> before invocation) the number of dimensions of the created
ndarray is equal to the sum of the number of core output dimensions +
number of loop dimensions. The size of the core output dimensions is
derived from the relevant dimension of input ndarrays (as specified in the
function definition) and the sizes of the other dimensions are equal
to the size of the loop dimension it is derived from. The
automatically created ndarray will be physical (unless dataflow is in
operation).

=back

In this context, note that you can run into the problem with
assignment to ndarrays containing greater-than-one dummy dimensions (see above).
Although your output ndarray(s) didn't contain any dummy dimensions in the
first place they may end up with implicitly created dummy dimensions
according to I<R3>.

As an example, suppose we have a (here unspecified) PP-function with
the signature:

 func((m,n),(m,n,o),(m),[o](m,o))

and you call it with 3 ndarrays C<$x(5,3,10,11)>,
C<$y(5,3,2,10,1,12)>, and C<$z(5,1,11,12)> as

 func($x,$y,$z,($d=null))

then the number of loop dimensions is 3 (by C<R0+R1> from C<$y> and C<$z>) with
sizes C<(10,11,12)> (by R2); the two output core dimensions are C<(5,2)>
(from the signature of func) resulting in a 5-dimensional output ndarray
C<$c> of size C<(5,2,10,11,12)> (see R5) and (the automatically created) C<$d>
is derived from C<($x,$y,$z)> in a way that can be expressed in pdl
pseudo-code as

 $d(:,:,i,j,k) .= func($x(:,:,i,j),$y(:,:,:,i,0,k),$z(:,0,j,k))
    with 0<=i<10, 0<=j<=11, 0<=k<12

If we analyze the color to grey-scale conversion again with these rules
in mind we note another great advantage of implicit broadcasting.
We can call the conversion with an ndarray representing a pixel (C<im(3)>),
a line of rgb pixels (C<im(3,x)>), a proper color image (C<im(3,x,y)>) or a
whole stack of RGB images (C<im(3,x,y,z)>). As long as C<$im> is of the form
C<(3,...)> the automatically created output ndarray will contain the right
number of dimensions and contain the intensity data as we expect it
since the loops have been implicitly performed thanks to I<implicit
broadcasting>. You can easily convince yourself that calling with a color
pixel C<$grey> is 0D, with a line it turns out 1D C<grey(x)>, with an image
we get C<grey(x,y)> and finally we get a converted image stack C<grey(x,y,z)>.

Let's fill these general rules with some more life by going through a
couple of further examples. The reader may try to figure out equivalent
formulations with explicit for-looping and compare the flexibility of
those routines using implicit broadcasting to the explicit
formulation. Furthermore, especially when using several broadcast
dimensions it is a useful exercise to check the relative speed
by doing some benchmark tests (which we still have to do).

First in the row is a slightly reworked centroid example, now coded
with broadcasting in mind.

 # broadcast mult to calculate centroid coords, works for stacks as well
 $xc = sumover(($im*xvals($im->dim(0))->clump(2)) /
       sumover($im->clump(2));

Let's analyze what's going on step by step. First the product:

 $prod = $im*xvals($im->dim(0))

This will actually work for C<$im> being one, two, three, and higher
dimensional. If C<$im> is one-dimensional it's just an ordinary product
(in the sense that every element of C<$im> is multiplied with the
respective element of C<xvals(...)>), if C<$im> has more dimensions further
broadcasting is done by adding appropriate dummy dimensions to C<xvals(...)>
according to R3.
More importantly, the two L<sumover|PDL::Ufunc/sumover> operations show
a first example of how to make use of the dimension manipulating
commands. A quick look at sumover's signature will remind you that
it will only "gobble up" the first dimension of a given input ndarray. But
what if we want to really compute the sum over all elements of the
first two dimensions? Well, nothing keeps us from passing a virtual
ndarray into sumover which in this case is formed by clumping the first
two dimensions of the "parent ndarray" into one. From the point of view of
the parent ndarray the sum is now computed over the first two dimensions,
just as we wanted, though sumover has just done the job as specified
by its signature. Got it ?

Another little finesse of writing the code like that: we intentionally
used C<< sumover($pdl->clump(2)) >> instead of C<sum($pdl)> so that we can
either pass just an image C<(x,y)> or a stack of images C<(x,y,t)> into this
routine and get either just one x-coordinate or a vector of
x-coordinates (of size t) in return.

Another set of common operations are what one could call "projection
operations". These operations take a N-D ndarray as input and return a
(N-1)-D "projected" ndarray. These operations are often performed with
functions like L<sumover|PDL::Ufunc/sumover>,
L<prodover|PDL::Ufunc/prodover>, L<minimum|PDL::Ufunc/minimum> and
L<maximum|PDL::Ufunc/maximum>.
Using again images as examples we might want to calculate
the maximum pixel value for each line of an image or image stack. We
know how to do that

 # maxima of lines (as function of line number and time)
 maximum($stack,($ret=null));

But what if you want to calculate maxima per column when implicit
broadcasting always applies the core functionality to the first dimension
and broadcasts over all others? How can we achieve that instead the
core functionality is applied to the second dimension and broadcasting is
done over the others. Can you guess it? Yes, we make a virtual ndarray
that has the second dimension of the "parent ndarray" as its first
dimension using the C<mv> command.

 # maxima of columns (as function of column number and time)
 maximum($stack->mv(1,0),($ret=null));

and calculating all the sums of sub-slices over the third dimension
is now almost too easy

 # sums of pixels in time (assuming time is the third dim)
 sumover($stack->mv(2,0),($ret=null));

Finally, if you want to apply the operation to all elements (like max over
all elements or sum over all elements) regardless of the dimensions of
the ndarray in question C<clump> comes in handy. As an example look at a
definition of C<sum> (summarised from F<Basic/Ufunc/ufunc.pd>):

 sub sum {
   PDL::Ufunc::sumover($name->clump(-1),($tmp=null));
   return $tmp; # return a 0D ndarray
 }

We have already mentioned that all basic operations support broadcasting
and assignment is no exception. So here are a couple of broadcasted
assignments

 pdl> $im = zeroes(byte, 10,20)
 pdl> $line = exp(-rvals(10)**2/9)
 # broadcasted assignment
 pdl> $im .= $line      # set every line of $im to $line
 pdl> $im2 .= 5         # set every element of $im2 to 5

By now you probably see how it works and what it does, don't you?

To finish the examples in this paragraph here is a function to create
an RGB image from what is called a palette image. The palette image
consists of two parts: an image of indices into a color lookup table
and the color lookup table itself. [ describe how it works ] We
are going to use a PP-function we haven't encoutered yet in the previous
examples. It is the aptly named L<index|PDL::Slices/index> function, signature
C<((n),(),[o]())> (see B<Appendix A>) with the core functionality that
C<index(pdl (0,2,4,5),2,($ret=null))> will return the element with index
2 of the first input ndarray. In this case, C<$ret> will contain the value 4.
So here is the example:

 # a broadcasted index lookup to generate an RGB, or RGBA or YMCK image
 # from a palette image (represented by a lookup table $palette and
 # an color-index image $im)
 # you can say just dummy(0) since the rules of broadcasting make it fit
 pdl> index($palette->xchg(0,1),
               $im->indx->dummy(0,$palette->dim(0)),
               $res=null);

Let's go through it and explain the steps involved. Assuming we are
dealing with an RGB lookup-table $palette is of size C<(3,x)>. First we
exchange the dimensions of the palette so that looping is done over
the first dimension of C<$palette> (of size 3 that represent r, g, and b
components). Now looking at C<$im>, we add a dummy dimension of size
equal to the length of the number of components (in the case we are
discussing here we could have just used the number 3 since we have 3
color components). We can use a dummy dimension since for red, green
and blue color components we use the same index from the original
image,
e.g.
assuming a certain pixel of C<$im> had the value 4 then the
lookup should produce the triple

 [palette(0,4),palette(1,4),palette(2,4)]

for the new red, green and
blue components of the output image. Hopefully by now you have some
sort of idea what the above piece of code is supposed to do (it is
often actually quite complicated to describe in detail how a piece of
broadcasting code works; just go ahead and experiment a bit to get a
better feeling for it).

If you have read the broadcasting rules carefully, then you might have
noticed that we didn't have to explicitly state the size of the dummy
dimension that we created for C<$im>; when we create it with size 1 (the
default) the rules of broadcasting make it automatically fit to the
desired size (by rule R4, in our example the size would be 3 assuming
a palette of size C<(3,x)>). Since situations like this do occur often in
practice this is actually why rule R4 has been introduced (the part
that makes dimensions of size 1 fit to the broadcast loop dim size). So
we can just say

 pdl> index($palette->xchg(0,1),$im->indx->dummy(0),($res=null));

Again, you can convince yourself that this routine will create the
right output if called with a pixel (C<$im> is 0D), a line (C<$im> is 1D),
an image (C<$im> is 2D), ..., an RGB lookup table (palette is C<(3,x)>) and
RGBA lookup table (palette is C<(4,x)>, see e.g. OpenGL). This
flexibility is achieved by the rules of broadcasting which are made to do
the right thing in most situations.

To wrap it all up once again, the general idea is as follows. If you
want to achieve looping over certain dimensions and have the I<core functionality>
applied to another specified set of dimensions you use
the dimension manipulating commands to create a (or several)
I<virtual> ndarray(s) so that from the point of view of the I<parent>
ndarray(s) you get what you want (always having the signature of the
function in question and R1-R5 in mind!). Easy, isn't it ?

See L<PDL::Slices/dice>, and L<PDL::Slices/dice_axis> for more
indexed lookups, and L<PDL::Slices/indexND>, L<PDL::Slices/whereND>,
L<PDL::Slices/whichND> for even more powerful constructs.

=head2 Output auto-creation and PP-function calling conventions

At this point we have to divert to some technical detail that has to
do with the general calling conventions of PP-functions and the
automatic creation of output arguments.
Basically, there are two ways of invoking PDL routines, namely

 $result = func($x,$y);

and

 func($x,$y,$result);

If you are only using implicit broadcasting then the output variable can
be automatically created by PDL. You flag that to the PP-function by
setting the output argument to a special kind of ndarray that is returned
from a call to the function C<< PDL->null >> that returns an essentially
"empty" ndarray (for those interested in details there is a flag
in the C pdl structure for this). The dimensions
of the created ndarray are determined by the rules of implicit
broadcasting: the first dimensions are the core output dimensions to
which the broadcasting dimensions are appended (which are in turn
determined by the dimensions of the input ndarrays as described above).
So you can say

 func($x,$y,($result=PDL->null));

or

 $result = func($x,$y)

which are B<exactly> equivalent.

Be warned that you can I<not> use output auto-creation when using
explicit broadcasting (for reasons explained in the following section on
B<explicit broadcasting>, the second variant of broadcasting).

In "tight" loops you probably want to avoid the implicit creation of a
temporary ndarray in each step of the loop that comes along with the
"functional" style but rather say

 # create output ndarray of appropriate size only at first invocation
 $result = null;
 for (0...$n) {
      func($x,$y,$result); # in all but the first invocation $result
      func2($y);           # is defined and has the right size to
                           # take the output provided $y's dims don't change
      twiddle($result,$x); # do something from $result to $x for iteration
 }

The take-home message of this section once more: be aware of the
limitation on output creation when using B<explicit broadcasting>.

=head2 Explicit broadcasting

Having so far only talked about the first flavour of broadcasting it is
now about time to introduce the second variant. Instead of shuffling
around dimensions all the time and relying on the rules of implicit
broadcasting to get it all right you sometimes might want to specify in a
more explicit way how to perform the broadcast loop. It is probably not
too surprising that this variant of the game is called I<explicit broadcasting>.
Now, before we create the wrong impression: it is not
either I<implicit> or I<explicit>; the two flavours do mix. But more
about that later.

The two most used functions with explicit broadcasting are
L<broadcast|PDL::Core/PDL::broadcast>
and L<unbroadcast|PDL::Slices/unbroadcast>.
We start with an example that illustrates typical
usage of the former:

 [ # ** this is the worst possible example to start with ]
 #  but can be used to show that $mat += $line is different from
 #                               $mat->broadcast(0) += $line
 # explicit broadcasting to add a vector to each column of a matrix
 pdl> $mat  = zeroes(4,3)
 pdl> $line = pdl (3.1416,2,-2)
 pdl> $mat->broadcast(0) += $line

In this example, C<< $mat->broadcast(0) >> tells PDL that you want the second
dimension of this ndarray to be broadcast over first leading to a broadcast
loop that can be expressed as

 for (j=0; j<3; j++) {
    for (i=0; i<4; i++) {
        mat(i,j) += src(j);
    }
 }

C<broadcast> takes a list of numbers as arguments which explicitly
specify which dimensions to broadcast over first. With the introduction
of explicit broadcasting the dimensions of an ndarray are conceptually split into
three different groups the latter two of which we have already
encountered: broadcast dimensions, core dimensions and extra dimensions.

Conceptually, it is best to think of those dimensions of an ndarray that
have been specified in a call to C<broadcast> as being taken away from
the set of normal dimensions and put on a separate stack. So assuming
we have an ndarray C<x(4,7,2,8)> saying

 $y = $x->broadcast(2,1)

creates a new virtual ndarray of dimension C<y(4,8)> (which we call the
remaining dims) that also has 2 broadcast dimensions of size C<(2,7)>. For
the purposes of this document we write that symbolically as
C<y(4,8){2,7}>. An important difference to the previous examples where
only implicit broadcasting was used is the fact that the core dimensions
are matched against the I<remaining dimensions> which are not
necessarily the first dimensions of the ndarray. We will now specify how
the presence of broadcast dimensions changes the rules R1-R5 for
broadcast loops (which apply to the special case where none of the ndarray
arguments has any broadcast dimensions).

=over 4

=item B0

Core dimensions are matched against the first n I<remaining dimensions>
of the ndarray argument (note the difference to R1). Any
further I<remaining dimensions> are I<extra dimensions> and are used
to determine the I<implicit loop dimensions>.

=item B1a

The number of I<implicit loop dimensions> is equal to the maximal
number of extra dimensions taken over the set of ndarray arguments.

=item B1b

The number of I<explicit loop dimensions> is equal to the maximal
number of broadcast dimensions taken over the set of ndarray arguments.

=item B1c

The total number of I<loop dimensions> is equal to the sum of
I<explicit loop dimensions> and I<implicit loop dimensions>. In the
broadcast loop, I<explicit loop dimensions> are broadcasted over first
followed by I<implicit loop dimensions>.

=item B2

The size of each of the I<loop dimensions> is derived from the size of
the respective dimensions of the ndarray arguments. It is given by the
maximal size found in any ndarrays having this broadcast dimension (for
I<explicit loop dimensions>) or extra dimension (for
I<implicit loop dimensions>).

=item B3

This rule applies to any I<explicit loop dimension> as well as any
I<implicit loop dimension>. For all ndarrays that have a given
I<broadcast/extra dimension> the size must be equal to the size of the
respective I<explicit/implicit loop dimension> or 1; otherwise you
raise a runtime exception. If the size of a I<broadcast/extra dimension>
of an ndarray is one it is implicitly treated as a dummy
dimension of size equal to the I<explicit/implicit loop dimension>.

=item B4

If an ndarray doesn't have a I<broadcast/extra dimension> that corresponds to
an I<explicit/implicit loop dimension>, in the broadcast loop this
ndarray is treated as if having a dummy dimension of size equal to the
size of that loop dimension.

=item B4a

All ndarrays that do have I<broadcast dimensions> must have the same number of
broadcast dimensions.

=item B5

Output auto-creation cannot be used if any of the ndarray arguments has any
I<broadcast dimensions>. Otherwise R5 applies.

=back

The same restrictions apply with regard to implicit dummy dimensions
(created by application of B4) as already mentioned in the section
on implicit broadcasting: if any of the output ndarrays has an (explicit or
implicitly created) greater-than-one dummy dimension a runtime
exception will be raised.

Let us demonstrate these rules at work in a generic case.
Suppose we have a (here unspecified) PP-function with
the signature:

 func((m,n),(m),(),[o](m))

and you call it with 3 ndarrays C<a(5,3,10,11)>, C<b(3,5,10,1,12)>, C<c(10)> and an
output ndarray C<d(3,11,5,10,12)> (which can here I<not> be automatically
created) as

 func($x->broadcast(1,3),$y->broadcast(0,3),$c,$d->broadcast(0,1))

From the signature of func and the above call the ndarrays split into
the following groups of core, extra and broadcast dimensions (written in
the form C<pdl(core dims){broadcast dims}[extra dims]>):

 a(5,10){3,11}[] b(5){3,1}[10,12] c(){}[10] d(5){3,11}[10,12]

With this to help us along (it is in general helpful to write the
arguments down like this when you start playing with broadcasting and
want to keep track of what is going on) we further deduce
that the number of explicit loop dimensions is 2 (by B1b from C<$a> and C<$b>)
with sizes C<(3,11)> (by B2); 2 implicit loop dimensions (by B1a from C<$b>
and C<$d>) of size C<(10,12)> (by B2) and the elements of are computed from
the input ndarrays in a way that can be expressed in pdl pseudo-code as

 for (l=0;l<12;l++)
  for (k=0;k<10;k++)
   for (j=0;j<11;j++)         effect of treating it as dummy dim (index j)
    for (i=0;i<3;i++)                         |
       d(i,j,:,k,l) = func(a(:,i,:,j),b(i,:,k,0,l),c(k))

Ugh, this example was really not easy in terms of bookkeeping. It
serves mostly as an example how to figure out what's going on when you
encounter a complicated looking expression. But now it is really time
to show that broadcasting is useful by giving some more of our so called
"practical" examples.

[ The following examples will need some additional explanations in the
future. For the moment please try to live with the comments in the
code fragments. ]

Example 1:

 *** inverse of matrix represented by eigvecs and eigvals
 ** given a symmetrical matrix M = A^T x diag(lambda_i) x A
 **    =>  inverse M^-1 = A^T x diag(1/lambda_i) x A
 ** first $tmp = diag(1/lambda_i)*A
 ** then  A^T * $tmp by broadcasted inner product
 # index handling so that matrices print correct under pdl
 $inv .= $evecs*0;  # just copy to get appropriately sized output
 $tmp .= $evecs;    # initialise, no back-propagation
 $tmp->broadcast(0) /= $evals;    #  broadcasted division
 # and now a matrix multiplication in disguise
 PDL::Primitive::inner($evecs->xchg(0,1)->broadcast(-1,1),
                       $tmp->broadcast(0,-1),
                       $inv->broadcast(0,1));
 # alternative for matrix mult using implicit broadcasting,
 # first xchg only for transpose
 PDL::Primitive::inner($evecs->xchg(0,1)->dummy(1),
                       $tmp->xchg(0,1)->dummy(2),
                       ($inv=null));

Example 2:

 # outer product by broadcasted multiplication
 # stress that we need to do it with explicit call to my_biop1
 # when using explicit broadcasting
 $res=zeroes($x->dim(0),$y->dim(0));
 my_biop1($x->broadcast(0,-1),$y->broadcast(-1,0),$res->(0,1),"*");
 # similar thing by implicit broadcasting with auto-created ndarray
 $res = $x->dummy(1) * $y->dummy(0);


Example 3:

 # different use of broadcast and unbroadcast to shuffle a number of
 # dimensions in one go without lots of calls to ->xchg and ->mv


 # use broadcast/unbroadcast to shuffle dimensions around
 # just try it out and compare the child ndarray with its parent
 $trans = $x->broadcast(4,1,0,3,2)->unbroadcast;

Example 4:

 # calculate a couple of bounding boxes
 # $bb will hold BB as [xmin,xmax],[ymin,ymax],[zmin,zmax]
 # we use again broadcast and unbroadcast to shuffle dimensions around
 pdl> $bb = zeroes(double, 2,3 );
 pdl> minimum($vertices->broadcast(0)->clump->unbroadcast(1), $bb->slice('(0),:'));
 pdl> maximum($vertices->broadcast(0)->clump->unbroadcast(1), $bb->slice('(1),:'));


Example 5:

 # calculate a self-rationed (i.e. self normalized) sequence of images
 # uses explicit broadcasting and an implicitly broadcasted division
 $stack = read_image_stack();
 # calculate the average (per pixel average) of the first $n+1 images
 $aver = zeroes([stack->dims]->[0,1]);  # make the output ndarray
 sumover($stack->slice(":,:,0:$n")->broadcast(0,1),$aver);
 $aver /= ($n+1);
 $stack /= $aver;  # normalize the stack by doing a broadcasted division
 # implicit versus explicit
 # alternatively calculate $aver with implicit broadcasting and auto-creation
 sumover($stack->slice(":,:,0:$n")->mv(2,0),($aver=null));
 $aver /= ($n+1);
 #

=head2 Implicit versus explicit broadcasting

In this paragraph we are going to illustrate when explicit broadcasting
is preferable over implicit broadcasting and vice versa. But then again,
this is probably not the best way of putting the case since you already
know: the two flavours do mix. So, it's more about how to get the best
of both worlds and, anyway, in the best of Perl traditions: TIMTOWTDI !

[ Sorry, this still has to be filled in in a later release; either
refer to above examples or choose some new ones ]

Finally, this may be a good place to justify all the technical detail
we have been going on about for a couple of pages: why broadcasting ?

Well, code that uses broadcasting should be (considerably) faster than
code that uses explicit for-loops (or similar Perl constructs) to achieve
the same functionality. Especially on supercomputers (with vector
computing facilities/parallel processing) PDL broadcasting is
implemented in a way that takes advantage of the additional facilities
of these machines. Furthermore, it is a conceptually simple
construct (though technical details might get involved at times) and
can I<greatly> reduce the syntactical complexity of PDL code (but keep
the admonition for documentation in mind). Once you
are comfortable with the I<broadcasting> way of thinking (and coding) it
shouldn't be too difficult to understand code that somebody else
has written than (provided they gave
you an idea what expected input dimensions are, etc.). As a general tip to
increase the performance of your code: if you have to introduce a loop
into your code try to reformulate the problem so that you can use
broadcasting to perform the loop (as with anything there are exceptions
to this rule of thumb; but the authors of this document tend to
think that these are rare cases ;).

=head1 PDL::PP

=head2 An easy way to define functions that are aware of indexing and broadcasting (and the universe and everything)

PDL:PP is part of the PDL distribution. It is used to generate
functions that are aware of indexing and broadcasting rules from very
concise descriptions. It can be useful for you if you want to write
your own functions or if you want to interface functions from an
external library so  that they support indexing and broadcasting (and
maybe dataflow as well, see L<PDL::Dataflow>). For further details
check L<PDL::PP>.

=head1 Appendix A

=head2 signatures of standard PDL::PP compiled functions

A selection of signatures of PDL primitives to show how many
dimensions PP compiled functions gobble up (and therefore you can
figure out what will be broadcasted over). Most of those functions are
the basic ones defined in C<primitive.pd>

 # functions in primitive.pd
 #
 sumover	((n),[o]())
 prodover	((n),[o]())
 axisvalues     ([o](n))				inplace
 inner		((n),(n),[o]())
 outer		((n),(m),[o](n,m))
 innerwt	((n),(n),(n),[o]())
 inner2         ((m),(m,n),(n),[o]())
 inner2t	((j,n),(n,m),(m,k),[o]())
 index		(1D,0D,[o])
 minimum	(1D,[o])
 maximum	(1D,[o])
 wstat		((n),(n),(),[o],())
 assgn		((),())

 # basic operations
 binary operations ((),(),[o]())
 unary operations  ((),[o]())

=head1 AUTHOR & COPYRIGHT

Copyright (C) 1997 Christian Soeller (c.soeller@auckland.ac.nz) & Tuomas
J. Lukka (lukka@fas.harvard.edu).
