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INDEXING(1)		       User Contributed Perl Documentation		      INDEXING(1)

       PDL::Indexing - how to index piddles.

       This manpage should serve as a first tutorial on the indexing and threading features of

       This manpage is still in alpha development and not yet complete. "Meta" comments that
       point out deficiencies/omissions of this document will be surrounded by square brackets
       ([]), e.g. [ Hopefully I will be able to remove this paragraph at some time in the future
       ]. Furthermore, it is possible that there are errors in the code examples. Please report
       any errors to Christian Soeller (c.soeller@auckland.ac.nz).

       Still to be done are (please bear with us and/or ask on the mailing list, see PDL::FAQ):

       o    document perl level threading

       o    threadids

       o    update and correct description of slice

       o    new functions in slice.pd (affine, lag, splitdim)

       o    reworking of paragraph on explicit threading

Indexing and threading 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 a pdl object in a very flexible
       way. Threading provides efficient implicit looping functionality (since the loops are
       implemented as optimized C code).

       Pdl objects (later often called "pdls") are perl objects that represent multidimensional
       arrays and operations on those. In contrast to simple perl @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 pdls.

       pdls can have children

       Central to many of the indexing capabilities of PDL are the relation of "parent" and
       "child" between pdls. Many of the indexing commands create a new pdl from an existing pdl.
       The new pdl is the "child" and the old one is the "parent". The data of the new pdl is
       defined by a transformation that specifies how to generate (compute) its data from the
       parent's data. The relation between the child pdl and its parent are often bidirectional,
       meaning that changes in the child's data are propagated back to the parent. (Note: You
       see, we are aiming in our terminology already towards the new dataflow features. The kind
       of dataflow that is used by the indexing commands (about which you will learn in a minute)
       is always in operation, not only when you have explicitly switched on dataflow in your pdl
       by saying "$a->doflow". For further information about data flow check the dataflow man-

       Another way to interpret the pdls 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". After these introductory remarks that hopefully pre-
       pared you for what is coming (rather than confuse you too much) we are going to dive right
       in and start with a description of the indexing commands and some typical examples how
       they might be used in PDL programs. We will further illustrate the pointer/dataflow analo-
       gies 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 trans-
       formation forwards and backwards as necessary. The other is to consider the child piddle a
       ``virtual'' piddle, which only stores a pointer to the parent and access information so
       that routines which use the child piddle actually directly access the data in the parent.
       If the virtual piddle is given to a routine which cannot use it, PDL transparently physi-
       calizes the virtual piddle before letting the routine use it.

       Currently (1.94_01) all transformations which are ``affine'', i.e. the indices of the data
       item in the parent piddle are determined by a linear transformation (+ constant) from the
       indices of the child piddle result in virtual piddles. All other indexing routines (e.g.
       "->index(...)") result in physical piddles.  All routines compiled by PP can accept affine
       piddles (except those routines that pass pointers to external library functions).

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

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

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

       Slicing pdls

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

	perldl> $im = sequence(5,5)
	perldl> 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]

	perldl> help vars
	PDL variables in package main::

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

       [ here it might be appropriate to quickly talk about the "help vars" command that provides
       information about pdls in the interactive "perldl" shell that comes with pdl.  ]

       Now suppose we want to create a 1-D pdl that just references one line of the image, say
       line 2; or a pdl 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 a pdl 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:

	perldl> $line = $im->slice(':,(2)')
	perldl> $even = $im->slice(':,1:-1:2')
	perldl> $area = $im->slice('3:4,3:1')
	perldl> 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" pdls are children of $im or in the other (largely equivalent) interpre-
       tation pointers to data of $im.	Operations on those virtual pdls access only those por-
       tions of the data as specified by the argument to slice. So we can just print line 2:

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

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

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

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

	perldl> $im++
	perldl> p $line
	[11 12 13 14 15]
	perldl> $line += 2
	perldl> 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 pdls change the parent physical pdl
       and vice versa (however, the basic "=" assignment doesn't, use ".=" to obtain that effect.
       See below for the reasons).  The virtual child pdls are something like "live links" to the
       "original" parent pdl. As previously said, they can be thought of to work similiar 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 pdl) 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 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 = $a->slice('ind0,ind1...')

       where "ind0" specifies what to do with index No 0 of the pdl $a, etc. Each element of the
       comma separated list can have one of the following forms:

       ':'   Use the whole dimension

       'n'   Use only index "n". The dimension of this index in the resulting virtual pdl is 1.
	     An example involving those first two index formats:

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

	       [ 3]
	       [ 8]

	      perldl> p $row

	       [1 2 3 4 5]

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

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

       '(n)' Use only index "n". This dimension is removed from the resulting pdl (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.

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

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

	     Spot the difference to the previous example?

       'n1:n2' or 'n1:n2:n3'
	     Take the range of indices from "n1" to "n2" or (second form) take the range of
	     indices from "n1" to "n2" with step "n3". An example for the use of this format is
	     the previous definition of the subimage composed of even lines.

	      perldl> $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 "n2" is
	     smaller than "n1" (in the example -1 is equivalent to index 4) the elements in the
	     virtual pdl are effectively reverted with respect to its parent.

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

	     Now, this is really something a bit strange on first sight. What is a dummy dimen-
	     sion? 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 "m" elements) with an
	     "(1,m)" or "(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.
	     "slice('*5,:')"). This works in the same way as a call to the dummy function creates
	     a new dummy dimension.  So read on and check its explanation below.

	     [Not yet implemented ??????]  With an argument like this you make generalised diago-
	     nals. The diagonal will be dimension no. "i" of the new output pdl and (if optional
	     part in brackets specified) will extend along the range of indices specified of the
	     respective parent pdl's dimension. In general an argument like this only makes sense
	     if there are other arguments like this in the same call to slice. The part in brack-
	     ets is optional for this type of argument. All arguments of this type that specify
	     the same target dimension "i" have to relate to the same number of indices in their
	     parent dimension. The best way to explain it is probably to give an example, here we
	     make a pdl that refers to the elements along the space diagonal of its parent pdl (a

	      $cube = zeroes(5,5,5);
	      $sdiag = $cube->slice('(=0),(=0),(=0)');

	     The above command creates a virtual pdl that represents the diagonal along the par-
	     ents' 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=1),(4),(5:4=1),(=1)');

	     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<5, 0<=j<2

       [ work in the new index function: "$b = $a->index($c);" ???? ]

       There are different kinds of assignments in PDL

       The previous examples have already shown that virtual pdls can be used to operate on or
       access portions of data of a parent pdl. They can also be used as lvalues in assignments
       (as the use of "++" in some of the examples above has already demonstrated). For explicit
       assignments to the data represented by a virtual pdl you have to use the overloaded ".="
       operator (which in this context we call propagated assignment). Why can't you use the nor-
       mal assignment operator "="?

       Well, you definitely still can use the '=' operator but it wouldn't do what you want. This
       is due to the fact that the '=' operator cannot be overloaded in the same way as other
       assignment operators. If we tried to use '=' to try to assign data to a portion of a phys-
       ical pdl through a virtual pdl we wouldn't achieve the desired effect (instead the vari-
       able representing the virtual pdl (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 [this is actually not yet
       clear and subject of discussions in the PDL developers mailing list]. In that sense it
       would break the connection of the pdl to the parent [ isn't this behaviour in a sense the
       opposite of what happens in dataflow, where ".=" breaks the connection to the parent? ].


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

	perldl> p $line
	[1 1 1 1 1]

       But using ".="

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

	perldl> print $line
	[1 1 1 1 1]

       Also, you can substitute

	perldl> $line .= 0;

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

       Related to the assignment feature is a little trap for the unwary: since perl currently
       does not allow subroutines to return lvalues the following shortcut of the above is
       flagged as a compile time error:

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

       instead you have to say something like

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

       We hope that future versions of perl will allow the simpler syntax (i.e. allow subroutines
       to return lvalues).  [Note: perl v5.6.0 does allow this, but it is an experimental fea-
       ture. However, early reports suggest it works in simple situations]

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

	# revert the elements of the first line of $a
	($tmp = $a->slice(':,(1)')) .= $a->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 not do what you
       wanted.	So the semantics are currently undefined for now and liable to change anytime. To
       obtain the desired behaviour, use

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

       which makes a physical copy of the slice or

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

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

       Other functions that manipulate dimensions

       Having talked extensively about the 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 threading which we will talk about later). Here are a list
       and some examples how to use them.

	   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 "(X,x,Y)"
	   ("0<=x<size_of_dummy_dim") just map to the element with index "(X,Y)" of the parent
	   pdl (where "X" and "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
	   threading; but using dummy dimensions the code would also work in a threadless world;
	   though once you have worked with PDL threads you wouldn't want to live without them

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

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

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

	   The rest is then summing the results of the pixelwise product together and normalising
	   with the sum of all pixel values in the original image thereby calculating the x-coor-
	   dinate 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
	   greyscale 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 greyscale 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 piddles with dummy dimensions is especially sensitive to
	   the internal representation. If the piddle can be represented as a virtual affine
	   (``vaffine'') piddle, only the control structures are stored. But if $b in

	    $a = zeroes(10000);
	    $b = $a->dummy(1,10000);

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

	   replaces two dimensions (which have to be of equal size) by one dimension that refer-
	   ences 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
	    $e = zeroes(float, 3, 3); # make everything zero
	    ($tmp = $e->diagonal(0,1)) .= 1; # set the elements along the diagonal to 1
	    print $e;

	   Or the other diagonal:

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

	   (Did you notice how we used the slice function to revert the sequence of lines before
	   setting the diagonal of the new child, thereby setting the cross diagonal of the par-
	   ent ?)  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

       "xchg" and "mv"
	   xchg exchanges or "transposes" the two  specified dimensions.  A straightforward exam-

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

	   $prod should now be pretty close to the unity matrix if $a is an orthogonal matrix.
	   Often "xchg" will be used in the context of threading but more about that later.

	   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 pdl:

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

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

	   collapses several dimensions into one. Its only argument specifies how many dimensions
	   of the source pdl should be collapsed (starting from the first). An (admittedly unre-
	   alistic) example is a 3D pdl 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 grab-
	   ber. So to have it again as an array of time sequences you say

	    perldl> $seqs = $stack->clump(2)
	    perldl> 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 threading.

       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.


       moves the dimension 1 of $a 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 "mv"
       function. I think you get the idea (in spite of my convoluted explanations).

       Propagated assignments ('.=') and dummy dimensions

       A sublety related to indexing is the assignment to pdls containing dummy dimensions of
       size greater than 1. These assignments (using ".=") are forbidden since several elements
       of the lvalue pdl 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:

	$a = pdl [1,2,3];
	$b = $a->dummy(1,4);
	$b .= yvals(zeroes(3,4));

       can produce unexpected results and the results are explicitly undefined by PDL because
       when PDL gets parallel computing features, the current result may well change.

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

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

       [ this will have to wait a bit ]

	XXXXX being memory efficient
	XXXXX in the context of threading
	XXXXX very flexible and powerful way of accessing portions of pdl data
	      (in much more general way than sec, etc allow)
	XXXXX efficient implementation
	XXXXX difference to section/at, etc.

       How to make things physical again

       [ XXXXX fill in later when everything has settled a bit more ]

	** When needed (xsub routine interfacing C lib function)
	** How achieved (->physical)
	** How to test (isphysical (explain how it works currently))
	** ->copy and ->sever

       In the previous paragraph on indexing we have already mentioned the term occasionally but
       now its really time to talk explicitly about "threading" with pdls. The term threading has
       many different meanings in different fields of computing. 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 pdls 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 thread-
       ing.  The other ingredient for threading (apart from the pdls involved) is a function that
       is threading aware (generally, these are PDL::PP compiled functions) and that the pdls are
       "threaded" over.  So much about the terminology and now let's try to shed some light on
       what it all means.

       Implicit threading - a first example

       There are two slightly different variants of threading. We start with what we call
       "implicit threading". Let's pick a practical example that involves looping of a function
       over many elements of a pdl. Suppose we have an RGB image that we want to convert to
       greyscale. The RGB image is represented by a 3-dim pdl "im(3,x,y)" where the first dimen-
       sion contains the three color components of each pixel and "x" and "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 greyvalue (to be a realistic example it should represent the rela-
       tive intensity with which our color insensitive eye cells would detect that color to
       achieve what we would call a natural conversion from color to greyscale). An approximation
       that works quite well is to compute the grey intensity from each RGB triplet (r,g,b) as a
       weighted sum

	greyvalue = 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 pdl 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 greyscale image

       Now we write the same using threading (noting that "inner" is a threading aware function
       defined in the PDL::Primitive package)

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

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

       How does the example work ?

       The first thing to note is that every function that is threading aware (these are without
       exception functions compiled from concise descriptions by PDL::PP, later just called
       PP-functions) expects a defined (minimum) number of dimensions (we call them core dimen-
       sions) from each of its pdl arguments. The inner function expects two one-dimensional
       (input) parameters from which it calculates a zero-dimensional (output) parameter. We
       write that symbolically as "inner((n),(n),[o]())" and call it "inner"'s 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 "outer((n),(m),[o](n,m))". The "[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 pdl.

       Here is the point when threading finally enters the game. If you call PP-functions with
       pdls that have more than the required core dimensions the first dimensions of the pdl
       arguments are used as the core dimensions and the additional extra dimensions are threaded
       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, $im is 3D, more specifi-
       cally "(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 $w. The second dimension
       is the first thread dimension (of size "x") and the third is here the second thread dimen-
       sion (of size "y"). The output pdl is automatically created (as requested by setting $grey
       to "null" prior to invocation). The output dimensions are obtained by appending the loop
       dimensions (here "(x,y)") to the core output dimensions (here 0D) to yield the final
       dimensions of the autocreated pdl (here "0D+2D=2D" to yield a 2D output of size "(x,y)").

       So the above command calls the core functioniality that computes the inner product of two
       1D vectors "x*y" times with $w and all 1D slices of the form "(':,(i),(j)')" of $im and
       sets the respective elements of the output pdl "$grey(i,j)" to the result of each computa-
       tion. 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 pdl with the right dimensions and sets the ele-
       ments indeed to the result of the computation for each pixel of the input image.

       When even more pdls and extra dimensions are involved things get a bit more complicated.
       We will first give the general rules how the thread dimensions depend on the dimensions of
       input pdls enabling you to figure out the dimensionality of an autocreated output pdl (for
       any given set of input pdls and core dimensions of the PP-function in question). The gen-
       eral 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 threading comes in extremly handy).

       A call for coding discipline

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

       In order to preserve human readability, PLEASE comment any nontrivial expression in your
       code involving threading.  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) = @_;
	  my $res;
	  return $res;

       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)

       How to figure out the loop dimensions

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

       R0    Core dimensions are identified with the first N dimensions of the respective pdl
	     argument (and are required). Any further dimensions are extra dimensions and used to
	     determine the loop dimensions.

       R1    The number of (implicit) loop dimensions is equal to the maximal number of extra
	     dimensions taken over the set of pdl arguments.

       R2    The size of each of the loop dimensions is derived from the size of the respective
	     dimensions of the pdl arguments. The size of a loop dimension is given by the maxi-
	     mal size found in any of the pdls having this extra dimension.

       R3    For all pdls 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 a pdl is one it is implic-
	     itly treated as a dummy dimension of size equal to that loop dim size when perform-
	     ing the thread loop.

       R4    If a pdl doesn't have a loop dimension, in the thread loop this pdl is treated as if
	     having a dummy dimension of size equal to the size of that loop dimension.

       R5    If output autocreation is used (by setting the relevant pdl to "PDL->null" before
	     invocation) the number of dimensions of the created pdl 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 pdls (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 pdl will be
	     physical (unless dataflow is in operation).

       In this context, note that you can run into the problem with assignment to pdls containing
       greater-than-one dummy dimensions (see above).  Although your output pdl(s) didn't contain
       any dummy dimensions in the first place they may end up with implicitly created dummy
       dimensions according to R4.

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


       and you call it with 3 pdls "a(5,3,10,11)", "b(5,3,2,10,1,12)", and "c(5,1,11,12)" as


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

	$d(:,:,i,j,k) .= func($a(:,:,i,j),$b(:,:,:,i,0,k),$c(:,0,j,k))
	   with 0<=i<10, 0<=j<=11, 0<=k<12

       If we analyze the color to greyscale conversion again with these rules in mind we note
       another great advantage of implicit threading.  We can call the conversion with a pdl rep-
       resenting a pixel (im(3)), a line of rgb pixels ("im(3,x)"), a proper color image
       ("im(3,x,y)") or a whole stack of RGB images ("im(3,x,y,z)"). As long as $im is of the
       form "(3,...)" the automatically created output pdl will contain the right number of
       dimensions and contain the intensity data as we exspect it since the loops have been
       implicitly performed thanks to implicit threading. You can easily convince yourself that
       calling with a color pixel $grey is 0D, with a line it turns out 1D grey(x), with an image
       we get "grey(x,y)" and finally we get a converted image stack "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-loop-
       ing and compare the flexibility of those routines using implicit threading to the explicit
       formulation. Furthermore, especially when using several thread dimensions it is a useful
       exercise to check the relative speed by doing some benchmark tests (which we still have to

       First in the row is a slightly reworked centroid example, now coded with threading in

	# threaded mult to calculate centroid coords, works for stacks as well
	$xc = sumover(($im*xvals(($im->dims)[0]))->clump(2)) /

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

	$prod = $im*xvals(zeroes(($im->dims)[0]))

       This will actually work for $im being one, two, three, and higher dimensional. If $im is
       one-dimensional it's just an ordinary product (in the sense that every element of $im is
       multiplied with the respective element of "xvals(...)"), if $im has more dimensions fur-
       ther threading is done by adding appropriate dummy dimensions to "xvals(...)"  according
       to R4.  More importantly, the two 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 pdl. 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 pdl into sumover which in this case is
       formed by clumping the first two dimensions of the "parent pdl" into one. From the point
       of view of the parent pdl 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
       "sumover($pdl->clump(2))" instead of "sum($pdl)" so that we can either pass just an image
       "(x,y)" or a stack of images "(x,y,t)" into this routine and get either just one x-coor-
       diante 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 pdl as input and return a (N-1)-D "projected" pdl. These operations
       are often performed with functions like sumover, prodover, minimum and 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)

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

	# maxima of columns (as function of column number and time)

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

	# sums of pixles in time (assuming time is the third dim)

       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 pdl in question "clump" comes
       in handy. As an example look at the definition of "sum" (as defined in "Basic.pm"):

	sub sum {
	  return $tmp->at(); # return a perl number, not a 0D pdl

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

	perldl> $im = zeroes(byte, 10,20)
	perldl> $line = exp(-rvals(10)**2/9)
	# threaded assignment
	perldl> $im .= $line	  # set every line of $im to $line
	perldl> $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 exam-
       ples. It is the aptly named index function, signature "((n),(),[o]())" (see Appendix B)
       with the core functionality that "index(pdl (0,2,4,5),2,($ret=null))" will return the ele-
       ment with index 2 of the first input pdl. In this case, $ret will contain the value 4.  So
       here is the example:

	# a threaded 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 $im)
	# you can say just dummy(0) since the rules of threading make it fit
	perldl> index($palette->xchg(0,1),

       Let's go through it and explain the steps involved. Assuming we are dealing with an RGB
       lookup-table $palette is of size "(3,x)". First we exchange the dimensions of the palette
       so that looping is done over the first dimension of $palette (of size 3 that represent r,
       g, and b components). Now looking at $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 $im had the value 4 then the lookup should produce the


       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 threading code works; just go ahead
       and experiment a bit to get a better feeling for it).

       If you have read the threading rules carefully, then you might have noticed that we didn't
       have to explicitely state the size of the dummy dimension that we created for $im; when we
       create it with size 1 (the default) the rules of threading make it automatically fit to
       the desired size (by rule R3, in our example the size would be 3 assuming a palette of
       size "(3,x)"). Since situations like this do occur often in practice this is actually why
       rule R3 has been introduced (the part that makes dimensions of size 1 fit to the thread
       loop dim size). So we can just say

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

       Again, you can convince yourself that this routine will create the right output if called
       with a pixel ($im is 0D), a line ($im is 1D), an image ($im is 2D), ..., an RGB lookup ta-
       ble (palette is "(3,x)") and RGBA lookup table (palette is "(4,x)", see e.g. OpenGL). This
       flexibility is achieved by the rules of threading 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 loop-
       ing over certain dimensions and have the core functionality applied to another specified
       set of dimensions you use the dimension manipulating commands to create a (or several)
       virtual pdl(s) so that from the point of view of the parent pdl(s) you get what you want
       (always having the signature of the function in question and R1-R5 in mind!). Easy, isn't
       it ?

       Output autocreation 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.  Basi-
       cally, there are two ways of invoking pdl routines, namely

	$result = func($a,$b);



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



	$result = func($a,$b)

       which are exactly equivalent.

       Be warned that you can not use output autocreation when using explicit threading (for rea-
       sons explained in the following section on explicit threading, the second variant of

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

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

       The take-home message of this section once more: be aware of the limitation on output cre-
       ation when using explicit threading.

       Explicit threading

       Having so far only talked about the first flavour of threading 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 threading to get it all right you sometimes might want to
       specify in a more explicit way how to perform the thread loop. It is probably not too sur-
       prising that this variant of the game is called explicit threading.  Now, before we create
       the wrong impression: it is not either implicit or explicit; the two flavours do mix. But
       more about that later.

       The two most used functions with explicit threading are thread and unthread.  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->thread(0) += $line
	# explicit threading to add a vector to each column of a matrix
	perldl> $mat  = zeroes(4,3)
	perldl> $line = pdl (3.1416,2,-2)
	perldl> ($tmp = $mat->thread(0)) += $line

       In this example, "$mat->thread(0)" tells PDL that you want the second dimension of this
       pdl to be threaded over first leading to a thread loop that can be expressed as

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

       "thread" takes a list of numbers as arguments which explicitly specify which dimensions to
       thread over first. With the introduction of explicit threading the dimensions of a pdl are
       conceptually split into three different groups the latter two of which we have already
       encountered: thread dimensions, core dimensions and extra dimensions.

       Conceptually, it is best to think of those dimensions of a pdl that have been specified in
       a call to "thread" as being taken away from the set of normal dimensions and put on a sep-
       arate stack. So assuming we have a pdl "a(4,7,2,8)" saying

	$b = $a->thread(2,1)

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

       T0  Core dimensions are matched against the first n remaining dimensions of the pdl argu-
	   ment (note the difference to R1). Any further remaining dimensions are extra dimen-
	   sions and are used to determine the implicit loop dimensions.

       T1a The number of implicit loop dimensions is equal to the maximal number of extra dimen-
	   sions taken over the set of pdl arguments.

       T1b The number of explicit loop dimensions is equal to the maximal number of thread dimen-
	   sions taken over the set of pdl arguments.

       T1c The total number of loop dimensions is equal to the sum of explicit loop dimensions
	   and implicit loop dimensions. In the thread loop, explicit loop dimensions are
	   threaded over first followed by implicit loop dimensions.

       T2  The size of each of the loop dimensions is derived from the size of the respective
	   dimensions of the pdl arguments. It is given by the maximal size found in any pdls
	   having this thread dimension (for explicit loop dimensions) or extra dimension (for
	   implicit loop dimensions).

       T3  This rule applies to any explicit loop dimension as well as any implicit loop dimen-
	   sion. For all pdls that have a given thread/extra dimension the size must be equal to
	   the size of the respective explicit/implicit loop dimension or 1; otherwise you raise
	   a runtime exception. If the size of a thread/extra dimension of a pdl is one it is
	   implicitly treated as a dummy dimension of size equal to the explicit/implicit loop

       T4  If a pdl doesn't have a thread/extra dimension that corresponds to an
	   explicit/implicit loop dimension, in the thread loop this pdl is treated as if having
	   a dummy dimension of size equal to the size of that loop dimension.

       T4a All pdls that do have thread dimensions must have the same number of thread dimen-

       T5  Output autocreation cannot be used if any of the pdl arguments has any thread dimen-
	   sions. Otherwise R5 applies.

       The same restrictions apply with regard to implicit dummy dimensions (created by applica-
       tion of T4) as already mentioned in the section on implicit threading: if any of the out-
       put pdls has an (explicit or implicitly created) greater-than-one dummy dimension a run-
       time exception will be raised.

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


       and you call it with 3 pdls "a(5,3,10,11)", "b(3,5,10,1,12)", "c(10)" and an output pdl
       "d(3,11,5,10,12)" (which can here not be automatically created) as


       From the signature of func and the above call the pdls split into the following groups of
       core, extra and thread dimensions (written in the form "pdl(core dims){thread dims}[extra

	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 threading and want to keep track of what is going on) we fur-
       ther deduce that the number of explicit loop dimensions is 2 (by T1b from $a and $b) with
       sizes "(3,11)" (by T2); 2 implicit loop dimensions (by T1a from $b and $d) of size
       "(10,12)" (by T2) and the elements of are computed from the input pdls 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))

       Uhhmpf, this example was really not easy in terms of bookeeping. It serves mostly as an
       example how to figure out what's going on when you encounter a complicated looking expres-
       sion. But now it is really time to show that threading 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 threaded inner product
	# index handling so that matrices print correct under pdl
	$inv .= $evecs*0;  # just copy to get appropriately sized output
	$tmp .= $evecs;    # initialise, no backpropagation
	($tmp2 = $tmp->thread(0)) /= $evals;	#  threaded division
	# and now a matrix multiplication in disguise
	# alternative for matrix mult using implicit threading,
	# first xchg only for transpose

       Example 2:

	# outer product by threaded multiplication
	# stress that we need to do it with explicit call to my_biop1
	# when using explicit threading
	# similiar thing by implicit threading with autocreated pdl
	$res = $a->dummy(1) * $b->dummy(0);

       Example 3:

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

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

       Example 4:

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

       Example 5:

	# calculate a self-ratioed (i.e. self normalized) sequence of images
	# uses explicit threading and an implicitly threaded 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 pdl
	$aver /= ($n+1);
	$stack /= $aver;  # normalize the stack by doing a threaded divison
	# implicit versus explicit
	# alternatively calculate $aver with implicit threading and autocreation
	$aver /= ($n+1);

       Implicit versus explicit threading

       In this paragraph we are going to illustrate when explicit threading is preferrable over
       implicit threading 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 threading ?

       Well, code that uses threading should be (considerably) faster than code that uses
       explicit for-loops (or similar perl constructs) to achieve the same functionality. Espe-
       cially on supercomputers (with vector computing facilities/parallel processing) PDL
       threading will be implemented in a way that takes advantage of the additional facilities
       of these machines. Furthermore, it is a conceptually simply construct (though technical
       details might get involved at times) and can greatly reduce the syntactical complexity of
       PDL code (but keep the admonition for documentation in mind). Once you are comfortable
       with the threading way of thinking (and coding) it shouldn't be too difficult to under-
       stand code that somebody else has written than (provided he gave you an idea what
       exspected 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 threading 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 ;).

       An easy way to define functions that are aware of indexing and threading (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 threading 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 exter-
       nal library so  that they support indexing and threading (and mabe dataflow as well, see
       PDL::Dataflow). For further details check PDL::PP.

Appendix A
       Affine transformations - a special class of simple and powerful transformations

       [ This is also something to be added in future releases. Do we already have the general
       make_affine routine in PDL ? It is possible that we will reference another appropriate
       manpage from here ]

Appendix B
       signatures of standard PDL::PP compiled functions

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

	# functions in primitive.pd
	sumover        ((n),[o]())
	prodover       ((n),[o]())
	axisvalues     ((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]())

       Copyright (C) 1997 Christian Soeller (c.soeller@auckland.ac.nz) & Tuomas J. Lukka
       (lukka@fas.harvard.edu). All rights reserved. Although destined for release as a man page
       with the standard PDL distribution, it is not public domain. Permission is granted to
       freely distribute verbatim copies of this document provided that no modifications outside
       of formatting be made, and that this notice remain intact.  You are permitted and encour-
       aged to use its code and derivatives thereof in your own source code for fun or for profit
       as you see fit.

perl v5.8.0				    2000-05-23				      INDEXING(1)
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