Provided by: pdl_2.4.7+dfsg-2ubuntu5_amd64 bug

NAME

       PDL::Primitive - primitive operations for pdl

DESCRIPTION

       This module provides some primitive and useful functions defined using PDL::PP and able to
       use the new indexing tricks.

       See PDL::Indexing for how to use indices creatively.  For explanation of the signature
       format, see PDL::PP.

SYNOPSIS

        # Pulls in PDL::Primitive, among other modules.
        use PDL;

        # Only pull in PDL::Primitive:
        use PDL::Primitive;

FUNCTIONS

   inner
         Signature: (a(n); b(n); [o]c())

       Inner product over one dimension

        c = sum_i a_i * b_i

       If "a() * b()" contains only bad data, "c()" is set bad. Otherwise "c()" will have its bad
       flag cleared, as it will not contain any bad values.

   outer
         Signature: (a(n); b(m); [o]c(n,m))

       outer product over one dimension

       Naturally, it is possible to achieve the effects of outer product simply by threading over
       the ""*"" operator but this function is provided for convenience.

       outer does handle bad values.  It will set the bad-value flag of all output piddles if the
       flag is set for any of the input piddles.

   x
        Signature: (a(i,x), b(z,i),[o]c(x,z))

       Matrix multiplication

       PDL overloads the "x" operator (normally the repeat operator) for matrix multiplication.
       The number of columns (size of the 0 dimension) in the left-hand argument must normally
       equal the number of rows (size of the 1 dimension) in the right-hand argument.

       Row vectors are represented as (N x 1) two-dimensional PDLs, or you may be sloppy and use
       a one-dimensional PDL.  Column vectors are represented as (1 x N) two-dimensional PDLs.

       Threading occurs in the usual way, but as both the 0 and 1 dimension (if present) are
       included in the operation, you must be sure that you don't try to thread over either of
       those dims.

       EXAMPLES

       Here are some simple ways to define vectors and matrices:

        pdl> $r = pdl(1,2);                # A row vector
        pdl> $c = pdl([[3],[4]]);          # A column vector
        pdl> $c = pdl(3,4)->(*1);          # A column vector, using NiceSlice
        pdl> $m = pdl([[1,2],[3,4]]);      # A 2x2 matrix

       Now that we have a few objects prepared, here is how to matrix-multiply them:

        pdl> print $r x $m                 # row x matrix = row
        [
         [ 7 10]
        ]

        pdl> print $m x $r                 # matrix x row = ERROR
        PDL: Dim mismatch in matmult of [2x2] x [2x1]: 2 != 1

        pdl> print $m x $c                 # matrix x column = column
        [
         [ 5]
         [11]
        ]

        pdl> print $m x 2                  # Trivial case: scalar mult.
        [
         [2 4]
         [6 8]
        ]

        pdl> print $r x $c                 # row x column = scalar
        [
         [11]
        ]

        pdl> print $c x $r                 # column x row = matrix
        [
         [3 6]
         [4 8]
        ]

       INTERNALS

       The mechanics of the multiplication are carried out by the matmult method.

   matmult
         Signature: (a(t,h); b(w,t); [o]c(w,h))

       Matrix multiplication

       Notionally, matrix multiplication $a x $b is equivalent to the threading expression

           $a->dummy(1)->inner($b->xchg(0,1)->dummy(2),$c);

       but for large matrices that breaks CPU cache and is slow.  Instead, matmult calculates its
       result in 32x32x32 tiles, to keep the memory footprint within cache as long as possible on
       most modern CPUs.

       For usage, see x, a description of the overloaded 'x' operator

       matmult ignores the bad-value flag of the input piddles.  It will set the bad-value flag
       of all output piddles if the flag is set for any of the input piddles.

   innerwt
         Signature: (a(n); b(n); c(n); [o]d())

       Weighted (i.e. triple) inner product

        d = sum_i a(i) b(i) c(i)

       innerwt does handle bad values.  It will set the bad-value flag of all output piddles if
       the flag is set for any of the input piddles.

   inner2
         Signature: (a(n); b(n,m); c(m); [o]d())

       Inner product of two vectors and a matrix

        d = sum_ij a(i) b(i,j) c(j)

       Note that you should probably not thread over "a" and "c" since that would be very
       wasteful. Instead, you should use a temporary for "b*c".

       inner2 does handle bad values.  It will set the bad-value flag of all output piddles if
       the flag is set for any of the input piddles.

   inner2d
         Signature: (a(n,m); b(n,m); [o]c())

       Inner product over 2 dimensions.

       Equivalent to

        $c = inner($a->clump(2), $b->clump(2))

       inner2d does handle bad values.  It will set the bad-value flag of all output piddles if
       the flag is set for any of the input piddles.

   inner2t
         Signature: (a(j,n); b(n,m); c(m,k); [t]tmp(n,k); [o]d(j,k)))

       Efficient Triple matrix product "a*b*c"

       Efficiency comes from by using the temporary "tmp". This operation only scales as "N**3"
       whereas threading using inner2 would scale as "N**4".

       The reason for having this routine is that you do not need to have the same thread-
       dimensions for "tmp" as for the other arguments, which in case of large numbers of
       matrices makes this much more memory-efficient.

       It is hoped that things like this could be taken care of as a kind of closures at some
       point.

       inner2t does handle bad values.  It will set the bad-value flag of all output piddles if
       the flag is set for any of the input piddles.

   crossp
         Signature: (a(tri=3); b(tri); [o] c(tri))

       Cross product of two 3D vectors

       After

        $c = crossp $a, $b

       the inner product "$c*$a" and "$c*$b" will be zero, i.e. $c is orthogonal to $a and $b

       crossp does not process bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   norm
         Signature: (vec(n); [o] norm(n))

       Normalises a vector to unit Euclidean length

       norm does handle bad values.  It will set the bad-value flag of all output piddles if the
       flag is set for any of the input piddles.

   indadd
         Signature: (a(); int ind(); [o] sum(m))

       Threaded Index Add: Add "a" to the "ind" element of "sum", i.e:

        sum(ind) += a

       Simple Example:

         $a = 2;
         $ind = 3;
         $sum = zeroes(10);
         indadd($a,$ind, $sum);
         print $sum
         #Result: ( 2 added to element 3 of $sum)
         # [0 0 0 2 0 0 0 0 0 0]

       Threaded Example:

         $a = pdl( 1,2,3);
         $ind = pdl( 1,4,6);
         $sum = zeroes(10);
         indadd($a,$ind, $sum);
         print $sum."\n";
         #Result: ( 1, 2, and 3 added to elements 1,4,6 $sum)
         # [0 1 0 0 2 0 3 0 0 0]

       The routine barfs if any of the indices are bad.

   conv1d
         Signature: (a(m); kern(p); [o]b(m); int reflect)

       1D convolution along first dimension

       The m-th element of the discrete convolution of an input piddle $a of size $M, and a
       kernel piddle $kern of size $P, is calculated as

                                     n = ($P-1)/2
                                     ====
                                     \
         ($a conv1d $kern)[m]   =     >      $a_ext[m - n] * $kern[n]
                                     /
                                     ====
                                     n = -($P-1)/2

       where $a_ext is either the periodic (or reflected) extension of $a so it is equal to $a on
       " 0..$M-1 " and equal to the corresponding periodic/reflected image of $a outside that
       range.

         $con = conv1d sequence(10), pdl(-1,0,1);

         $con = conv1d sequence(10), pdl(-1,0,1), {Boundary => 'reflect'};

       By default, periodic boundary conditions are assumed (i.e. wrap around).  Alternatively,
       you can request reflective boundary conditions using the "Boundary" option:

         {Boundary => 'reflect'} # case in 'reflect' doesn't matter

       The convolution is performed along the first dimension. To apply it across another
       dimension use the slicing routines, e.g.

         $b = $a->mv(2,0)->conv1d($kernel)->mv(0,2); # along third dim

       This function is useful for threaded filtering of 1D signals.

       Compare also conv2d, convolve, fftconvolve, fftwconv, rfftwconv

       conv1d does not process bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   in
         Signature: (a(); b(n); [o] c())

       test if a is in the set of values b

          $goodmsk = $labels->in($goodlabels);
          print pdl(4,3,1)->in(pdl(2,3,3));
         [0 1 0]

       "in" is akin to the is an element of of set theory. In priciple, PDL threading could be
       used to achieve its functionality by using a construct like

          $msk = ($labels->dummy(0) == $goodlabels)->orover;

       However, "in" doesn't create a (potentially large) intermediate and is generally faster.

       in does not process bad values.  It will set the bad-value flag of all output piddles if
       the flag is set for any of the input piddles.

   uniq
       return all unique elements of a piddle

       The unique elements are returned in ascending order.

         PDL> p pdl(2,2,2,4,0,-1,6,6)->uniq
         [-1 0 2 4 6]     # 0 is returned 2nd (sorted order)

         PDL> p pdl(2,2,2,4,nan,-1,6,6)->uniq
         [-1 2 4 6 nan]   # NaN value is returned at end

       Note: The returned pdl is 1D; any structure of the input piddle is lost.  "NaN" values are
       never compare equal to any other values, even themselves.  As a result, they are always
       unique. "uniq" returns the NaN values at the end of the result piddle.  This follows the
       Matlab usage.

       See uniqind if you need the indices of the unique elements rather than the values.

       Bad values are not considered unique by uniq and are ignored.

        $a=sequence(10);
        $a=$a->setbadif($a%3);
        print $a->uniq;
        [0 3 6 9]

   uniqind
       Return the indices of all unique elements of a piddle The order is in the order of the
       values to be consistent with uniq. "NaN" values never compare equal with any other value
       and so are always unique.  This follows the Matlab usage.

         PDL> p pdl(2,2,2,4,0,-1,6,6)->uniqind
         [5 4 1 3 6]     # the 0 at index 4 is returned 2nd, but...

         PDL> p pdl(2,2,2,4,nan,-1,6,6)->uniqind
         [5 1 3 6 4]     # ...the NaN at index 4 is returned at end

       Note: The returned pdl is 1D; any structure of the input piddle is lost.

       See uniq if you want the unique values instead of the indices.

       Bad values are not considered unique by uniqind and are ignored.

   uniqvec
       Return all unique vectors out of a collection

         NOTE: If any vectors in the input piddle have NaN values
         they are returned at the end of the non-NaN ones.  This is
         because, by definition, NaN values never compare equal with
         any other value.

         NOTE: The current implementation does not sort the vectors
         containing NaN values.

       The unique vectors are returned in lexicographically sorted ascending order. The 0th
       dimension of the input PDL is treated as a dimensional index within each vector, and the
       1st and any higher dimensions are taken to run across vectors. The return value is always
       2D; any structure of the input PDL (beyond using the 0th dimension for vector index) is
       lost.

       See also uniq for a uniqe list of scalars; and qsortvec for sorting a list of vectors
       lexicographcally.

       If a vector contains all bad values, it is ignored as in uniq.  If some of the values are
       good, it is treated as a normal vector. For example, [1 2 BAD] and [BAD 2 3] could be
       returned, but [BAD BAD BAD] could not.  Vectors containing BAD values will be returned
       after any non-NaN and non-BAD containing vectors, followed by the NaN vectors.

   hclip
         Signature: (a(); b(); [o] c())

       clip (threshold) $a by $b ($b is upper bound)

       hclip does handle bad values.  It will set the bad-value flag of all output piddles if the
       flag is set for any of the input piddles.

   lclip
         Signature: (a(); b(); [o] c())

       clip (threshold) $a by $b ($b is lower bound)

       lclip does handle bad values.  It will set the bad-value flag of all output piddles if the
       flag is set for any of the input piddles.

   clip
       Clip (threshold) a piddle by (optional) upper or lower bounds.

        $b = $a->clip(0,3);
        $c = $a->clip(undef, $x);

       clip handles bad values since it is just a wrapper around hclip and lclip.

   wtstat
         Signature: (a(n); wt(n); avg(); [o]b(); int deg)

       Weighted statistical moment of given degree

       This calculates a weighted statistic over the vector "a".  The formula is

        b() = (sum_i wt_i * (a_i ** degree - avg)) / (sum_i wt_i)

       Bad values are ignored in any calculation; $b will only have its bad flag set if the
       output contains any bad data.

   statsover
         Signature: (a(n); w(n); float+ [o]avg(); float+ [o]prms(); int+ [o]median(); int+ [o]min(); int+ [o]max(); float+ [o]adev(); float+ [o]rms())

       Calculate useful statistics over a dimension of a piddle

         ($mean,$prms,$median,$min,$max,$adev,$rms) = statsover($piddle, $weights);

       This utility function calculates various useful quantities of a piddle. These are:

       ·  the mean:

            MEAN = sum (x)/ N

          with "N" being the number of elements in x

       ·  RMS deviation from the mean:

            RMS = sqrt(sum( (x-mean(x))^2 )/N)

          (also known as the root-mean-square deviation, or the square root of the variance)

       ·  the median

          The median is the 50th percentile data value.  Median is found by medover, so WEIGHTING
          IS IGNORED FOR THE MEDIAN CALCULATION.

       ·  the minimum

       ·  the maximum

       ·  the absolute deviation:

            ADEV = sqrt(sum( abs(x-mean(x)) )/N)

          (This is also called the standard deviation)

       ·  the population RMS deviation from the mean:

            PRMS = sqrt( sum( (x-mean(x))^2 )/(N-1)

          The population deviation is the best-estimate of the deviation of the population from
          which a sample is drawn.

       This operator is a projection operator so the calculation will take place over the final
       dimension. Thus if the input is N-dimensional each returned value will be N-1 dimensional,
       to calculate the statistics for the entire piddle either use "clump(-1)" directly on the
       piddle or call "stats".

       Bad values are simply ignored in the calculation, effectively reducing the sample size.
       If all data are bad then the output data are marked bad.

   stats
       Calculates useful statistics on a piddle

        ($mean,$prms,$median,$min,$max,$adev,$rms) = stats($piddle,[$weights]);

       This utility calculates all the most useful quantities in one call.  It works the same way
       as "statsover", except that the quantities are calculated considering the entire input PDL
       as a single sample, rather than as a collection of rows. See "statsover" for definitions
       of the returned quantities.

       Bad values are handled; if all input values are bad, then all of the output values are
       flagged bad.

   histogram
         Signature: (in(n); int+[o] hist(m); double step; double min; int msize => m)

       Calculates a histogram for given stepsize and minimum.

        $h = histogram($data, $step, $min, $numbins);
        $hist = zeroes $numbins;  # Put histogram in existing piddle.
        histogram($data, $hist, $step, $min, $numbins);

       The histogram will contain $numbins bins starting from $min, each $step wide. The value in
       each bin is the number of values in $data that lie within the bin limits.

       Data below the lower limit is put in the first bin, and data above the upper limit is put
       in the last bin.

       The output is reset in a different threadloop so that you can take a histogram of
       "$a(10,12)" into "$b(15)" and get the result you want.

       For a higher-level interface, see hist.

        pdl> p histogram(pdl(1,1,2),1,0,3)
        [0 2 1]

       histogram does handle bad values.  It will set the bad-value flag of all output piddles if
       the flag is set for any of the input piddles.

   whistogram
         Signature: (in(n); float+ wt(n);float+[o] hist(m); double step; double min; int msize => m)

       Calculates a histogram from weighted data for given stepsize and minimum.

        $h = whistogram($data, $weights, $step, $min, $numbins);
        $hist = zeroes $numbins;  # Put histogram in existing piddle.
        whistogram($data, $weights, $hist, $step, $min, $numbins);

       The histogram will contain $numbins bins starting from $min, each $step wide. The value in
       each bin is the sum of the values in $weights that correspond to values in $data that lie
       within the bin limits.

       Data below the lower limit is put in the first bin, and data above the upper limit is put
       in the last bin.

       The output is reset in a different threadloop so that you can take a histogram of
       "$a(10,12)" into "$b(15)" and get the result you want.

        pdl> p whistogram(pdl(1,1,2), pdl(0.1,0.1,0.5), 1, 0, 4)
        [0 0.2 0.5 0]

       whistogram does handle bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   histogram2d
         Signature: (ina(n); inb(n); int+[o] hist(ma,mb); double stepa; double mina; int masize => ma;
                            double stepb; double minb; int mbsize => mb;)

       Calculates a 2d histogram.

        $h = histogram2d($datax, $datay, $stepx, $minx,
              $nbinx, $stepy, $miny, $nbiny);
        $hist = zeroes $nbinx, $nbiny;  # Put histogram in existing piddle.
        histogram2d($datax, $datay, $hist, $stepx, $minx,
              $nbinx, $stepy, $miny, $nbiny);

       The histogram will contain $nbinx x $nbiny bins, with the lower limits of the first one at
       "($minx, $miny)", and with bin size "($stepx, $stepy)".  The value in each bin is the
       number of values in $datax and $datay that lie within the bin limits.

       Data below the lower limit is put in the first bin, and data above the upper limit is put
       in the last bin.

        pdl> p histogram2d(pdl(1,1,1,2,2),pdl(2,1,1,1,1),1,0,3,1,0,3)
        [
         [0 0 0]
         [0 2 2]
         [0 1 0]
        ]

       histogram2d does handle bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   whistogram2d
         Signature: (ina(n); inb(n); float+ wt(n);float+[o] hist(ma,mb); double stepa; double mina; int masize => ma;
                            double stepb; double minb; int mbsize => mb;)

       Calculates a 2d histogram from weighted data.

        $h = whistogram2d($datax, $datay, $weights,
              $stepx, $minx, $nbinx, $stepy, $miny, $nbiny);
        $hist = zeroes $nbinx, $nbiny;  # Put histogram in existing piddle.
        whistogram2d($datax, $datay, $weights, $hist,
              $stepx, $minx, $nbinx, $stepy, $miny, $nbiny);

       The histogram will contain $nbinx x $nbiny bins, with the lower limits of the first one at
       "($minx, $miny)", and with bin size "($stepx, $stepy)".  The value in each bin is the sum
       of the values in $weights that correspond to values in $datax and $datay that lie within
       the bin limits.

       Data below the lower limit is put in the first bin, and data above the upper limit is put
       in the last bin.

        pdl> p whistogram2d(pdl(1,1,1,2,2),pdl(2,1,1,1,1),pdl(0.1,0.2,0.3,0.4,0.5),1,0,3,1,0,3)
        [
         [  0   0   0]
         [  0 0.5 0.9]
         [  0 0.1   0]
        ]

       whistogram2d does handle bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   fibonacci
         Signature: ([o]x(n))

       Constructor - a vector with Fibonacci's sequence

       fibonacci does not process bad values.  It will set the bad-value flag of all output
       piddles if the flag is set for any of the input piddles.

   append
         Signature: (a(n); b(m); [o] c(mn))

       append two or more piddles by concatenating along their first dimensions

        $a = ones(2,4,7);
        $b = sequence 5;
        $c = $a->append($b);  # size of $c is now (7,4,7) (a jumbo-piddle ;)

       "append" appends two piddles along their first dims. Rest of the dimensions must be
       compatible in the threading sense. Resulting size of first dim is the sum of the sizes of
       the first dims of the two argument piddles - ie "n + m".

       Similar functions include glue (below) and cat.

       append does not process bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   glue
         $c = $a->glue(<dim>,$b,...)

       Glue two or more PDLs together along an arbitrary dimension (N-D append).

       Sticks $a, $b, and all following arguments together along the specified dimension.  All
       other dimensions must be compatible in the threading sense.

       Glue is permissive, in the sense that every PDL is treated as having an infinite number of
       trivial dimensions of order 1 -- so "$a->glue(3,$b)" works, even if $a and $b are only one
       dimensional.

       If one of the PDLs has no elements, it is ignored.  Likewise, if one of them is actually
       the undefined value, it is treated as if it had no elements.

       If the first parameter is a defined perl scalar rather than a pdl, then it is taken as a
       dimension along which to glue everything else, so you can say "$cube =
       PDL::glue(3,@image_list);" if you like.

       "glue" is implemented in pdl, using a combination of xchg and append.  It should probably
       be updated (one day) to a pure PP function.

       Similar functions include append (above) and cat.

   axisvalues
         Signature: ([o,nc]a(n))

       Internal routine

       "axisvalues" is the internal primitive that implements axisvals and alters its argument.

       axisvalues does not process bad values.  It will set the bad-value flag of all output
       piddles if the flag is set for any of the input piddles.

   random
       Constructor which returns piddle of random numbers

        $a = random([type], $nx, $ny, $nz,...);
        $a = random $b;

       etc (see zeroes).

       This is the uniform distribution between 0 and 1 (assumedly excluding 1 itself). The
       arguments are the same as "zeroes" (q.v.) - i.e. one can specify dimensions, types or give
       a template.

       You can use the perl function srand to seed the random generator. For further details
       consult Perl's  srand documentation.

   randsym
       Constructor which returns piddle of random numbers

        $a = randsym([type], $nx, $ny, $nz,...);
        $a = randsym $b;

       etc (see zeroes).

       This is the uniform distribution between 0 and 1 (excluding both 0 and 1, cf random). The
       arguments are the same as "zeroes" (q.v.) - i.e. one can specify dimensions, types or give
       a template.

       You can use the perl function srand to seed the random generator. For further details
       consult Perl's  srand documentation.

   grandom
       Constructor which returns piddle of Gaussian random numbers

        $a = grandom([type], $nx, $ny, $nz,...);
        $a = grandom $b;

       etc (see zeroes).

       This is generated using the math library routine "ndtri".

       Mean = 0, Stddev = 1

       You can use the perl function srand to seed the random generator. For further details
       consult Perl's  srand documentation.

   vsearch
         Signature: (i(); x(n); int [o]ip())

       routine for searching 1D values i.e. step-function interpolation.

        $inds = vsearch($vals, $xs);

       Returns for each value of $vals the index of the least larger member of $xs (which need to
       be in increasing order). If the value is larger than any member of $xs, the index to the
       last element of $xs is returned.

       This function is useful e.g. when you have a list of probabilities for events and want to
       generate indices to events:

        $a = pdl(.01,.86,.93,1); # Barnsley IFS probabilities cumulatively
        $b = random 20;
        $c = vsearch($b, $a); # Now, $c will have the appropriate distr.

       It is possible to use the cumusumover function to obtain cumulative probabilities from
       absolute probabilities.

       needs major (?) work to handles bad values

   interpolate
         Signature: (xi(); x(n); y(n); [o] yi(); int [o] err())

       routine for 1D linear interpolation

        ( $yi, $err ) = interpolate($xi, $x, $y)

       Given a set of points "($x,$y)", use linear interpolation to find the values $yi at a set
       of points $xi.

       "interpolate" uses a binary search to find the suspects, er..., interpolation indices and
       therefore abscissas (ie $x) have to be strictly ordered (increasing or decreasing).  For
       interpolation at lots of closely spaced abscissas an approach that uses the last index
       found as a start for the next search can be faster (compare Numerical Recipes "hunt"
       routine). Feel free to implement that on top of the binary search if you like. For out of
       bounds values it just does a linear extrapolation and sets the corresponding element of
       $err to 1, which is otherwise 0.

       See also interpol, which uses the same routine, differing only in the handling of
       extrapolation - an error message is printed rather than returning an error piddle.

       needs major (?) work to handles bad values

   interpol
        Signature: (xi(); x(n); y(n); [o] yi())

       routine for 1D linear interpolation

        $yi = interpol($xi, $x, $y)

       "interpol" uses the same search method as interpolate, hence $x must be strictly ordered
       (either increasing or decreasing).  The difference occurs in the handling of out-of-bounds
       values; here an error message is printed.

   interpND
       Interpolate values from an N-D piddle, with switchable method

         $source = 10*xvals(10,10) + yvals(10,10);
         $index = pdl([[2.2,3.5],[4.1,5.0]],[[6.0,7.4],[8,9]]);
         print $source->interpND( $index );

       InterpND acts like indexND, collapsing $index by lookup into $source; but it does
       interpolation rather than direct sampling.  The interpolation method and boundary
       condition are switchable via an options hash.

       By default, linear or sample interpolation is used, with constant value outside the
       boundaries of the source pdl.  No dataflow occurs, because in general the output is
       computed rather than indexed.

       All the interpolation methods treat the pixels as value-centered, so the "sample" method
       will return "$a->(0)" for coordinate values on the set [-0.5,0.5), and all methods will
       return "$a->(1)" for a coordinate value of exactly 1.

       Recognized options:

       method
          Values can be:

          ·  0, s, sample, Sample (default for integer source types)

             The nearest value is taken. Pixels are regarded as centered on their respective
             integer coordinates (no offset from the linear case).

          ·  1, l, linear, Linear (default for floating point source types)

             The values are N-linearly interpolated from an N-dimensional cube of size 2.

          ·  3, c, cube, cubic, Cubic

             The values are interpolated using a local cubic fit to the data.  The fit is
             constrained to match the original data and its derivative at the data points.  The
             second derivative of the fit is not continuous at the data points.  Multidimensional
             datasets are interpolated by the successive-collapse method.

             (Note that the constraint on the first derivative causes a small amount of ringing
             around sudden features such as step functions).

          ·  f, fft, fourier, Fourier

             The source is Fourier transformed, and the interpolated values are explicitly
             calculated from the coefficients.  The boundary condition option is ignored --
             periodic boundaries are imposed.

             If you pass in the option "fft", and it is a list (ARRAY) ref, then it is a stash
             for the magnitude and phase of the source FFT.  If the list has two elements then
             they are taken as already computed; otherwise they are calculated and put in the
             stash.

       b, bound, boundary, Boundary
          This option is passed unmodified into indexND, which is used as the indexing engine for
          the interpolation.  Some current allowed values are 'extend', 'periodic', 'truncate',
          and 'mirror' (default is 'truncate').

       bad
          contains the fill value used for 'truncate' boundary.  (default 0)

       fft
          An array ref whose associated list is used to stash the FFT of the source data, for the
          FFT method.

   one2nd
       Converts a one dimensional index piddle to a set of ND coordinates

        @coords=one2nd($a, $indices)

       returns an array of piddles containing the ND indexes corresponding to the one dimensional
       list indices. The indices are assumed to correspond to array $a clumped using "clump(-1)".
       This routine is used in whichND, but is useful on its own occasionally.

        pdl> $a=pdl [[[1,2],[-1,1]], [[0,-3],[3,2]]]; $c=$a->clump(-1)
        pdl> $maxind=maximum_ind($c); p $maxind;
        6
        pdl> print one2nd($a, maximum_ind($c))
        0 1 1
        pdl> p $a->at(0,1,1)
        3

   which
         Signature: (mask(n); int [o] inds(m))

       Returns indices of non-zero values from a 1-D PDL

        $i = which($mask);

       returns a pdl with indices for all those elements that are nonzero in the mask. Note that
       the returned indices will be 1D. If you feed in a multidimensional mask, it will be
       flattened before the indices are calculated.  See also whichND for multidimensional masks.

       If you want to index into the original mask or a similar piddle with output from "which",
       remember to flatten it before calling index:

         $data = random 5, 5;
         $idx = which $data > 0.5; # $idx is now 1D
         $bigsum = $data->flat->index($idx)->sum;  # flatten before indexing

       Compare also where for similar functionality.

       SEE ALSO:

       which_both returns separately the indices of both zero and nonzero values in the mask.

       where returns associated values from a data PDL, rather than indices into the mask PDL.

       whichND returns N-D indices into a multidimensional PDL.

        pdl> $x = sequence(10); p $x
        [0 1 2 3 4 5 6 7 8 9]
        pdl> $indx = which($x>6); p $indx
        [7 8 9]

       which does handle bad values.  It will set the bad-value flag of all output piddles if the
       flag is set for any of the input piddles.

   which_both
         Signature: (mask(n); int [o] inds(m); int [o]notinds(q))

       Returns indices of zero and nonzero values in a mask PDL

        ($i, $c_i) = which_both($mask);

       This works just as which, but the complement of $i will be in $c_i.

        pdl> $x = sequence(10); p $x
        [0 1 2 3 4 5 6 7 8 9]
        pdl> ($small, $big) = which_both ($x >= 5); p "$small\n $big"
        [5 6 7 8 9]
        [0 1 2 3 4]

       which_both does handle bad values.  It will set the bad-value flag of all output piddles
       if the flag is set for any of the input piddles.

   where
       Use a mask to select values from one or more data PDLs

       "where" accepts one or more data piddles and a mask piddle.  It returns a list of output
       piddles, corresponding to the input data piddles.  Each output piddle is a 1-dimensional
       list of values in its corresponding data piddle. The values are drawn from locations where
       the mask is nonzero.

       The output PDLs are still connected to the original data PDLs, for the purpose of
       dataflow.

       "where" combines the functionality of which and index into a single operation.

       BUGS:

       There is no "whereND", and probably should be.  While "where" works OK for most
       N-dimensional cases, it does not thread properly over (for example) the (N+1)th dimension
       in data that is compared to an N-dimensional mask.

        $i = $x->where($x+5 > 0); # $i contains those elements of $x
                                  # where mask ($x+5 > 0) is 1
        $i .= -5;  # Set those elements (of $x) to -5. Together, these
                   # commands clamp $x to a maximum of -5.

       It is also possible to use the same mask for several piddles with the same call:

        ($i,$j,$k) = where($x,$y,$z, $x+5>0);

       Note: $i is always 1-D, even if $x is >1-D.

       WARNING: The first argument (the values) and the second argument (the mask) currently have
       to have the exact same dimensions (or horrible things happen). You *cannot* thread over a
       smaller mask, for example.

   whichND
       Return the coordinates of non-zero values in a mask.

       WhichND returns the N-dimensional coordinates of each nonzero value in a mask PDL with any
       number of dimensions.

       For historical reasons the return value is different in list and scalar context.  In
       scalar context, you get back a PDL containing coordinates suitable for use in indexND or
       range; in list context, the coordinates are broken out into separate PDLs.

        $coords = whichND($mask);

       returns a PDL containing the coordinates of the elements that are non-zero in $mask,
       suitable for use in indexND.  The 0th dimension contains the full coordinate listing of
       each point; the 1st dimension lists all the points.  For example, if $mask has rank 4 and
       100 matching elements, then $coords has dimension 4x100.

        @coords=whichND($mask);

       returns a perl list of piddles containing the coordinates of the elements that are non-
       zero in $mask.  Each element corresponds to a particular index dimension.  For example, if
       $mask has rank 4 and 100 matching elements, then @coords has 4 elements, each of which is
       a pdl of size 100.

       SEE ALSO:

       which finds coordinates of nonzero values in a 1-D mask.

       where extracts values from a data PDL that are associated with nonzero values in a mask
       PDL.

        pdl> $a=sequence(10,10,3,4)
        pdl> ($x, $y, $z, $w)=whichND($a == 203); p $x, $y, $z, $w
        [3] [0] [2] [0]
        pdl> print $a->at(list(cat($x,$y,$z,$w)))
        203

   setops
       Implements simple set operations like union and intersection

          Usage: $set = setops($a, <OPERATOR>, $b);

       The operator can be "OR", "XOR" or "AND". This is then applied to $a viewed as a set and
       $b viewed as a set. Set theory says that a set may not have two or more identical
       elements, but setops takes care of this for you, so "$a=pdl(1,1,2)" is OK. The functioning
       is as follows:

       "OR"
           The resulting vector will contain the elements that are either in $a or in $b or both.
           This is the union in set operation terms

       "XOR"
           The resulting vector will contain the elements that are either in $a or $b, but not in
           both. This is

                Union($a, $b) - Intersection($a, $b)

           in set operation terms.

       "AND"
           The resulting vector will contain the intersection of $a and $b, so the elements that
           are in both $a and $b. Note that for convenience this operation is also aliased to
           intersect

       It should be emphasized that these routines are used when one or both of the sets $a, $b
       are hard to calculate or that you get from a separate subroutine.

       Finally IDL users might be familiar with Craig Markwardt's "cmset_op.pro" routine which
       has inspired this routine although it was written independently However the present
       routine has a few less options (but see the exampels)

       You will very often use these functions on an index vector, so that is what we will show
       here. We will in fact something slightly silly. First we will find all squares that are
       also cubes below 10000.

       Create a sequence vector:

         pdl> $x = sequence(10000)

       Find all odd and even elements:

         pdl> ($even, $odd) = which_both( ($x % 2) == 0)

       Find all squares

         pdl> $squares= which(ceil(sqrt($x)) == floor(sqrt($x)))

       Find all cubes (being careful with roundoff error!)

         pdl> $cubes= which(ceil($x**(1.0/3.0)) == floor($x**(1.0/3.0)+1e-6))

       Then find all squares that are cubes:

         pdl> $both = setops($squares, 'AND', $cubes)

       And print these (assumes that "PDL::NiceSlice" is loaded!)

         pdl> p $x($both)
          [0 1 64 729 4096]

       Then find all numbers that are either cubes or squares, but not both:

         pdl> $cube_xor_square = setops($squares, 'XOR', $cubes)

         pdl> p $cube_xor_square->nelem()
          112

       So there are a total of 112 of these!

       Finally find all odd squares:

         pdl> $odd_squares = setops($squares, 'AND', $odd)

       Another common occurance is to want to get all objects that are in $a and in the
       complement of $b. But it is almost always best to create the complement explicitly since
       the universe that both are taken from is not known. Thus use which_both if possible to
       keep track of complements.

       If this is impossible the best approach is to make a temporary:

       This creates an index vector the size of the universe of the sets and set all elements in
       $b to 0

         pdl> $tmp = ones($n_universe); $tmp($b)=0;

       This then finds the complement of $b

         pdl> $C_b = which($tmp == 1);

       and this does the final selection:

         pdl> $set = setops($a, 'AND', $C_b)

   intersect
       Calculate the intersection of two piddles

          Usage: $set = intersect($a, $b);

       This routine is merely a simple interface to setops. See that for more information

       Find all numbers less that 100 that are of the form 2*y and 3*x

        pdl> $x=sequence(100)
        pdl> $factor2 = which( ($x % 2) == 0)
        pdl> $factor3 = which( ($x % 3) == 0)
        pdl> $ii=intersect($factor2, $factor3)
        pdl> p $x($ii)
        [0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96]

AUTHOR

       Copyright (C) Tuomas J. Lukka 1997 (lukka@husc.harvard.edu). Contributions by Christian
       Soeller (c.soeller@auckland.ac.nz), Karl Glazebrook (kgb@aaoepp.aao.gov.au), Craig
       DeForest (deforest@boulder.swri.edu) and Jarle Brinchmann (jarle@astro.up.pt) All rights
       reserved. There is no warranty. You are allowed to redistribute this software /
       documentation under certain conditions. For details, see the file COPYING in the PDL
       distribution. If this file is separated from the PDL distribution, the copyright notice
       should be included in the file.

       Updated for CPAN viewing compatibility by David Mertens.