NAME
PDL::Ufunc - primitive ufunc operations for pdl
DESCRIPTION
This module provides some primitive and useful functions defined using PDL::PP based on functionality of what are sometimes called ufuncs (for example NumPY and Mathematica talk about these). It collects all the functions generally used to reduce or accumulate along a dimension. These all do their job across the first dimension but by using the slicing functions you can do it on any dimension.
The PDL::Reduce module provides an alternative interface to many of the functions in this module.
SYNOPSIS
use PDL::Ufunc;FUNCTIONS
prodover
Signature: (a(n); int+ [o]b())Project via product to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the product along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = prodover($x);$spectrum = prodover $image->transposeprodover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
cprodover
Signature: (a(n); cdouble [o]b())Project via product to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the product along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = cprodover($x);$spectrum = cprodover $image->transposeUnlike "prodover", the calculations are performed in complex double precision.
cprodover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
dprodover
Signature: (a(n); double [o]b())Project via product to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the product along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = dprodover($x);$spectrum = dprodover $image->transposeUnlike "prodover", the calculations are performed in double precision.
dprodover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
cumuprodover
Signature: (a(n); int+ [o]b(n))Cumulative product
This function calculates the cumulative product along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
The sum is started so that the first element in the cumulative product is the first element of the parameter.
$y = cumuprodover($x);$spectrum = cumuprodover $image->transposecumuprodover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
ccumuprodover
Signature: (a(n); cdouble [o]b(n))Cumulative product
This function calculates the cumulative product along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
The sum is started so that the first element in the cumulative product is the first element of the parameter.
$y = ccumuprodover($x);$spectrum = ccumuprodover $image->transposeUnlike "cumuprodover", the calculations are performed in complex double precision.
ccumuprodover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
dcumuprodover
Signature: (a(n); double [o]b(n))Cumulative product
This function calculates the cumulative product along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
The sum is started so that the first element in the cumulative product is the first element of the parameter.
$y = dcumuprodover($x);$spectrum = dcumuprodover $image->transposeUnlike "cumuprodover", the calculations are performed in double precision.
dcumuprodover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
sumover
Signature: (a(n); int+ [o]b())Project via sum to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the sum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = sumover($x);$spectrum = sumover $image->transposesumover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
csumover
Signature: (a(n); cdouble [o]b())Project via sum to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the sum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = csumover($x);$spectrum = csumover $image->transposeUnlike "sumover", the calculations are performed in complex double precision.
csumover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
dsumover
Signature: (a(n); double [o]b())Project via sum to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the sum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = dsumover($x);$spectrum = dsumover $image->transposeUnlike "sumover", the calculations are performed in double precision.
dsumover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
cumusumover
Signature: (a(n); int+ [o]b(n))Cumulative sum
This function calculates the cumulative sum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
The sum is started so that the first element in the cumulative sum is the first element of the parameter.
$y = cumusumover($x);$spectrum = cumusumover $image->transposecumusumover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
ccumusumover
Signature: (a(n); cdouble [o]b(n))Cumulative sum
This function calculates the cumulative sum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
The sum is started so that the first element in the cumulative sum is the first element of the parameter.
$y = ccumusumover($x);$spectrum = ccumusumover $image->transposeUnlike "cumusumover", the calculations are performed in complex double precision.
ccumusumover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
dcumusumover
Signature: (a(n); double [o]b(n))Cumulative sum
This function calculates the cumulative sum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
The sum is started so that the first element in the cumulative sum is the first element of the parameter.
$y = dcumusumover($x);$spectrum = dcumusumover $image->transposeUnlike "cumusumover", the calculations are performed in double precision.
dcumusumover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
andover
Signature: (a(n); int+ [o]b())Project via and to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the and along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = andover($x);$spectrum = andover $image->transposeIf a() contains only bad data (and its bad flag is set), b() is set bad. Otherwise b() will have its bad flag cleared, as it will not contain any bad values.
bandover
Signature: (a(n); [o]b())Project via bitwise and to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the bitwise and along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = bandover($x);$spectrum = bandover $image->transposeIf a() contains only bad data (and its bad flag is set), b() is set bad. Otherwise b() will have its bad flag cleared, as it will not contain any bad values.
borover
Signature: (a(n); [o]b())Project via bitwise or to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the bitwise or along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = borover($x);$spectrum = borover $image->transposeIf a() contains only bad data (and its bad flag is set), b() is set bad. Otherwise b() will have its bad flag cleared, as it will not contain any bad values.
firstnonzeroover
Signature: (a(n); [o]b())Project via first non-zero value to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the first non-zero value along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = firstnonzeroover($x);$spectrum = firstnonzeroover $image->transposeIf a() contains only bad data (and its bad flag is set), b() is set bad. Otherwise b() will have its bad flag cleared, as it will not contain any bad values.
orover
Signature: (a(n); int+ [o]b())Project via or to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the or along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = orover($x);$spectrum = orover $image->transposeIf a() contains only bad data (and its bad flag is set), b() is set bad. Otherwise b() will have its bad flag cleared, as it will not contain any bad values.
zcover
Signature: (a(n); int+ [o]b())Project via == 0 to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the == 0 along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = zcover($x);$spectrum = zcover $image->transposeIf a() contains only bad data (and its bad flag is set), b() is set bad. Otherwise b() will have its bad flag cleared, as it will not contain any bad values.
diffover
Signature: (x(t); [o]dx(t))Differencing. DX(t) = X(t) - X(t-1), DX(0) = X(0). Can be done inplace.
Unlike "diff2", output vector is same length. Originally by Maggie J. Xiong.
diffover does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
diff2
Signature: (a(n); [o]o(nminus1=CALC($SIZE(n) - 1)))Numerically differentiates a vector along 0th dimension.
By using "xchg" in PDL::Slices etc. it is possible to use any dimension. Unlike "diffover", output vector is one shorter.
print pdl(q[3 4 2 3 2 3 1])->diff2;
# [1 -2 1 -1 1 -2]On bad value, output value is set bad. On next good value, output value is difference between that and last good value.
intover
Signature: (a(n); float+ [o]b())Project via integral to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the integral along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = intover($x);$spectrum = intover $image->transposeNotes:
intover uses a point spacing of one (i.e., delta-h==1). You will need to scale the result to correct for the true point delta.
For n > 3, these are all O(h^4) (like Simpson's rule), but are integrals between the end points assuming the pdl gives values just at these centres: for such `functions', sumover is correct to O(h), but is the natural (and correct) choice for binned data, of course.
intover ignores the bad-value flag of the input ndarrays. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
average
Signature: (a(n); int+ [o]b())Project via average to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the average along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = average($x);$spectrum = average $image->transposeaverage processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
avgover
Synonym for "average".
caverage
Signature: (a(n); cdouble [o]b())Project via average to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the average along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = caverage($x);$spectrum = caverage $image->transposeUnlike average, the calculation is performed in complex double precision.
caverage processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
cavgover
Synonym for "caverage".
daverage
Signature: (a(n); double [o]b())Project via average to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the average along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = daverage($x);$spectrum = daverage $image->transposeUnlike average, the calculation is performed in double precision.
daverage processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
davgover
Synonym for "daverage".
minimum
Signature: (a(n); [o]c())Project via minimum to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the minimum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = minimum($x);$spectrum = minimum $image->transposeOutput is set bad if no elements of the input are non-bad, otherwise the bad flag is cleared for the output ndarray.
Note that NaNs are considered to be valid values and will "win" over non-NaN; see isfinite and badmask for ways of masking NaNs.
minover
Synonym for "minimum".
minimum_ind
Signature: (a(n); indx [o] c())Like minimum but returns the first matching index rather than the value
Output is set bad if no elements of the input are non-bad, otherwise the bad flag is cleared for the output ndarray.
Note that NaNs are considered to be valid values and will "win" over non-NaN; see isfinite and badmask for ways of masking NaNs.
minover_ind
Synonym for "minimum_ind".
minimum_n_ind
Signature: (a(n); indx [o]c(m); PDL_Indx m_size => m)Returns the index of first m_size minimum elements. As of 2.077, you can specify how many by either passing in an ndarray of the given size (DEPRECATED - will be converted to indx if needed and the input arg will be set to that), or just the size, or a null and the size.
minimum_n_ind($pdl, $out = zeroes(5)); # DEPRECATED
$out = minimum_n_ind($pdl, 5);
minimum_n_ind($pdl, $out = null, 5);Output bad flag is cleared for the output ndarray if sufficient non-bad elements found, else remaining slots in $c() are set bad.
Note that NaNs are considered to be valid values and will "win" over non-NaN; see isfinite and badmask for ways of masking NaNs.
minover_n_ind
Synonym for "minimum_n_ind".
maximum
Signature: (a(n); [o]c())Project via maximum to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the maximum along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = maximum($x);$spectrum = maximum $image->transposeOutput is set bad if no elements of the input are non-bad, otherwise the bad flag is cleared for the output ndarray.
Note that NaNs are considered to be valid values and will "win" over non-NaN; see isfinite and badmask for ways of masking NaNs.
maxover
Synonym for "maximum".
maximum_ind
Signature: (a(n); indx [o] c())Like maximum but returns the first matching index rather than the value
Output is set bad if no elements of the input are non-bad, otherwise the bad flag is cleared for the output ndarray.
Note that NaNs are considered to be valid values and will "win" over non-NaN; see isfinite and badmask for ways of masking NaNs.
maxover_ind
Synonym for "maximum_ind".
maximum_n_ind
Signature: (a(n); indx [o]c(m); PDL_Indx m_size => m)Returns the index of first m_size maximum elements. As of 2.077, you can specify how many by either passing in an ndarray of the given size (DEPRECATED - will be converted to indx if needed and the input arg will be set to that), or just the size, or a null and the size.
maximum_n_ind($pdl, $out = zeroes(5)); # DEPRECATED
$out = maximum_n_ind($pdl, 5);
maximum_n_ind($pdl, $out = null, 5);Output bad flag is cleared for the output ndarray if sufficient non-bad elements found, else remaining slots in $c() are set bad.
Note that NaNs are considered to be valid values and will "win" over non-NaN; see isfinite and badmask for ways of masking NaNs.
maxover_n_ind
Synonym for "maximum_n_ind".
minmaximum
Signature: (a(n); [o]cmin(); [o] cmax(); indx [o]cmin_ind(); indx [o]cmax_ind())Find minimum and maximum and their indices for a given ndarray;
pdl> $x=pdl [[-2,3,4],[1,0,3]]
pdl> ($min, $max, $min_ind, $max_ind)=minmaximum($x)
pdl> p $min, $max, $min_ind, $max_ind
[-2 0] [4 3] [0 1] [2 2]See also "minmax", which clumps the ndarray together.
If a() contains only bad data, then the output ndarrays will be set bad, along with their bad flag. Otherwise they will have their bad flags cleared, since they will not contain any bad values.
minmaxover
Synonym for "minmaximum".
avg
Return the average of all elements in an ndarray.
See the documentation for "average" for more information.
$x = avg($data);This routine handles bad values.
sum
Return the sum of all elements in an ndarray.
See the documentation for "sumover" for more information.
$x = sum($data);This routine handles bad values.
prod
Return the product of all elements in an ndarray.
See the documentation for "prodover" for more information.
$x = prod($data);This routine handles bad values.
davg
Return the average (in double precision) of all elements in an ndarray.
See the documentation for "daverage" for more information.
$x = davg($data);This routine handles bad values.
dsum
Return the sum (in double precision) of all elements in an ndarray.
See the documentation for "dsumover" for more information.
$x = dsum($data);This routine handles bad values.
dprod
Return the product (in double precision) of all elements in an ndarray.
See the documentation for "dprodover" for more information.
$x = dprod($data);This routine handles bad values.
zcheck
Return the check for zero of all elements in an ndarray.
See the documentation for "zcover" for more information.
$x = zcheck($data);This routine handles bad values.
and
Return the logical and of all elements in an ndarray.
See the documentation for "andover" for more information.
$x = and($data);This routine handles bad values.
band
Return the bitwise and of all elements in an ndarray.
See the documentation for "bandover" for more information.
$x = band($data);This routine handles bad values.
or
Return the logical or of all elements in an ndarray.
See the documentation for "orover" for more information.
$x = or($data);This routine handles bad values.
bor
Return the bitwise or of all elements in an ndarray.
See the documentation for "borover" for more information.
$x = bor($data);This routine handles bad values.
min
Return the minimum of all elements in an ndarray.
See the documentation for "minimum" for more information.
$x = min($data);This routine handles bad values.
max
Return the maximum of all elements in an ndarray.
See the documentation for "maximum" for more information.
$x = max($data);This routine handles bad values.
median
Return the median of all elements in an ndarray.
See the documentation for "medover" for more information.
$x = median($data);This routine handles bad values.
mode
Return the mode of all elements in an ndarray.
See the documentation for "modeover" for more information.
$x = mode($data);This routine handles bad values.
oddmedian
Return the oddmedian of all elements in an ndarray.
See the documentation for "oddmedover" for more information.
$x = oddmedian($data);This routine handles bad values.
any
Return true if any element in ndarray set
Useful in conditional expressions:
if (any $x>15) { print "some values are greater than 15\n" }See "or" for comments on what happens when all elements in the check are bad.
all
Return true if all elements in ndarray set
Useful in conditional expressions:
if (all $x>15) { print "all values are greater than 15\n" }See "and" for comments on what happens when all elements in the check are bad.
minmax
Returns a list with minimum and maximum values of an ndarray.
($mn, $mx) = minmax($pdl);This routine does not broadcast over the dimensions of $pdl; it returns the minimum and maximum values of the whole ndarray. See "minmaximum" if this is not what is required. The two values are returned as Perl scalars, and therefore ignore whether the values are bad.
pdl> $x = pdl [1,-2,3,5,0]
pdl> ($min, $max) = minmax($x);
pdl> p "$min $max\n";
-2 5medover
Signature: (a(n); [o]b(); [t]tmp(n))Project via median to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the median along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = medover($x);$spectrum = medover $image->transposemedover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
oddmedover
Signature: (a(n); [o]b(); [t]tmp(n))Project via oddmedian to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the oddmedian along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = oddmedover($x);$spectrum = oddmedover $image->transposeThe median is sometimes not a good choice as if the array has an even number of elements it lies half-way between the two middle values - thus it does not always correspond to a data value. The lower-odd median is just the lower of these two values and so it ALWAYS sits on an actual data value which is useful in some circumstances.
oddmedover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
modeover
Signature: (data(n); [o]out(); [t]sorted(n))Project via mode to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the mode along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = modeover($x);$spectrum = modeover $image->transposeThe mode is the single element most frequently found in a discrete data set.
It only makes sense for integer data types, since floating-point types are demoted to integer before the mode is calculated.
modeover treats BAD the same as any other value: if BAD is the most common element, the returned value is also BAD.
modeover does not process bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
pctover
Signature: (a(n); p(); [o]b(); [t]tmp(n))Project via specified percentile to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the specified percentile along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = pctover($x);$spectrum = pctover $image->transposeThe specified percentile must be between 0.0 and 1.0. When the specified percentile falls between data points, the result is interpolated. Values outside the allowed range are clipped to 0.0 or 1.0 respectively. The algorithm implemented here is based on the interpolation variant described at http://en.wikipedia.org/wiki/Percentile as used by Microsoft Excel and recommended by NIST.
pctover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
oddpctover
Signature: (a(n); p(); [o]b(); [t]tmp(n))Project via specified percentile to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the specified percentile along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = oddpctover($x);$spectrum = oddpctover $image->transposeThe specified percentile must be between 0.0 and 1.0. When the specified percentile falls between two values, the nearest data value is the result. The algorithm implemented is from the textbook version described first at http://en.wikipedia.org/wiki/Percentile.
oddpctover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
pct
Return the specified percentile of all elements in an ndarray. The specified percentile (p) must be between 0.0 and 1.0. When the specified percentile falls between data points, the result is interpolated.
$x = pct($data, $pct);oddpct
Return the specified percentile of all elements in an ndarray. The specified percentile (p) must be between 0.0 and 1.0. When the specified percentile falls between data points, the nearest data value is the result.
$x = oddpct($data, $pct);qsort
Signature: (a(n); [o]b(n))Quicksort a vector into ascending order.
print qsort random(10);Bad values are moved to the end of the array:
pdl> p $y
[42 47 98 BAD 22 96 74 41 79 76 96 BAD 32 76 25 59 BAD 96 32 BAD]
pdl> p qsort($y)
[22 25 32 32 41 42 47 59 74 76 76 79 96 96 96 98 BAD BAD BAD BAD]qsorti
Signature: (a(n); indx [o]indx(n))Quicksort a vector and return index of elements in ascending order.
$ix = qsorti $x;
print $x->index($ix); # Sorted listBad elements are moved to the end of the array:
pdl> p $y
[42 47 98 BAD 22 96 74 41 79 76 96 BAD 32 76 25 59 BAD 96 32 BAD]
pdl> p $y->index( qsorti($y) )
[22 25 32 32 41 42 47 59 74 76 76 79 96 96 96 98 BAD BAD BAD BAD]qsortvec
Signature: (a(n,m); [o]b(n,m))Sort a list of vectors lexicographically.
The 0th dimension of the source ndarray is dimension in the vector; the 1st dimension is list order. Higher dimensions are broadcasted over.
print qsortvec pdl([[1,2],[0,500],[2,3],[4,2],[3,4],[3,5]]);
[
[ 0 500]
[ 1 2]
[ 2 3]
[ 3 4]
[ 3 5]
[ 4 2]
]Vectors with bad components are moved to the end of the array:
pdl> p $p = pdl("[0 0] [-100 0] [BAD 0] [100 0]")->qsortvec
[
[-100 0]
[ 0 0]
[ 100 0]
[ BAD 0]
]qsortveci
Signature: (a(n,m); indx [o]indx(m))Sort a list of vectors lexicographically, returning the indices of the sorted vectors rather than the sorted list itself.
As with qsortvec, the input PDL should be an NxM array containing M separate N-dimensional vectors. The return value is an integer M-PDL containing the M-indices of original array rows, in sorted order.
As with qsortvec, the zeroth element of the vectors runs slowest in the sorted list.
Additional dimensions are broadcasted over: each plane is sorted separately, so qsortveci may be thought of as a collapse operator of sorts (groan).
Vectors with bad components are moved to the end of the array as for "qsortvec".
magnover
Signature: (a(n); float+ [o]b())Project via Euclidean (aka Pythagorean) distance to N-1 dimensions
This function reduces the dimensionality of an ndarray by one by taking the Euclidean (aka Pythagorean) distance along the 1st dimension.
By using xchg etc. it is possible to use any dimension.
$y = magnover($x);$spectrum = magnover $image->transposeMinimum float precision output. See also "inner" in PDL::Primitive, "norm" in PDL::Primitive.
magnover processes bad values. It will set the bad-value flag of all output ndarrays if the flag is set for any of the input ndarrays.
AUTHOR
Copyright (C) Tuomas J. Lukka 1997 (lukka@husc.harvard.edu). Contributions by Christian Soeller (c.soeller@auckland.ac.nz) and Karl Glazebrook (kgb@aaoepp.aao.gov.au). 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.