PDL::Func - useful functions

 use PDL::Func;
 use PDL::Math;

 # somewhat pointless way to estimate cos and sin,
 # but is shows that you can thread if you want to
 # (and the library lets you)
 #
 my $obj = PDL::Func->init( Interpolate => "Hermite" );
 # 
 my $x = pdl( 0 .. 45 ) * 4 * 3.14159 / 180;
 my $y = cat( sin($x), cos($x) );
 $obj->set( x => $x, y => $y, bc => "simple" );
 #
 my $xi = pdl( 0.5, 1.5, 2.5 );
 my $yi = $obj->interpolate( $xi );
 #
 print "sin( $xi ) equals ", $yi->slice(':,(0)'), "\n";
 sin( [0.5 1.5 2.5] ) equals  [0.87759844 0.070737667 -0.80115622]
 #
 print "cos( $xi ) equals ", $yi->slice(':,(1)'), "\n";
 cos( [0.5 1.5 2.5] ) equals  [ 0.4794191 0.99768655 0.59846449]
 #
 print sin($xi), "\n", cos($xi), "\n";
 [0.47942554 0.99749499 0.59847214]
 [0.87758256 0.070737202 -0.80114362]

This module aims to contain useful functions. Honest.

This module aims to provide a relatively-uniform interface to the various interpolation methods available to PDL. The idea is that a different interpolation scheme can be used just by changing an attribute of a CWPDL::Func object. Some interpolation schemes (as exemplified by the SLATEC library) also provide additional functionality, such as integration and gradient estimation.

Throughout this documentation, CW$x and CW$y refer to the function to be interpolated whilst CW$xi and CW$yi are the interpolated values.

The avaliable types, or schemes, of interpolation are listed below. Also given are the valid attributes for each scheme: the flag value indicates whether it can be set (s), got (g), and if it is required (r) for the method to work.

Interpolate => Linear

An extravagent way of calling the linear interpolation routine PDL::Primitive::interpolate. The valid attributes are:

 Attribute    Flag  Description
 x            sgr   x positions of data
 y            sgr   function values at x positions
 err          g     error flag

Interpolate => Hermite

Use the piecewice cubic Hermite interpolation routines from the SLATEC library. Only available if PDL::Slatec is installed. The valid attributes are:

 Attribute    Flag  Description
 x            sgr   x positions of data
 y            sgr   function values at x positions
 bc           sgr   boundary conditions
 g            g     estimated gradient at x positions
 err          g     error flag

Given the initial set of points CW(x,y), an estimate of the gradient is made at these points, using the given boundary conditions. The gradients are stored in the CWg attribute, accessible via:

 $gradient = $obj->get( 'g' );

However, as this gradient is only calculated 'at the last moment', CWg will only contain data after one of CWinterpolate, CWgradient, or CWintegrate is used.

If your data is monotonic, and you are not too bothered about edge effects, then the default value of CWbc of CWsimple is for you. Otherwise, take a look at the description of PDL::Slatec::chic and use a hash reference for the CWbc attribute, with the following keys:

monotonic

0 if the interpolant is to be monotonic in each interval (so the gradient will be 0 at each switch point), otherwise the gradient is calculated using a 3-point difference formula at switch points. If > 0 then the interpolant is forced to lie close to the data, if < 0 no such control is imposed. Default = 0.

start

A perl list of one or two elements. The first element defines how the boundary condition for the start of the array is to be calculated; it has a range of CW-5 .. 5, as given for the CWic parameter of chic. The second element, only used if options 2, 1, -1, or 2 are chosen, contains the value of the CWvc parameter. Default = [ 0 ].

end

As for CWstart, but for the end of the data.

An example would be

 $obj->set( bc => { start => [ 1, 0 ], end => [ 1, -1 ] } )

which sets the first derivative at the first point to 0, and at the last point to -1.

The CWstatus method provides a simple mechanism to check if the previous method was successful. If the function returns an error flag, then it is stored in the CWerr attribute. To find out which routine was used, use the CWroutine method.

 $obj = PDL::Func->init( Interpolate => "Hermite", x => $x, y => $y );
 $obj = PDL::Func->init( { x => $x, y => $y } );

Create a PDL::Func object, which can interpolate, and possibly integrate and calculate gradients of a dataset.

If not specified, the value of Interpolate is taken to be CWLinear, which means the interpolation is performed by PDL::Primitive::interpolate. A value of CWHermite uses piecewise cubic Hermite functions, which also allows the integral and gradient of the data to be estimated.

Options can either be provided directly to the method, as in the first example, or within a hash reference, as shown in the second example.

 my $nset = $obj->set( x = $newx, $y => $newy );
 my $nset = $obj->set( { x = $newx, $y => $newy } );

Set attributes for a PDL::Func object.

The return value gives the number of the supplied attributes which were actually set.

 my $x         = $obj->get( x );
 my ( $x, $y ) = $obj->get( qw( x y ) );

Get attributes from a PDL::Func object.

Given a list of attribute names, return a list of their values; in scalar mode return a scalar value. If the supplied list contains an unknown attribute, CWget returns a value of CWundef for that attribute.

 my $scheme = $obj->scheme;

Return the type of interpolation of a PDL::Func object.

Returns either CWLinear or CWHermite.

 my $status = $obj->status;

Returns the status of a PDL::Func object.

This method provides a high-level indication of the success of the last method called (except for CWget which is ignored). Returns 1 if everything is okay, 0 if there has been a serious error, and -1 if there was a problem which was not serious. In the latter case, CW$obj->get("err") may provide more information, depending on the particular scheme in use.

 my $name = $obj->routine;

Returns the name of the last routine called by a PDL::Func object.

This is mainly useful for decoding the value stored in the CWerr attribute.

 $obj->attributes;
 PDL::Func->attributes;

Print out the flags for the attributes of a PDL::Func object.

Useful in case the documentation is just too opaque!

 PDL::Func->attributes;
 Flags  Attribute
  SGR    x
  SGR    y
  G      err

 my $yi = $obj->interpolate( $xi );

Returns the interpolated function at a given set of points (PDL::Func).

A status value of -1, as returned by the CWstatus method, means that some of the CW$xi points lay outside the range of the data. The values for these points were calculated by extrapolation (the details depend on the scheme being used).

 my $gi          = $obj->gradient( $xi );
 my ( $yi, $gi ) = $obj->gradient( $xi );

Returns the derivative and, optionally, the interpolated function for the CWHermite scheme (PDL::Func).

 my $ans = $obj->integrate( index => pdl( 2, 5 ) );
 my $ans = $obj->integrate( x => pdl( 2.3, 4.5 ) );

Integrate the function stored in the PDL::Func object, if the scheme is CWHermite.

The integration can either be between points of the original CWx array (CWindex), or arbitrary x values (CWx). For both cases, a two element piddle should be given, to specify the start and end points of the integration.

index

The values given refer to the indices of the points in the CWx array.

x

The array contains the actual values to integrate between.

If the CWstatus method returns a value of -1, then one or both of the integration limits did not lie inside the CWx array. Caveat emptor with the result in such a case.

It should be relatively easy to provide an interface to other interpolation routines, such as those provided by the Gnu Scientific Library (GSL), or the B-spline routines in the SLATEC library.

In the documentation, the methods are preceeded by CWPDL::Func:: to avoid clashes with functions such as CWset when using the CWhelp or CWapropos commands within perldl.

Amalgamated CWPDL::Interpolate and CWPDL::Interpolate::Slatec to form CWPDL::Func. Comments greatly appreciated on the current implementation, as it is not too sensible.

Thanks to Robin Williams, Halldór Olafsson, and Vince McIntyre.

Robin is working on a new version, that improves on the current version a lot. No time scale though!

Copyright (C) 2000,2001 Doug Burke (dburke@cfa.harvard.edu). All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution.