man PDL::Func (3): interpolation, integration, & gradient estimation (differentiation) of functions

SYNOPSIS

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]

DESCRIPTION

This module aims to contain useful functions. Honest.

INTERPOLATION AND MORE

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 "PDL::Func" object. Some interpolation schemes (as exemplified by the SLATEC library) also provide additional functionality, such as integration and gradient estimation.

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

The available 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 "(x,y)", an estimate of the gradient is made at these points, using the given boundary conditions. The gradients are stored in the "g" attribute, accessible via:

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

However, as this gradient is only calculated 'at the last moment', "g" will only contain data after one of "interpolate", "gradient", or "integrate" is used.

Boundary conditions for the Hermite routines

If your data is monotonic, and you are not too bothered about edge effects, then the default value of "bc" of "simple" is for you. Otherwise, take a look at the description of PDL::Slatec::chic and use a hash reference for the "bc" 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 "-5 .. 5", as given for the "ic" parameter of chic. The second element, only used if options 2, 1, -1, or 2 are chosen, contains the value of the "vc" parameter. Default = [ 0 ].
end: As for "start", 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.

Errors

The "status" 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 "err" attribute. To find out which routine was used, use the "routine" method.

FUNCTIONS

init

 $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 "Linear", which means the interpolation is performed by PDL::Primitive::interpolate. A value of "Hermite" 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.

set

 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.

get

 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, "get" returns a value of "undef" for that attribute.

scheme

 my $scheme = $obj->scheme;

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

Returns either "Linear" or "Hermite".

status

 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 "get" 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, "$obj->get("err")" may provide more information, depending on the particular scheme in use.

routine

 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 "err" attribute.

attributes

 $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

interpolate

 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 "status" method, means that some of the $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).

gradient

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

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

integrate

 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 "Hermite".

The integration can either be between points of the original "x" array ("index"), or arbitrary x values ("x"). 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 "x" array.
x: The array contains the actual values to integrate between.

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

TODO

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 preceded by "PDL::Func::" to avoid clashes with functions such as "set" when using the "help" or "apropos" commands within perldl or pdl2.

HISTORY

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

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

AUTHOR

Copyright (C) 2000,2001 Doug Burke ([email protected]). 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.