Provided by: pdl_2.007-5_amd64

#### NAME

```       PDL::Func - 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.

```

#### INTERPOLATIONANDMORE

```       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

Throughout this documentation, \$x and \$y refer to the function to be interpolated whilst
\$xi and \$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 "(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:

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

```   PDL::Func::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

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.

PDL::Func::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.

PDL::Func::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.

PDL::Func::scheme
my \$scheme = \$obj->scheme;

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

Returns either "Linear" or "Hermite".

PDL::Func::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,
use.

PDL::Func::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.

PDL::Func::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

PDL::Func::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).

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).

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 preceeded 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, HalldÃ³r Olafsson, and Vince McIntyre.

```

#### AUTHOR

```       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.
```