# $Id: Spline.pm,v 1.1 1995/12/26 17:28:17 willijar Exp $

=head1 NAME

    Math::Spline  - Cubic Spline Interpolation of data

=head1 SYNOPSIS

    require Math::Spline;
    $spline=new Math::Spline(\@x,\@y)
    $y_interp=$spline->evaluate($x);

    use Math::Spline qw(spline linsearch binsearch);
    use Math::Derivative qw(Derivative2);
    @y2=Derivative2(\@x,\@y);
    $index=binsearch(\@x,$x);
    $index=linsearch(\@x,$x,$index);
    $y_interp=spline(\@x,\@y,\@y2,$index,$x);

=head1 DESCRIPTION

This package provides cubic spline interpolation of numeric data. The
data is passed as references to two arrays containing the x and y
ordinates. It may be used as an exporter of the numerical functions
or, more easily as a class module.

The B<Math::Spline> class constructor B<new> takes references to the
arrays of x and y ordinates of the data. An interpolation is performed
using the B<evaluate> method, which, when given an x ordinate returns
the interpolate y ordinate at that value.

The B<spline> function takes as arguments references to the x and y
ordinate array, a reference to the 2nd derivatives (calculated using
B<Derivative2>, the low index of the interval in which to interpolate
and the x ordinate in that interval. Returned is the interpolated y
ordinate. Two functions are provided to look up the appropriate index
in the array of x data. For random calls B<binsearch> can be used -
give a reference to the x ordinates and the x loopup value it returns
the low index of the interval in the data in which the value
lies. Where the lookups are strictly in ascending sequence (e.g. if
interpolating to produce a higher resolution data set to draw a curve)
the B<linsearch> function may more efficiently be used. It performs
like B<binsearch>, but requires a third argument being the previous
index value, which is incremented if necessary.

=head1 NOTE

requires Math::Derivative module

=head1 EXAMPLE

    require Math::Spline;
    my @x=(1,3,8,10);
    my @y=(1,2,3,4);
    $spline=new Math::Spline(\@x,\@y);
    print $spline->evaluate(5)."\n";

produces the output

2.44

=head1 HISTORY

$Log: Spline.pm,v $
Revision 1.1  1995/12/26 17:28:17  willijar
Initial revision


=head1 BUGS

Bug reports or constructive comments are welcome.

=head1 AUTHOR

John A.R. Williams <J.A.R.Williams@aston.ac.uk>

=head1 SEE ALSO

"Numerical Recipies: The Art of Scientific Computing"
W.H. Press, B.P. Flannery, S.A. Teukolsky, W.T. Vetterling.
Cambridge University Press. ISBN 0 521 30811 9.

=cut

require Exporter;
package Math::Spline;
@ISA=qw(Exporter);
@EXPORT_OK=qw(linsearch binsearch spline);
use Carp;
use Math::Derivative qw(Derivative2);
use strict;

sub new {
  my $type=shift;
  my $self=[];
  push @{$self},shift; # x
  push @{$self},shift; # y
  my $y2=[Derivative2($self->[0],$self->[1])];
  push @{$self},$y2;
  bless $self,$type;
}

sub evaluate {
  my ($self,$v)=@_;
  my $idx=binsearch($self->[0],$v);
  spline($self->[0],$self->[1],$self->[2],$idx,$v);
}

sub spline { 
  my ($x,$y,$y2,$i,$v)=@_;
  my ($klo,$khi)=($i,$i+1);
  my $h=$x->[$khi]-$x->[$klo];
  if ($h==0) { croak "Zero interval in spline data.\n"; }
  my $a=($x->[$khi]-$v)/$h;
  my $b=($v-$x->[$klo])/$h;
  return $a*$y->[$klo] + $b*$y->[$khi]
      +(($a*$a*$a-$a)*$y2->[$klo]
	+($b*$b*$b-$b)*$y2->[$khi])*($h*$h)/6.0;
}

sub binsearch { # binary search routine finds index just below value
  my ($x,$v)=@_;
  my ($klo,$khi)=(0,$#{$x});
  my $k;
  while (($khi-$klo)>1) {
    $k=int(($khi+$klo)/2);
    if ($x->[$k]>$v) { $khi=$k; } else { $klo=$k; }
  }
  return $klo;
}

sub linsearch { # more efficient if repetatively doint it
  my ($x,$v,$khi)=@_; $khi+=1;
  my $n=$#{$x};
  while($v>$x->[$khi] and $khi<$n) { $khi++; }
  $_[2]=$khi-1;
}

1;