@cindex interpolation
@cindex spline

This chapter describes functions for performing interpolation.  The
library provides a variety of interpolation methods, including Cubic
splines and Akima splines.  The interpolation types are interchangeable,
allowing different methods to be used without recompiling.
Interpolations can be defined for both normal and periodic boundary
conditions.  Additional functions are available for computing
derivatives and integrals of interpolating functions.

These interpolation methods produce curves that pass through each
datapoint.  To interpolate noisy data with a smoothing curve see
@ref{Basis Splines}.

The functions described in this section are declared in the header files
@file{gsl_interp.h} and @file{gsl_spline.h}.

@menu
* Introduction to Interpolation::  
* Interpolation Functions::     
* Interpolation Types::         
* Index Look-up and Acceleration::  
* Evaluation of Interpolating Functions::  
* Higher-level Interface::      
* Interpolation Example programs::  
* Interpolation References and Further Reading::  
@end menu

@node Introduction to Interpolation
@section Introduction

Given a set of data points @math{(x_1, y_1) \dots (x_n, y_n)} the
routines described in this section compute a continuous interpolating
function @math{y(x)} such that @math{y(x_i) = y_i}.  The interpolation
is piecewise smooth, and its behavior at the end-points is determined by
the type of interpolation used.

@node Interpolation Functions
@section Interpolation Functions

The interpolation function for a given dataset is stored in a
@code{gsl_interp} object.  These are created by the following functions.

@deftypefun {gsl_interp *} gsl_interp_alloc (const gsl_interp_type * @var{T}, size_t @var{size})
@tpindex gsl_interp
This function returns a pointer to a newly allocated interpolation
object of type @var{T} for @var{size} data-points.
@end deftypefun

@deftypefun int gsl_interp_init (gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], size_t @var{size})
This function initializes the interpolation object @var{interp} for the
data (@var{xa},@var{ya}) where @var{xa} and @var{ya} are arrays of size
@var{size}.  The interpolation object (@code{gsl_interp}) does not save
the data arrays @var{xa} and @var{ya} and only stores the static state
computed from the data.  The @var{xa} data array is always assumed to be
strictly ordered, with increasing @math{x} values; 
the behavior for other arrangements is not defined.
@end deftypefun

@deftypefun void gsl_interp_free (gsl_interp * @var{interp})
This function frees the interpolation object @var{interp}.
@end deftypefun

@node Interpolation Types
@section Interpolation Types
@tpindex gsl_interp_type
The interpolation library provides six interpolation types:

@deffn {Interpolation Type} gsl_interp_linear
@cindex linear interpolation
Linear interpolation.  This interpolation method does not require any
additional memory.
@end deffn

@deffn {Interpolation Type} gsl_interp_polynomial
@cindex polynomial interpolation
Polynomial interpolation.  This method should only be used for
interpolating small numbers of points because polynomial interpolation
introduces large oscillations, even for well-behaved datasets.  The
number of terms in the interpolating polynomial is equal to the number
of points.
@end deffn

@deffn {Interpolation Type} gsl_interp_cspline
@cindex cubic splines
Cubic spline with natural boundary conditions.  The resulting curve is
piecewise cubic on each interval, with matching first and second
derivatives at the supplied data-points.  The second derivative is
chosen to be zero at the first point and last point.
@end deffn

@deffn {Interpolation Type} gsl_interp_cspline_periodic
Cubic spline with periodic boundary conditions.  The resulting curve
is piecewise cubic on each interval, with matching first and second
derivatives at the supplied data-points.  The derivatives at the first
and last points are also matched.  Note that the last point in the
data must have the same y-value as the first point, otherwise the
resulting periodic interpolation will have a discontinuity at the
boundary.

@end deffn

@deffn {Interpolation Type} gsl_interp_akima
@cindex Akima splines
Non-rounded Akima spline with natural boundary conditions.  This method
uses the non-rounded corner algorithm of Wodicka.
@end deffn

@deffn {Interpolation Type} gsl_interp_akima_periodic
Non-rounded Akima spline with periodic boundary conditions.  This method
uses the non-rounded corner algorithm of Wodicka.
@end deffn

The following related functions are available:

@deftypefun {const char *} gsl_interp_name (const gsl_interp * @var{interp})
This function returns the name of the interpolation type used by @var{interp}.
For example,

@example
printf ("interp uses '%s' interpolation.\n", 
        gsl_interp_name (interp));
@end example

@noindent
would print something like,

@example
interp uses 'cspline' interpolation.
@end example
@end deftypefun

@deftypefun {unsigned int} gsl_interp_min_size (const gsl_interp * @var{interp})
This function returns the minimum number of points required by the
interpolation type of @var{interp}.  For example, Akima spline interpolation
requires a minimum of 5 points.
@end deftypefun

@node Index Look-up and Acceleration
@section Index Look-up and Acceleration

The state of searches can be stored in a @code{gsl_interp_accel} object,
which is a kind of iterator for interpolation lookups.  It caches the
previous value of an index lookup.  When the subsequent interpolation
point falls in the same interval its index value can be returned
immediately.

@deftypefun size_t gsl_interp_bsearch (const double @var{x_array}[], double @var{x}, size_t @var{index_lo}, size_t @var{index_hi})
This function returns the index @math{i} of the array @var{x_array} such
that @code{x_array[i] <= x < x_array[i+1]}.  The index is searched for
in the range [@var{index_lo},@var{index_hi}].  @inlinefn{}
@end deftypefun

@deftypefun {gsl_interp_accel *} gsl_interp_accel_alloc (void)
@tpindex gsl_interp_accel
This function returns a pointer to an accelerator object, which is a
kind of iterator for interpolation lookups.  It tracks the state of
lookups, thus allowing for application of various acceleration
strategies.
@end deftypefun

@deftypefun size_t gsl_interp_accel_find (gsl_interp_accel * @var{a}, const double @var{x_array}[], size_t @var{size}, double @var{x})
This function performs a lookup action on the data array @var{x_array}
of size @var{size}, using the given accelerator @var{a}.  This is how
lookups are performed during evaluation of an interpolation.  The
function returns an index @math{i} such that @code{x_array[i] <= x <
x_array[i+1]}.  @inlinefn{}
@end deftypefun

@deftypefun int gsl_interp_accel_reset (gsl_interp_accel * @var{acc});
This function reinitializes the accelerator object @var{acc}.  It
should be used when the cached information is no longer
applicable---for example, when switching to a new dataset.
@end deftypefun

@deftypefun void gsl_interp_accel_free (gsl_interp_accel* @var{acc})
This function frees the accelerator object @var{acc}.
@end deftypefun

@node Evaluation of Interpolating Functions
@section Evaluation of Interpolating Functions

@deftypefun  double gsl_interp_eval (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_interp_eval_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc}, double * @var{y})
These functions return the interpolated value of @var{y} for a given
point @var{x}, using the interpolation object @var{interp}, data arrays
@var{xa} and @var{ya} and the accelerator @var{acc}.
@end deftypefun

@deftypefun double gsl_interp_eval_deriv (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_interp_eval_deriv_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d})
These functions return the derivative @var{d} of an interpolated
function for a given point @var{x}, using the interpolation object
@var{interp}, data arrays @var{xa} and @var{ya} and the accelerator
@var{acc}. 
@end deftypefun

@deftypefun double gsl_interp_eval_deriv2 (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_interp_eval_deriv2_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d2})
These functions return the second derivative @var{d2} of an interpolated
function for a given point @var{x}, using the interpolation object
@var{interp}, data arrays @var{xa} and @var{ya} and the accelerator
@var{acc}. 
@end deftypefun

@deftypefun double gsl_interp_eval_integ (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{a}, double @var{b}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_interp_eval_integ_e (const gsl_interp * @var{interp}, const double @var{xa}[], const double @var{ya}[], double @var{a}, double @var{b}, gsl_interp_accel * @var{acc}, double * @var{result})
These functions return the numerical integral @var{result} of an
interpolated function over the range [@var{a}, @var{b}], using the
interpolation object @var{interp}, data arrays @var{xa} and @var{ya} and
the accelerator @var{acc}.
@end deftypefun

@node Higher-level Interface
@section Higher-level Interface

The functions described in the previous sections required the user to
supply pointers to the @math{x} and @math{y} arrays on each call.  The
following functions are equivalent to the corresponding
@code{gsl_interp} functions but maintain a copy of this data in the
@code{gsl_spline} object.  This removes the need to pass both @var{xa}
and @var{ya} as arguments on each evaluation. These functions are
defined in the header file @file{gsl_spline.h}.

@deftypefun {gsl_spline *} gsl_spline_alloc (const gsl_interp_type * @var{T}, size_t @var{size})
@tpindex gsl_spline
@end deftypefun

@deftypefun int gsl_spline_init (gsl_spline * @var{spline}, const double @var{xa}[], const double @var{ya}[], size_t @var{size})
@end deftypefun

@deftypefun void gsl_spline_free (gsl_spline * @var{spline})
@end deftypefun

@deftypefun {const char *} gsl_spline_name (const gsl_spline * @var{spline})
@end deftypefun

@deftypefun {unsigned int} gsl_spline_min_size (const gsl_spline * @var{spline})
@end deftypefun

@deftypefun double gsl_spline_eval (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_spline_eval_e (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc}, double * @var{y})
@end deftypefun

@deftypefun double gsl_spline_eval_deriv (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_spline_eval_deriv_e (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d})
@end deftypefun

@deftypefun double gsl_spline_eval_deriv2 (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_spline_eval_deriv2_e (const gsl_spline * @var{spline}, double @var{x}, gsl_interp_accel * @var{acc}, double * @var{d2})
@end deftypefun

@deftypefun double gsl_spline_eval_integ (const gsl_spline * @var{spline}, double @var{a}, double @var{b}, gsl_interp_accel * @var{acc})
@deftypefunx int gsl_spline_eval_integ_e (const gsl_spline * @var{spline}, double @var{a}, double @var{b}, gsl_interp_accel * @var{acc}, double * @var{result})
@end deftypefun

@node Interpolation Example programs
@section Examples

The following program demonstrates the use of the interpolation and
spline functions.  It computes a cubic spline interpolation of the
10-point dataset @math{(x_i, y_i)} where @math{x_i = i + \sin(i)/2} and
@math{y_i = i + \cos(i^2)} for @math{i = 0 \dots 9}.

@example
@verbatiminclude examples/interp.c
@end example

@noindent
The output is designed to be used with the @sc{gnu} plotutils
@code{graph} program,

@example
$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps
@end example

@iftex
@sp 1
@center @image{interp2,3.4in}
@end iftex

@noindent
The result shows a smooth interpolation of the original points.  The
interpolation method can be changed simply by varying the first argument of
@code{gsl_spline_alloc}.

The next program demonstrates a periodic cubic spline with 4 data
points.  Note that the first and last points must be supplied with 
the same y-value for a periodic spline.

@example
@verbatiminclude examples/interpp.c
@end example

@noindent

The output can be plotted with @sc{gnu} @code{graph}.

@example
$ ./a.out > interp.dat
$ graph -T ps < interp.dat > interp.ps
@end example

@iftex
@sp 1
@center @image{interpp2,3.4in}
@end iftex

@noindent
The result shows a periodic interpolation of the original points. The
slope of the fitted curve is the same at the beginning and end of the
data, and the second derivative is also.

@node Interpolation References and Further Reading
@section References and Further Reading

Descriptions of the interpolation algorithms and further references can
be found in the following books:

@itemize @asis
@item C.W. Ueberhuber,
@cite{Numerical Computation (Volume 1), Chapter 9 ``Interpolation''},
Springer (1997), ISBN 3-540-62058-3.

@item D.M. Young, R.T. Gregory
@cite{A Survey of Numerical Mathematics (Volume 1), Chapter 6.8},
Dover (1988), ISBN 0-486-65691-8.
@end itemize

@noindent
