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+@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.
+
+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})
+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; 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
+
+The interpolation library provides five 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}].
+@end deftypefun
+
+@deftypefun {gsl_interp_accel *} gsl_interp_accel_alloc (void)
+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]}.
+@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})
+@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 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