interp
======

cubic spline evaluation function



Calling Sequence
~~~~~~~~~~~~~~~~


::

    [yp [,yp1 [,yp2 [,yp3]]]]=interp(xp, x, y, d [, out_mode])




Arguments
~~~~~~~~~

:xp real vector or matrix
: :x,y,d real vectors of the same size defining a cubic spline or sub-
  spline function (called `s` in the following)
: :out_mode (optional) string defining the evaluation of `s` outside
  the [x1,xn] interval
: :yp vector or matrix of same size than `xp`, elementwise evaluation
  of `s` on `xp` (yp(i)=s(xp(i) or yp(i,j)=s(xp(i,j))
: :yp1, yp2, yp3 vectors (or matrices) of same size than `xp`,
  elementwise evaluation of the successive derivatives of `s` on `xp`
:



Description
~~~~~~~~~~~

Given three vectors `(x,y,d)` defining a spline or sub-spline function
(see `splin`_) with `yi=s(xi), di = s'(xi)` this function evaluates
`s` (and `s', s'', s'''` if needed) at `xp(i)` :

The `out_mode` parameter set the evaluation rule for extrapolation,
i.e. for `xp(i)` not in [x1,xn] :

:"by_zero" an extrapolation by zero is done
: :"by_nan" extrapolation by Nan
: :"C0" the extrapolation is defined as follows :
: :"natural" the extrapolation is defined as follows ( *p_i* being the
  polynomial defining `s` on [x_i,x_{i+1}]) :
: :"linear" the extrapolation is defined as follows :
: :"periodic" `s` is extended by periodicity.
:



Examples
~~~~~~~~


::

    // see the examples of splin and lsq_splin
    
    // an example showing C2 and C1 continuity of spline and subspline
    a = -8; b = 8;
    x = `linspace`_(a,b,20)';
    y = `sinc`_(x);
    dk = `splin`_(x,y);  // not_a_knot
    df = `splin`_(x,y, "fast");
    xx = `linspace`_(a,b,800)';
    [yyk, yy1k, yy2k] = interp(xx, x, y, dk); 
    [yyf, yy1f, yy2f] = interp(xx, x, y, df); 
    `clf`_()
    `subplot`_(3,1,1)
    `plot2d`_(xx, [yyk yyf])
    `plot2d`_(x, y, style=-9)
    `legends`_(["not_a_knot spline","fast sub-spline","interpolation points"],...
            [1 2 -9], "ur",%f)
    `xtitle`_("spline interpolation")
    `subplot`_(3,1,2)
    `plot2d`_(xx, [yy1k yy1f])
    `legends`_(["not_a_knot spline","fast sub-spline"], [1 2], "ur",%f)
    `xtitle`_("spline interpolation (derivatives)")
    `subplot`_(3,1,3)
    `plot2d`_(xx, [yy2k yy2f])
    `legends`_(["not_a_knot spline","fast sub-spline"], [1 2], "lr",%f)
    `xtitle`_("spline interpolation (second derivatives)")
    
    // here is an example showing the different extrapolation possibilities
    x = `linspace`_(0,1,11)';
    y = `cosh`_(x-0.5);
    d = `splin`_(x,y);
    xx = `linspace`_(-0.5,1.5,401)';
    yy0 = interp(xx,x,y,d,"C0");
    yy1 = interp(xx,x,y,d,"linear");
    yy2 = interp(xx,x,y,d,"natural");
    yy3 = interp(xx,x,y,d,"periodic");  
    `clf`_()
    `plot2d`_(xx,[yy0 yy1 yy2 yy3],style=2:5,frameflag=2,leg="C0@linear@natural@periodic")
    `xtitle`_(" different way to evaluate a spline outside its domain")




See Also
~~~~~~~~


+ `splin`_ cubic spline interpolation
+ `lsq_splin`_ weighted least squares cubic spline fitting




History
~~~~~~~
Version Description 5.4.0 previously, imaginary part of input
arguments were implicitly ignored.
.. _splin: splin.html
.. _lsq_splin: lsq_splin.html