lsq_splin
=========

weighted least squares cubic spline fitting



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


::

    [y, d] = lsq_splin(xd, yd [, wd], x)




Arguments
~~~~~~~~~

:xd, yd vectors of the same size, data to be fitted by a cubic spline
: :wd (optional) a vector of same format than `xd` and `yd`, weights
  of the least square fit.
: :x a strictly increasing (row or column) vector, breakpoints of the
  cubic spline
: :y, d vectors of same format than `x`, the triplet (x,y,d) defines
  the approximated cubic spline.
:



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

This function computes an approximated cubic spline *s* for the data
`xd, yd, wd` (in the following *m* is supposed to be the length of
these vectors) and from a choice of the spline breakpoints `x` (for
instance if you want n breakpoints uniformly chosen you may use
`x=linspace(min(xd),max(xd),n))`). If *S* is the space of all cubic
splines functions with breakpoints *x1 < x2 < ... < xn* then the
resulting spline *s* is such that:

for all *fin S*, i.e. realizes the minimum of the sum of the squared
errors over all functions of *S*.

The spline *s* is completely defined by the triplet `(x,y,d)` ( `y`
and `d` are the vectors of the spline ordinates and first derivatives
at the xi 's : *yi=s(xi)* and *di=s'(xi)*) and its evaluation at some
points must be done by the `interp`_ function.



Remarks
~~~~~~~

When `wd` is not given, all the points have the same weight 1.

A point *(xd(k),yd(k))* is considered in the fit if *xd(k) in [x1,xn]
and wd(k) > 0*. In particular you can put a null (or even negative)
weight to all data points you want to ignore in the fitting. When the
total number of points taken into account in the fit procedure is
(strictly) less than 4 an error is issued.

The vector `xd` do not need to be in increasing order.

Depending on the number and on the positions of the *xd(k)* 's and on
the choice of the *x(i)* 's there may be several solutions but only
one is selected. When this occurs a warning message is displayed in
the Scilab command window. This function is intended to be used when
*m* is much larger than *n* and in this case no such problem may
occurred.



Examples
~~~~~~~~


::

    // this is an artificial example where the data xd and yd
    // are build from a perturbed sin function
    a = 0; b = 2*%pi;
    sigma = 0.1;  // standard deviation of the gaussian noise
    m = 200;       // number of experimental points
    xd = `linspace`_(a,b,m)';
    yd = `sin`_(xd) + `grand`_(xd,"nor",0,sigma);
    
    n = 6; // number of breakpoints
    x = `linspace`_(a,b,n)';
    
    // compute the spline
    [y, d] = lsq_splin(xd, yd, x);  // use equal weights
    
    // plotting
    ye = `sin`_(xd);
    ys = `interp`_(xd, x, y, d);
    `clf`_()
    `plot2d`_(xd,[ye yd ys],style=[2 -2 3], ...
           leg="exact function@experimental measures (gaussian perturbation)@fitted spline")
    `xtitle`_("a least square spline")
    `show_window`_()




See Also
~~~~~~~~


+ `interp`_ cubic spline evaluation function
+ `splin`_ cubic spline interpolation


.. _interp: interp.html
.. _splin: splin.html