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<title>Overview of B-splines - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">38.1 Overview</h3>
<p><a name="index-basis-splines_002c-overview-2425"></a>
B-splines are commonly used as basis functions to fit smoothing
curves to large data sets. To do this, the abscissa axis is
broken up into some number of intervals, where the endpoints
of each interval are called <dfn>breakpoints</dfn>. These breakpoints
are then converted to <dfn>knots</dfn> by imposing various continuity
and smoothness conditions at each interface. Given a nondecreasing
knot vector
<!-- {$t = \{t_0, t_1, \dots, t_{n+k-1}\}$} -->
t = {t_0, t_1, <small class="dots">...</small>, t_{n+k-1}},
the n basis splines of order k are defined by
<pre class="example"> B_(i,1)(x) = (1, t_i <= x < t_(i+1)
(0, else
B_(i,k)(x) = [(x - t_i)/(t_(i+k-1) - t_i)] B_(i,k-1)(x) + [(t_(i+k) - x)/(t_(i+k) - t_(i+1))] B_(i+1,k-1)(x)
</pre>
<p>for i = 0, <small class="dots">...</small>, n-1. The common case of cubic B-splines
is given by k = 4. The above recurrence relation can be
evaluated in a numerically stable way by the de Boor algorithm.
<p>If we define appropriate knots on an interval [a,b] then
the B-spline basis functions form a complete set on that interval.
Therefore we can expand a smoothing function as
<pre class="example"> f(x) = \sum_i c_i B_(i,k)(x)
</pre>
<p>given enough (x_j, f(x_j)) data pairs. The c_i can
be readily obtained from a least-squares fit.
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