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1.4.4 Splines
Starting from an approximation on an inhomogeneous mesh, the idea of
splines is to provide a global interpolation that is continuous up to
a certain derivative.
For example, take a cubic Hermite spline with the 4 parameters
, which
completely determine all 4 coefficients of the cubic polynomial

(1.4.4#eq.1) 
It is straight forward to calculate the first and second derives

(1.4.4#eq.2) 

(1.4.4#eq.3) 
Starting with only sampled values and no information about the curvature,
are usually calculated so as to guarantee a smooth
interpolation. Most common is to require that
be continuous
from one interval to the next, leading to a tridiagonal system that holds
for

(1.4.4#eq.4) 
and can be solved efficiently with
operations.
Two free parameters at the boundaries are used to impose the
boundary conditions by choosing
and
(1.4.4#eq.2).
Figure:
Approximation of
with cubic Hermite splines.

Figure (1.4.4#fig.1) illustrates the procedure and shows the excellent
quality of a cubic approximation until it breaks down at the limit of 2
mesh points per wavelength.
SYLLABUS Previous: 1.4.3 Finite elements
Up: 1.4 Numerical discretization
Next: 1.4.5 Harmonic functions
