next up previous contents index
suivant: Interpolation with spline functions monter: Natural splines: spline précédent: Definition   Table des matières   Index

Theorem

The set of spline functions of degree l on $ \sigma_{n}^{}$ is a $ \mathbb {R}$-vectorial subspace of dimension n + l.

Proof
On [a, x1], s is a polynomial A of degree less or equal to l, hence on [a, x1], s = A(x) = a0 + a1x + ...alxl and A is a linear combinaison of 1, x,...xl.
On [x1, x2], s is a polynomial B of degree less or equal to l, hence on [x1, x2], s = B(x) = b0 + b1x + ...blxl.
s has continuous derivatives up to order l - 1, hence :

$\displaystyle \forall$0 $\displaystyle \leq$ j $\displaystyle \leq$ l - 1,    B(j)(x1) - A(j)(x1) = 0

therefore B(x) - A(x) = $ \alpha_{1}^{}$(x - x1)l or B(x) = A(x) + $ \alpha_{1}^{}$(x - x1)l.
Define the function :

q1(x) = $\displaystyle \left\{\vphantom{
\begin{array}{rcl}
0 & \mbox{sur} & [a,x_1] \\
(x-x_1)^l & \mbox{sur} & [x_1,b]\\
\end{array}
}\right.$$\displaystyle \begin{array}{rcl}
0 & \mbox{sur} & [a,x_1] \\
(x-x_1)^l & \mbox{sur} & [x_1,b]\\
\end{array}$

Hence :

s|[a, x2] = a0 + a1x + ...alxl + $\displaystyle \alpha_{1}^{}$q1(x)

On [x2, x3], s is a polynomial C of degree less or equal than l, hence on [x2, x3], s = C(x) = c0 + c1x + ...clxl.
s has continuous derivatives until l - 1, hence :

$\displaystyle \forall$0 $\displaystyle \leq$ j $\displaystyle \leq$ l - 1,    C(j)(x2) - B(j)(x2) = 0

therefore C(x) - B(x) = $ \alpha_{2}^{}$(x - x2)l or C(x) = B(x) + $ \alpha_{2}^{}$(x - x2)l.
Define the function :

q2(x) = $\displaystyle \left\{\vphantom{
\begin{array}{rcl}
0 & \mbox{on} & [a,x_2] \\
(x-x_2)^l & \mbox{on} & [x_2,b]\\
\end{array}
}\right.$$\displaystyle \begin{array}{rcl}
0 & \mbox{on} & [a,x_2] \\
(x-x_2)^l & \mbox{on} & [x_2,b]\\
\end{array}$

Hence : s|[a, x3] = a0 + a1x + ...alxl + $ \alpha_{1}^{}$q1(x) + $ \alpha_{2}^{}$q2(x)
And so on, the functions are defined by :

$\displaystyle \forall$1 $\displaystyle \leq$ j $\displaystyle \leq$ n - 1,qj(x) = $\displaystyle \left\{\vphantom{
\begin{array}{rcl}
0 & \mbox{on} & [a,x_j] \\
(x-x_j)^l & \mbox{on} & [x_j,b]\\
\end{array}
}\right.$$\displaystyle \begin{array}{rcl}
0 & \mbox{on} & [a,x_j] \\
(x-x_j)^l & \mbox{on} & [x_j,b]\\
\end{array}$

hence,

s|[a, b] = a0 + a1x + ...alxl + $\displaystyle \alpha_{1}^{}$q1(x) + .... + $\displaystyle \alpha_{{n-1}}^{}$qn-1(x)

and s is a linear combination of n + l independant functions 1, x,..xl, q1,..qn-1.


next up previous contents index
suivant: Interpolation with spline functions monter: Natural splines: spline précédent: Definition   Table des matières   Index
giac documentation written by Renée De Graeve and Bernard Parisse