Theorem

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 + ...a_lx^l and A is a linear combinaison of 1, x,...x^l.
On [x₁, x₂], s is a polynomial B of degree less or equal to l, hence on [x₁, x₂], s = B(x) = b₀ + b₁x + ...b_lx^l.
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 + ...a_lx^l + $\displaystyle \alpha_{1}^{}$ 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 + ...c_lx^l.
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 + ...a_lx^l + $\alpha_{1}^{}$ q₁(x) + $\alpha_{2}^{}$ q₂(x)
And so on, the functions are defined by :

$\displaystyle \forall$ 1 $\displaystyle \leq$ j $\displaystyle \leq$ n - 1,q_j(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 + ...a_lx^l + $\displaystyle \alpha_{1}^{}$ q₁(x) + .... + $\displaystyle \alpha_{{n-1}}^{}$ q_n-1(x)

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

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