Editing Spline interpolation (section)

==Example==
[[Image:Cubic splines three points.svg|frame|right|Interpolation with cubic "natural" splines between three points]]
In case of three points the values for <math>k_0, k_1, k_2</math> are found by solving the [[Tridiagonal matrix|tridiagonal linear equation system]]
:<math>
\begin{bmatrix}
 a_{11} & a_{12} & 0       \\
 a_{21} & a_{22} & a_{23}  \\
 0      & a_{32} & a_{33}  \\
\end{bmatrix}
\begin{bmatrix}
 k_0 \\
 k_1 \\
 k_2 \\
\end{bmatrix}
=
\begin{bmatrix}
 b_1 \\
 b_2 \\
 b_3 \\
\end{bmatrix}
</math>
with
:<math>a_{11} = \frac{2}{x_1 - x_0},</math>
:<math>a_{12} = \frac{1}{x_1 - x_0},</math>
:<math>a_{21} = \frac{1}{x_1 - x_0},</math>
:<math>a_{22} = 2 \left(\frac{1}{x_1 - x_0} + \frac{1}{{x_2 - x_1}}\right),</math>
:<math>a_{23} = \frac{1}{{x_2 - x_1}},</math>
:<math>a_{32} = \frac{1}{x_2 - x_1},</math>
:<math>a_{33} = \frac{2}{x_2 - x_1},</math>
:<math>b_1 = 3 \frac{y_1 - y_0}{(x_1 - x_0)^2},</math>
:<math>b_2 = 3 \left(\frac{y_1 - y_0}{{(x_1 - x_0)}^2} + \frac{y_2 - y_1}{{(x_2 - x_1)}^2}\right),</math>
:<math>b_3 = 3 \frac{y_2 - y_1}{(x_2 - x_1)^2}.</math>

For the three points
:<math>(-1,0.5),\ (0,0),\ (3,3),</math>
one gets that
:<math>k_0 = -0.6875,\ k_1 = -0.1250,\ k_2 = 1.5625,</math>
and from ({{EquationNote|10}}) and ({{EquationNote|11}}) that
:<math>a_1 =  k_0(x_1 - x_0) - (y_1 - y_0) = -0.1875,</math>
:<math>b_1 = -k_1(x_1 - x_0) + (y_1 - y_0) = -0.3750,</math>
:<math>a_2 =  k_1(x_2 - x_1) - (y_2 - y_1) = -3.3750,</math>
:<math>b_2 = -k_2(x_2 - x_1) + (y_2 - y_1) = -1.6875.</math>

In the figure, the spline function consisting of the two cubic polynomials <math>q_1(x)</math> and <math>q_2(x)</math> given by ({{EquationNote|9}}) is displayed.