I hesitate here because of an understanding with a calculation problems.
I want to calculate an interpolation using the Vandermonde matrix. see: http://en.wikipedia.org/wiki/Vandermonde_matrix My polynomial has degree 2. I have the following coordinates p1 (1,4), p2 (2,6), p3 (4,1). I want a formula exhibition that interpolates these points. An interpolation point is difend as $p(u,g_i)$
The general matrix form is like this: T*M*vector_g
$q(u)= \begin{matrix} (u ^ {2} & u & 1) \end{matrix} * \begin{matrix} a_1 & a_2 &a_3 \\b_2 & b_2 &b_2 \\c_1 & c_1 &c_2 \end{matrix} * \begin{matrix} g_0\\g_1\\g_2 \end{matrix} $
Explenation:
Monomvector T = ($u^{2}$ u 1 )
geometry vector vector_g = $\begin{matrix} g_0\\g_1\\g_2 \end{matrix} $
Vanemonde Matrix M = $\begin{matrix} a_1 & a_2 &a_3 \\b_2 & b_2 &b_2 \\c_1 & c_1 &c_2 \end{matrix} $
Now my understanding problem:
When i use the given parameter p1..p2 i have a linear equation like
p1 (1,4) => a + b + c = 4 p2 (2,6) => 4a + 2b + c = 6 p3 (4,1) => 16a + 4b + c = 4
and the result are the coefficients for a,b,c are $\frac{-3}{2} u^2 $\frac{13}{2}u -1
But my Professor wants the general matrix form like above .... all i have is following :
q(u) = (u^{2}$ u 1) * $(\frac{-3}{2} u^2 $\frac{13}{2}u -1)$ = g0
But when i calculate
p1 (1,4) => a + b + c = $g_0$
p2 (2,6) => 4a + 2b + c = $g_1$
p3 (4,1) => 16a + 4b + c = $g_2$
i get a wrong matrix but in the right form ....
if you need more information just tell