Vandermonde Matrix Based Interpolation Method

Suppose you want to compute the polinomial:

(1)
$$P(x)=a_0+a_1x+a_2x^2+...+a_nx^n$$

which passes through $n+1$ given points:

(2)
\begin{align} \left( {\begin{array}{*{20}{c}} {{x_0}} \\ {P({x_0})} \end{array}} \right),\left( {\begin{array}{*{20}{c}} {{x_1}} \\ {P({x_1})} \end{array}} \right),...,\left( {\begin{array}{*{20}{c}} {{x_n}} \\ {P({x_n})} \end{array}} \right) \end{align}

Since each pair should suffice the polynomial, the coefficients can be found by inversion of the Vandermonde matrix:

(3)
\begin{align} \left( {\begin{array}{*{20}{c}} {{a_0}} \\ {{a_1}} \\ \vdots \\ {{a_n}} \end{array}} \right) = {\left( {\begin{array}{*{20}{c}} 1&{{x_0}}& \ldots &{x_0^n} \\ 1&{{x_1}}& \ldots &{x_1^n} \\ \vdots & \vdots & \ddots & \vdots \\ 1&{{x_n}}& \ldots &{x_n^n} \end{array}} \right)^{ - 1}}\left( {\begin{array}{*{20}{c}} {P({x_0})} \\ {P({x_1})} \\ \vdots \\ {P({x_n})} \end{array}} \right) \end{align}
page revision: 4, last edited: 19 Dec 2011 17:36