Newton Polynomial - Main Idea

Main Idea

Solving an interpolation problem leads to a problem in linear algebra where we have to solve a system of linear equations. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Newton basis, we get a system of linear equations with a much simpler lower triangular matrix which can be solved faster.

For k + 1 data points we construct the Newton basis as

Using these polynomials as a basis for we have to solve

$\begin{bmatrix} 1 & & \ldots & & 0 \\ 1 & x_1-x_0 & & & \\ 1 & x_2-x_0 & (x_2-x_0)(x_2-x_1) & & \vdots \\ \vdots & \vdots & & \ddots & \\ 1 & x_k-x_0 & \ldots & \ldots & \prod_{j=0}^{k-1}(x_k - x_j) \end{bmatrix} \begin{bmatrix} a_0 \\ \vdots \\ a_{k} \end{bmatrix} = \begin{bmatrix} y_0 \\ \vdots \\ y_{k} \end{bmatrix}$

to solve the polynomial interpolation problem.

This system of equations can be solved recursively by solving

Read more about this topic: Newton Polynomial

Famous quotes containing the words main and/or idea:

“The main question raised by the thriller is not what kind of world we live in, or what reality is like, but what it has done to us.”
—Ralph Harper (b. 1915)

“The whole idea of image is so confused. On the one hand, Madison Avenue is worried about the image of the players in a tennis tour. On the other hand, sports events are often sponsored by the makers of junk food, beer, and cigarettes. What’s the message when an athlete who works at keeping her body fit is sponsored by a sugar-filled snack that does more harm than good?”
—Martina Navratilova (b. 1956)