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:

“What is done for science must also be done for art: accepting undesirable side effects for the sake of the main goal, and moreover diminishing their importance by making this main goal more magnificent. For one should reform forward, not backward: social illnesses, revolutions, are evolutions inhibited by a conserving stupidity.”
—Robert Musil (1880–1942)

“The idea which man forms of beauty imprints itself throughout his attire, rumples or stiffens his garments, rounds off or aligns his gestures, and, finally, even subtly penetrates the features of his face.”
—Charles Baudelaire (1821–1867)