Newton’s Progressive Build

Divided differences offer an incremental approach to polynomial interpolation. Starting from a set of sample points $(x_0,y_0)$, $(x_1,y_1)$, and so forth, we compute coefficients $c_0$, $c_1$, etc., that build the polynomial step by step. The first coefficient is simply $y_0$; subsequent ones involve differences of preceding values divided by differences in $x$ coordinates. The resulting polynomial takes the nested form $P\left(x\right)=c_0+c_1\left(x,-,x_0\right)+c_2\left(x,-,x_0\right)\left(x,-,x_1\right)+\dots$. Each new coefficient modifies the previous polynomial by a factor involving $x$ minus earlier nodes.

This structure simplifies adding more data points: you only need to compute new divided differences rather than refitting from scratch. The coefficients correspond to forward differences along one dimension of an array, making them straightforward to implement with nested loops or triangular tables.

The calculator accepts any number of points, as long as the $x$ coordinates are distinct. It constructs the divided-difference table, extracts the coefficients, and evaluates the polynomial at the desired $x$. This approach forms the backbone of many interpolation schemes used for data fitting and numerical approximation.

Unlike Lagrange polynomials, Newton’s form is easy to update when new observations arrive. It also clarifies how the polynomial evolves: each additional term bends the curve to pass through the next data point. From a geometric standpoint, the factors $\left(x,-,x_i\right)$ represent lines through previous nodes, while the coefficients encode slopes of increasing order.

To use this calculator, enter pairs like 0 1 or 2.5 3.1 on separate lines, then supply an evaluation value. The result displays the interpolated $y$ at that $x$. You can explore how adding points changes the polynomial’s shape and how extrapolation behaves outside the data range.