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# The Newton Polynomial Interpolation

This online calculator constructs Newton interpolating polynomial for given data points. The calculator also shows general form and simplified form, interpolates additional points, if entered, and plots a chart

This online calculator constructs Newton interpolation polynomial for a given set of data points. It also calculates an interpolated value for entered points and plots a chart.

### Usage

First, enter the data points, one point per line, in the form x f(x), separated by spaces. If you want to interpolate the function using interpolating polynomial, enter the interpolation points into the following field, as x values, separated by spaces.

You can also find some theory about the Newton interpolating polynomial below the calculator.

#### The Newton Polynomial Interpolation

Newton Polynomial

Newton Polynomial after simplification

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Newton Polynomial
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### Newton Polynomial Interpolation

General form of the Newton interpolating polynomial is:

$P_n(x)=f(x_0)+\sum_{k=1}^n \left( f(x_0, \dots , x_k) \cdot \prod_{i=0}^{k-1}{(x-x_i)} \right)$,

where n is polynomial degree,
$f(x_0, \dots , x_k)$ is _k_th divided difference, defined as
$f(x_i, x_{i+1}, \dots , x_{i+k})=\frac{f(x_{i+1}, x_{i+2} \dots , x_{i+k}) - f(x_i, x_{i+1}, \dots , x_{i+k-1})}{x_{i+k}-x_i}$.

The _k_th divided difference also can be expressed as:
$f(x_0, x_1, \dots , x_k)=\sum_{i=0}^k \left( \frac{f(x_i)}{ \prod_{j=0, j \neq i}^k (x_i-x_j) } \right)$.
That last form is used in the calculator.

In the Newton interpolation, additional basis polynomials and the corresponding coefficients can be calculated when more data points are to be used, and all existing basis polynomials and their coefficients remain unchanged. This is more suitable for manual calculations because, for example, additional points in the Lagrange interpolation require all basis polynomials to be recalculated.

Note that due to the polynomial interpolation's uniqueness, the Newton interpolation is the same as Lagrange interpolation. It is the same nth degree polynomial expressed in terms of different basis polynomials weighted by different coefficients.

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