# Newtons Interpolating Polynomials Method

by admin in Curve Fitting , Interpolation , Math, Statistics, and Optimization , MATLAB Family on April 27, 2019This code conduct an Interpolation for a given set of data using Newton’s Interpolating Polynomials method, where the coefficients of the polynomial are calculated using Newton’s divided differences method.

### Code Outputs:

- Printed Table of Interpolated Values
- Chart Compare of Interpolated Values with Given Points.

Xi F(Xi)

3.0000 0.2274

11.0000 0.0003

31.0000 0.5464

53.0000 0.5287

71.0000 0.3455

93.0000 0.3624

111.0000 0.7551

### Input Requirements:

- X Axis Points
- Y Axis Points

### About the Method:

**Newton polynomial**, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called **Newton’s divided differences interpolation polynomial** because the coefficients of the polynomial are calculated using Newton’s divided differences method.

### Definition:

Given a set of ** k + 1** data points

where no two ** x_{j}** are the same, the Newton interpolation polynomial is a linear combination of

**Newton basis polynomials**

with the Newton basis polynomials defined as

for ** j >** 0 and

**n0(x) = 1**.

The coefficients are defined as

where

is the notation for divided differences.

Thus the Newton polynomial can be written as

### Newton Forward Divided Difference Formula:

The Newton polynomial can be expressed in a simplified form when **x0, x1, …, xk** are arranged consecutively with equal spacing. Introducing the notation **h = xi+1 – xi** for each **i = 0,1,…, k-1** and **x = x0 + s.h**, the difference **x – xi **can be written as **(s – i).h**. So the Newton polynomial becomes

This is called the **Newton forward divided difference formula**

### Newton Backward Divided Difference Formula:

If the nodes are reordered as **xk, xk-1, …, x0**, the Newton polynomial becomes

If are equally spaced with and for *i* = 0, 1, …, *k*, then,

is called the **Newton backward divided difference formula.**

### Examples:

The divided differences can be written in the form of a table. For example, for a function *f* is to be interpolated on points **x0, …, xn**. Write

Then the interpolating polynomial is formed as above using the topmost entries in each column as coefficients. For example, suppose we are to construct the interpolating polynomial to ** f(x) = tan(x)** using divided differences, at the points:

Using six digits of accuracy, we construct the table:

Thus, the interpolating polynomial is

Given more digits of accuracy in the table, the first and third coefficients will be found to be zero.

Another example:

The sequence **f0 **such that **f0(1) = 6, f0(2) = 9, f0(3) = 2 and f0(4) =5**, i.e., they are **6, 9, 2, 5** from **x0 = 1** to **x3 = 4**.

You obtain the slope of order **1** in the following way:

As we have the slopes of order **1,** it’s possible to obtain the next order:

Finally, we define the slope of order **3:**

Once we have the slope, we can define the consequent polynomials:

- .
- .

### References:

* [1]* Numerical Methods for Scientists and Engineers, R.W. Hamming.

[2] Stetekluh, Jeff. ‘Algorithm for the Newton Form of the Interpolating Polynomial’.

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