Last Updated on October 21, 2024 by Maged kamel

## What is Newton-divided difference interpolation?

Newton has developed a new form for the interpolation of functions. In this post, we will be discussing Newton-divided difference interpolation.

A new form of Quadratic expression was adopted. For a Quadratic function where n=2, we need n+1 points, three points. Our n+1=3 points, namely (xo, x1, x2), and their y coordinates are (y_{o}, y_{1}, y_{2}).

The quadratic function will be written as Q(x) = b_{0}+b_{1}(x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}). It can be further expanded as Q(x) = b_{o}+b_{1}x- b_{1}*x_{o} +b_{2}(x^2-x*x_{1}-x_{0}*x+x_{0 }*x_{1}). Recalling our polynomial expressed as P2(x)= a_{o}+a_{1}*x+a_{2}*x^2.

Since both functions are the same, we will equate both. For the item of x, we have a_{1}*x=+b_{1}*x-b_{2}*x_{1}*x-b_{2}*x_{0}*x. a_{1}= b_{1} -b_{2}*x_{1} -b_{2}*x_{0}.

Similarly, For the item of x^2, we have a_{2}*x^2=b_{2}*x^2. Then for the value of the term a_{2}, it will be a_{2}= b_{2}

For the constant item, we have a_{0}= b_{0}-b_{1}*x_{0}+b_{2}*x_{0}*x_{1}. To get values of b_{0},b_{1},b_{2}, in terms of the given three Points. Use the first point(x_{0},y_{0}) to get the value of b_{0}. We can rewrite the Q(x_{0}) as: x=x_{0},y=y_{0}.

Q(x) = b_{0}+b_{1}(x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}). Q(x_{0}) =y_{0}= b_{0}+b_{1}(x_{0}-x_{0}) +b_{2}(x_{0}-x_{0}) (x_{0}-x_{1}). b_{0}=y_{0.}

Back to our equation of Q(x), Q(x) = y_{0}+b_{1}(x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}). For the second point(x_{1},y_{1}), we can rewrite the Q(x_{1}) as: x=x_{1},y=y_{1}. From the equation Q(x) = y_{0}+b_{1}(x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}).

### The expression of the first divided difference.

Q(x_{1}) =y_{1}= y_{0}+b_{1}(x_{1}-x_{0}) +b_{2}(x_{1}-x_{0}) (x_{1}-x_{1}). y_{1}= y_{0}+b_{1}(x_{1}-x_{0})+0 then b_{1}=(y_{1}-y_{0})/(x_{1}-x_{0}). This is the first divided difference written as f, bracket x_{0},x_{1}, then bracket.

Back to our equation of Q(x). Q(x) = y_{0}+b_{1}(x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}). Rewrite as: Q(x) = y_{0}+((y_{1}-y_{0})/(x_{1}-x_{0}))* (x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}). For the third point(x_{2,}y_{2}), we can rewrite the Q(x_{2}) as: x=x_{2},y=y_{2}. From the equation Q(x) = y_{0}+((y_{1}-y_{0})/(x_{1}-x_{0}))* (x-x_{0}) +b_{2}(x-x_{0}) (x-x_{1}).

Q(x_{2}) =y_{2}= y_{0}+((y_{1}-y_{0})/(x_{1}-x_{0}))* (x_{2}-x_{0}) +b_{2}(x_{2}-x_{0}) (x_{2}-x_{1}).

The following steps in the next slide picture will illustrate the procedure to get the value of b_{2}.

### Newton-divided difference or second-divided difference.

For the second divided-difference, which is written as f, bracket x_{0},x_{1},x2, then bracket.

This is the final expression for the quadratic polynomial using Newton-divided difference interpolation.

The polynomial is shown in the case of the first order.

The first divided difference is shown in the next slide. A line represents the first divided difference.

The polynomial is shown in the case of the first order. The first divided difference is shown in the next slide. A line represents the first divided difference.

For the Higher-order n value, an expression can be developed and made as a table.

We can make a table for the second divided difference. The next slide image shows the arrangement. The source is from Amos Gilat’s Numerical Methods for Engineers. The clouded area is for a Quadratic polynomial.

The main advantage of Newton’s divided difference interpolation is that we will not substitute it to make a solution for the n equations. In our case, n=2 is for a quadratic function.

The next post will solve two problems as practice problems for Newton-Divided difference polynomials.

This is a Wiki link for Numerical analysis.

This is a link to Holistic Numerical Methods-Newton divided Differences.