## What is Newton-divided difference interpolation?

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

A new form for 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.

You can click on any picture to enlarge then press the small arrow to review all the other images as a slide show.

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 and is 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, and is written as f, bracket x_{0},x_{1},x2, then bracket.

This is the final expression for the quadratic polynomial by 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. The first divided difference is represented by a line.

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

For the second divided difference, we can make a table form, The next slide image shows the arrangement. The source is from the Numerical Methods for Engineers by Amos Gilat. the clouded area is for Quadratic polynomial.

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

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

This is a link to the pdf file used to illustrate this post.