brief illustration- post 2c-numerical-interpolation

2c- What is Newton-divided difference interpolation?

Spread the love

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 (yo, y1, y2).

The quadratic function will be written as Q(x) = b0+b1(x-x0) +b2(x-x0) (x-x1). It can be further expanded as Q(x) = bo+b1x- b1*xo +b2(x^2-x*x1-x0*x+x0 *x1). Recalling our polynomial expressed as  P2(x)= ao+a1*x+a2*x^2.

Since both functions are the same, we will equate both. For the item of x, we have a1*x=+b1*x-b2*x1*x-b2*x0*x. a1= b1 -b2*x1 -b2*x0.

Introduction to Newton-divided difference interpolation.

Similarly, For the item of x^2, we have a2*x^2=b2*x^2. Then for the value of the term a2, it will be a2= b2

For the constant item, we have a0= b0-b1*x0+b2*x0*x1. To get values of b0,b1,b2, in terms of the given three Points. Use the first point(x0,y0) to get the value of b0. We can rewrite the Q(x0) as: x=x0,y=y0.

Q(x) = b0+b1(x-x0) +b2(x-x0) (x-x1). Q(x0) =y0= b0+b1(x0-x0) +b2(x0-x0) (x0-x1). b0=y0.

Back to our equation of Q(x), Q(x) = y0+b1(x-x0) +b2(x-x0) (x-x1). For the second point(x1,y1), we can rewrite the Q(x1) as: x=x1,y=y1. From the equation Q(x) = y0+b1(x-x0) +b2(x-x0) (x-x1).

The expression of the first divided difference.

Q(x1) =y1= y0+b1(x1-x0) +b2(x1-x0) (x1-x1). y1= y0+b1(x1-x0)+0 then b1=(y1-y0)/(x1-x0). This is the first divided difference written as f, bracket x0,x1, then bracket.

The expression of b1 in terms of the x1,y1 and x2,y2.

Back to our equation of Q(x). Q(x) = y0+b1(x-x0) +b2(x-x0) (x-x1). Rewrite as: Q(x) = y0+((y1-y0)/(x1-x0))* (x-x0) +b2(x-x0) (x-x1). For the third point(x2,y2), we can rewrite the Q(x2) as: x=x2,y=y2. From the equation Q(x) = y0+((y1-y0)/(x1-x0))* (x-x0) +b2(x-x0) (x-x1).
Q(x2) =y2= y0+((y1-y0)/(x1-x0))* (x2-x0) +b2(x2-x0) (x2-x1).

The following steps in the next slide picture will illustrate the procedure to get the value of b2.

The steps to get b2 value.

Newton-divided difference or second-divided difference.

For the second divided-difference, which is written as f, bracket x0,x1,x2, then bracket.

The expression for Newton the second divided difference.

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

The final expression of Newton divided -differnec for a quadratic function.

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.

Newton 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.

Newton second divided difference diagram

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.

Scroll to Top