Numerical Analysis/Divided differences

The Expanded Form of the Definition edit

The usual definition of divided differences is equivalent to the Expanded form







With help of a polynomial functions   with   this can be written as


Since we will need the Expanded form (expanded) for our other work below, we first prove that it is equivalent to the usual definition.

Proof of the expanded form edit

For  , (expanded) holds because


We now assume (expanded) holds for   and show that this implies it also holds for  . Thus by induction it holds for all  .

If the formula  , where  , then denoting     and  , we have


We have,




which gives


Hence, since the assertion holds for   and  , then by induction, the assertion holds for all positive integer  .

Symmetry property of divided differences edit

The divided differences have a number of special properties that can simplify work with them. One of the property is called the Symmetry Property which states that the Divided differences remain unaffected by permutations (rearrangement) of their variables.

Now we prove this symmetry property by showing that


When  , we have


Hence  , which is the symmetry of the first divided difference.

When  , we have


Hence   etc., which is the symmetry of the second divided difference.

Similarly, when   we have


Hence   etc., which is the symmetry of the third divided difference.

In general, we can use the Expanded Form (expanded) to obtain


Hence   etc., which is the symmetry of the   divided difference.

Computing the divided differences in tabular form edit

A difference table is again a convenient device for displaying differences, the standard diagonal form being used and thus the generation of the divided differences is outlined in Table below.


A Numerical Example 1 edit

For a function  , the divided differences are given by


find  .

A Numerical Example 2 edit

For a function  , the divided differences are given by


Determine the missing entries in the table.

Algorithm: Computing the Divided Differences edit

Algorithm: Newton's Divided-Differences edit

   Given the points  
   Step 1:  Initialize  
   Step 2:  
   Result: The diagonal,   now contains  

Relationship between Generalization of the Mean Value Theorem and the Derivatives edit

Generalization of the Mean Value Theorem edit

For any n + 1 pairwise distinct points x0, ..., xn in the domain of an n-times differentiable function f there exists an interior point


where the nth derivative of f equals   ! times the   divided difference at these points:


This is called the Generalized Mean Value Theorem. For   we have


for some   between   and  , which is exactly Mean Value Theorem. We have extended MVT to higher order derivatives as


What is the theorem telling us?

This theorem is telling us that the Newton's   divided difference is in some sense approximation to the   derivatives of   .

A Numerical Example edit

Let  ,  . Then, Show that


for some   between the minimum and maximum of   and  .

Quiz edit

1 find   where  


2 If   then, this is called symmetry of the

zero divided difference
first divided difference
second divided difference
third divided difference

3 Let  ,   Then,  


4 If   for some   between   and   then, this is exactly

Generalized Mean Value Theorem
Mean Value Theorem
Derivative of  
Rolle's Theorem

5 If   then, this is called

First Divided Difference
Second Divided Difference
Third Divided Difference
Fourth Divided Difference

Reference edit

  • Guide to Numerical Analysis by Peter R. Turner
  • Numerical Analysis by Richard L. Burden and J. Douglas Faires (EIGHT EDITION)
  • Elementary Numerical Analysis by Kendall Atkinson (Second Edition)
  • Applied Numerical Analysis by Gerald / Wheatley (Sixth Edition)
  • Theory and Problems of Numerical Analysis by Francis Scheid