Numerical Analysis/Divided differences

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The Expanded Form of the Definition[edit]

The usual definition of divided differences is equivalent to the

Expanded formString Module Error: function rep expects a number as second parameter, received "

"

 

 

 

 

(expanded)

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"

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,

and

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:  
          For 
             For 
               
             End
          End
   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

25
-25
50
-50

2

If then, this is called symmetry of the

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

3

Let , Then,

-0.2473009
0.2473009
-0.2474404
0.2474404

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