STEP 19

FINITE DIFFERENCES 2

Forward, backward, central difference notations

There are several different notations for the single set of finite differences, described in the preceding Step. We introduce each of these three notations in terms of the so-called shift operator, which we will define first.

1. The shift operator E

   Let be a set of values of the function f(x) The shift operator E is defined by:

   .

   Consequently,

   .

   and so on, i.e.,

   ,

   where k is any positive integer. Moreover, the last formula can be extended to negative integers, and indeed to all real values of j and k, so that, for example,

   ,

   and

   .

2. The forward difference operator Q

   If we define the forward difference operator Q by

   ,

   then

   ,

   which is the first-order forward difference at x_j. Similarly, we find that

   

   is the second-order forward difference at x_j, and so on. The forward difference of order k is

   ,

   where k is any integer.

3. The backward difference operator

   If we define the backward difference operator by

   ,

   then

   ,

   which is the first-order backward difference at x_j. Similarly,

   

   is the second-order backward difference at x_j, etc. The backward difference of order k is

   ,

   where k is any integer. Note that .

4. The central difference operator

   If we define the central difference operator by

   ,

   then

   ,

   which is the first-order central difference at x_j. Similarly,

   

   is the second-order central difference at x_j, etc. The central difference of order k is

   ,

   where k is any integer. Note that .

5. Differences display

   The role of the forward, central, and backward differences is displayed by the difference table:

   

   Although forward, central, and backward differences represent precisely the same data:
   

   1. Forward differences are useful near the start of a table, since they only involve tabulated function values below x_j ;
      
   2. Central differences are useful away from the ends of a table, where there are available tabulated function values above and below x_j;
      
   3. Backward differences are useful near the end of a table, since they only involve tabulated function values above x_j.

   Checkpoint

   1. What is the definition of the shift operator?
      
   2. How are the forward, backward, and central difference operators defined?
      
   3. When are the forward, backward, and central difference notations likely to be of special use?

   EXERCISES

   1. Construct a table of differences for the polynomial

      ;

      for x = 0(1)4. Use the table to obtain the values of :
      

      1. ;
         
      2. ;
         
      3. .
   2. For the difference table in Section 3 of STEP 18 of f (x) = e^x for x = 0.1(0.05)0.5 determine to six significant digits the quantities (taking x₀ = 0.1 ):
      1. ;
         
      2. ;
         
      3. ;
         
      4. ;
         
      5. ;
   3. Prove the statements:
      1. ;
         
      2. ;
         
      3. ;
         
      4. .

      Answers

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