Mathematical Experiment 8


  1. Let f be an arbitrary function. For any two distinct points x0,x1 set
  2. [ x0x1] =  f(x0)-f(x1)
    x0-x1
    .
    For any three distinct points x0,x1,x2 set
    [ x0x1x2] =  [ x0x1] -[x1x2]
    x0-x2
    .
    In general if x0,x1,¼,xn are distinct, set
    [ x0x1x2¼xn] =  [x0x1x2¼xn-1] -[ x1x2x3¼xn]
    x0-xn
    .
    It is a matter of algebraic substitution to verify that for a polynomial f(x) of degree n the identity
    f(x
    f(x0)+[ x0x1] (x-x0)+[x0x1x2] (x-x0)(x-x1
    +¼+[ x0x1x2¼xn](x-x0)(x-x1)¼(x-xn-1)
    holds for any distinct x0,x1,x2,¼,xn. The spreadsheet can then be applied to solve the polynomial interpolation problem based on this formula of divided differences. The basic procedure is to set up a table to compute the divided differences according to this scheme:

  3. Draw the graph of the interpolation polynomial satisfying the following condition with Maple:
  4. x
    3
    2
    6
    5
    1
    f(x)
    3
    -3
    1
    4
    2
  5. When the package ``networks'' is loaded, a number of new commands are added. The most important one is draw. This allows you to visualize the graph.

exp8.tex