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NEWTON'S BACKWARD DIFFERENCE FORMULA
This is another way of approximating a function with an nth degree polynomial passing through (n+1) equally spaced points.
As a particular case, lets again consider the linear approximation to f(x)
f1 - f0
f0x1 - f1x0
f(x) @ P1(x) =
x + 
x1 - x0
x1 - x0

 
x1 - x
x - x0
 f0 + 
f1 
x1 - x0
x1 - x0

 
x - x1
 (
x - x0
 )
=  f1 - 
f0 +
 - 1 f1 
x0 - x1
x - x0

 
x - x1
=  f1 - 
(f1- f0) 
x1 - x0
= f1 + sÑf1
where s = (x - x1) / (x1 - x0) and Ñf1 is the backward difference of  f  at x1.
The same can be obtained from the difference operators as follows.
f(x)  =  f (xn + 
x - xn
  h)
h
 
=  f(xn + sh)   =   Es f(xn),      s  =  ( x - xn )/ h

 
=  ( 1 - Ñ )s f( xn )

 
         = fn + sÑfn +
 s(s + 1) 
 Ñ2fn + . . . +
 s(s + 1) . . . (s + n -1)
 Ñnfn + . . .
2!
n!
Neglecting the (n+1) difference as onwards we get
 f(x)  @ Pn(x)  = fn + sÑfn +
 s(s + 1) 
 Ñ2fn + . . . +
 s(s + 1) . . . (s + n -1)
 Ñnfn
2!
n!
This polynomial is called the Newton - Gregory backward difference formula.
The error in this interpolation
 En(x)  = 
 s(s + 1) . . . (s + n)
 hn+1f(n+1)(x) ( x0 < (x) < xn )
(n + 1)!
Example :

Find f(0.15) using Newton backward difference table from the data

x
f(x)
 Ñf
Ñ2f 
 Ñ3f
 Ñ4f
0.1
0.09983
 
 
 
  
 
 
0.09884
 
 
  
0.2
0.19867 
 
-0.00199
 
  
 
 
0.09685
 
-0.00156
  
0.3
0.29552
 
-0.00355
 
0.00121
 
 
0.0939 
 
-0.00035
  
0.4
0.38942
 
 -0.0039
 
  
 
 
0.09
 
    
0.5
0.47943
 
      
s  =  ( x - xn ) / h  =  (0.15 - 0.5) / 0.1  =  -3.5
f(0.15)  = fn + sÑfn +
 s(s + 1) 
 Ñ2fn +  s(s + 1)(s + 2) Ñ3fn + 
 s(s + 1)(s + 2)(s + 3)
 Ñ4fn
2!
3!
              = .97943 + (-3.5)*.09 +
 (-3.5)(-3.5 + 1) 
 (-.0039)+  (-3.5)(-3.5 + 1)(-3.5 + 2) (-.00035) + 
2!
3!
 
(-3.5)(-3.5 + 1)(-3.5 + 2)(-3.5 + 3)
 (.00121) 
4!
=  0.97943 - 0.315 - 0.01706 + 0.000765625 + 0.00033086
= 0.14847
Error in the approximation
 E5(x)  = 
 -3.5(-3.5 +1)(-3.5 + 2)(-3.5 + 3)(-3.5 + 4)
 h5f5(x) ( 0.1 < x < 0.5 )
5!
= 2.734375 x 10-7 f5(x)
 
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations