Page 228 - Applied Numerical Methods Using MATLAB

DIFFERENCE APPROXIMATION FOR SECOND AND HIGHER DERIVATIVE 217
Now, we turn our attention to the high-order derivatives. But, instead of deriv-
ing the specific formulas, let’s make an algorithm to generate whatever difference
approximation formula we want. For instance, if we want to get the approxima-
tion formula of the second derivative based on the function values f 2 ,f 1 ,f 0 ,f −1 ,
and f −2 , we write

(2) c 2 f 2 + c 1 f 1 + c 0 f 0 + c −1 f −1 + c −2 f −2
D (x, h) = (5.3.3)
c4 2
h
and take the Taylor series expansion of f 2 ,f 1 ,f −1 ,and f −2 excluding f 0 on the
right-hand side of this equation to rewrite it as

(2)
D (x, h)
c4
(2h) (2h) (2h)
 2 3 4 
 (2) (3) (4) 
 c 2 f 0 + 2hf + f + f + f +· · ·
0 0 0 0 
 
 2 3! 4! 
 
 
 

 2 3 4 
 h h h 
 (2) (3) (4) 
 +c 1 f 0 + hf + f 0 + f 0 + f 0 +· · · + c 0 f 0 

0

1  2 3! 4! 
=
2 2 3 4
h  h (2) h (3) h (4) 
 +c −1 f 0 − hf + f 0 − f 0 + f 0 −· · · 
 
0


 2 3! 4! 
 
 
 

 2 3 4 
 (2h) (2h) (2h) 
 (2) (3) (4) 

 +c −2 f 0 − 2hf + f − f + f 
0 0 0 0 − ··· 
2 3! 4!

 
 (c 2 + c 1 + c 0 + c −1 + c −2 )f 0 + h(2c 2 + c 1 − c −1 − 2c −2 )f 
 0 
 

 2 2 
 2 1 1 2 
 2 (2) 
 +h c 2 + c 1 + c −1 + c −2 f 
 0 
2 2 2 2 
 

1  
3 3
= 2 1 1 2 (5.3.4)
2 3 (3)
h  +h c 2 + c 1 − c −1 − c −2 f 
 0 
 3! 3! 3! 3! 
 
 
 
4 4
 
 2 1 1 2 (4) 
 4 
 +h c 2 + c 1 + c −1 + c −2 f +· · · 
4! 4! 4! 4!
 0 
We should solve the following set of equations to determine the coefficients
c 2 ,c 1 ,c 0 ,c −1 ,and c −2 so as to make the expression conform to the second
(2)
derivative f at x + 0h = x.
0
1 1 1 1 1  c 2   
 
0
2 1 0 −1 −2
 
 c 1   0 
 2 2 
 2 /2! 1/2! 0 1/2!     (5.3.5)
2 /2! 
 c 0  =  1 


3
 3 1/3! 0 −1/3! −2 /3!   c −1   
0
2 /3!
4
4
2 /4! 1/4! 0 1/4! 2 /4! c −2 0