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8.7.4 Accuracy of a Truncated Taylor Series
In this subsection and subection 8.7.5, we illustrate the use of matrices as a
convenient constructional tool to state and manipulate problems with two
indices. In this application, we desire to verify the accuracy of the truncated
N
Taylor series S = ∑ x n as an approximation to the function y = exp(x), over
n=0 n!
the interval 0 ≤ x < 1.
Because this application’s purpose is to illustrate a constructional scheme,
we write the code lines as we are proceeding with the different computa-
tional steps:

1. We start by dividing the (0, 1) interval into equally spaced seg-
ments. This array is given by:

x=[0:0.01:1];
M=length(x);

2. Assume that we are truncating the series at the value N = 10:

N=10;

3. Construct the matrix W having the following form:

 x 2 x 3 x N 
 1 x 1 1 1 L 1 
 2! 2 3! 3 N! 
N
 x 2 x 2 L x 2 
 1 x 2 2! 3! N! 
 x 2 x 3 x N 
W = 1 x 3 3 3 L 3  (8.20)
 2! 3! N! 
 M M L 
 M M O M 
 2 3 N 
 x M x M x M 
  1 x M 2! 3! L N!  

Specify the size of W, and then give the induction rule to go from
one column to the next:

Wi j(, − 1 )
Wi j(, ) = x i( ) * (8.21)
j − 1

© 2001 by CRC Press LLC