A Note on a Recent Paper by J. S. Respondek ()
1. Introduction and Objectives
For convenience of the reader, we begin this section by introducing the following abstract from [1] concerning the algorithm in [2].
“In this paper, we give the comments on the article ‘Inversion of a Generalized Vandermonde Matrix’ by M. E. A. El Mikkawy, Int. J. Computer Math. 80 (2003), pp. 759-765. The article gives an algorithm for the elementary symmetric function’s calculation which contains a severe error. In these comments, we have proposed necessary corrections of that algorithm”.
Let us begin our discussion by giving the following definition and the algorithm for the elementary symmetric function’s in [2]:
Definition 1.1 ([2], p. 760). If the n parameters 
are distinct, then the elementary symmetric functions 
in 
are defined for j = 1(1) n by:
for i = 2(1) n.(1.1)
Denote by:
(1.2)
Algorithm (Algorithm 2.1 in [2], p. 760). For
, we may calculate the elements of the first column of the n × n matrix 
in (1.2) as follows:
Set 
For i = 2, 3, ∙∙∙, n
(1.3)
Next j Next i The elements in the remaining n − 1 columns of 
may be obtained by symmetry using
, i = 1(1)n, k = 2(1)n. (1.4)
The notation in (1.4) means that for specific i and k, 
may be obtained from the algebraic expression of 
by replacing each 
by 
in the expression of 
. 
In [1], Respondek claims that the above algorithm contains a severe error. In the next section, we are going to show that the algorithm, as it is in [2] and (1.3), can be implemented properly without causing any errors by using vectors (one-dimensional arrays) rather than using 2-dimensional arrays. To enable users and programmers of the algorithm to carry out the computations using all existing subscripts and superscripts in the algorithm, we give a correction in the first line of the algorithm. A Maple implementation for the algorithm, as it is in [2] and (1.3), is given as an example for symbolic programming.
2. Main Results
It is now time to show that the algorithm, as it is in [2] and (1.3), can be implemented properly without causing any errors by using vectors (one-dimensional arrays) rather than using 2-dimensional arrays. The following is a Maple implementation for the algorithm in [2] and (1.3). Some sample output for the case n = 4 is given.
> # Implementation of Algorithm 2.1 in [2], p. 760
> restart:
> n:=4: g :=array(1..n):
> g[1]:=1:
> for i from 2 to n do
g[i]:= c[i] * g[i-1]:
for j from i-1 to 2 by -1 do
g[j] := simplify(g[j] + c[i] * g[j-1]);
od:
od:
> sigma := array(1..n,1..n):
> for i to n do sigma[i,1] := g[i] ; od:
> for i to n do
for j from 2 to n do
sigma[i,j] := subs({c[j]=c[1]},sigma[i,1]);
od:
od:
> sigma = op(sigma);
To enable users and programmers of the algorithm to carry out the computations using all existing subscripts and superscripts in the algorithm, as actually done by J. S. Respondek [1], we give a correction in the first line of the algorithm. The modified algorithm is now given by:
A Modified Algorithm. For 
we may calculate the elements of the first column of the n × n matrix 
in (1.2) as follows:
Set
, i = 1(1) n − 1.
For i = 2, 3, ∙∙∙, n
(2.1)
Next j Next i