Arrays

A one-dimensional array is created using the array command

array( .. );

This creates an array with indices . Often is 1. Entries can be inserted into the array by assignment as for tables, e.g.

v := array(1..n);
v[1] := a;
for i from 2 to n do v[i] := a*v[i-1] mod n od;

One-dimensional arrays are like tables with one index which is restricted to be in a fixed integer range. Arrays are more efficient to access than tables and range checking on the indices is done. Here is an example of a one-dimensional array used to sort a sequence of numbers. For instance, suppose we are given an array = array(1..n) as above. The code presented here sorts into ascending order using the bubblesort algorithm:

for i to n-1 do
    for j from i+1 to n do
        if v[i] > v[j] then temp := v[i]; v[i] := v[j]; v[j] := temp fi
    od
od;

Another application of one-dimensional arrays is as an intermediate data structure. For example, suppose we represent a polynomial as a list of coefficients. Suppose we are given also a polynomial of degree . We can use an array to compute the product as follows

m := nops(a)-1; # degree of a
n := nops(b)-1; # degree of b
c := array(0..m+n); # allocate storage for the product
for i from 0 to m+n do c[i] := 0 od;
for i from 0 to m do
    for j from 0 to n do
        c[i+j] := c[i+j] + a[i+1]*b[j+1]
    od
od:
[seq(c[i],i=0..n+m)]; # put the product in a list

Two-dimensional arrays, and higher dimensional arrays work similarly. A two-dimensional array is created by the command

array( , .. );

As an example, suppose we are given the vector of symmetric polynomials in variables .

=-1.00.5plus##1`##1=12=^^M=12 v := array(1..4): v[1] := 1: v[2] := x[1] + x[2] + x[3]: v[3] := x[1]*x[2] + x[1]*x[3] + x[2]*x[3]: v[4] := x[1]*x[2]*x[3]:

plusplus -100 plus Let us construct a two-dimensional array , where is the derivative of wrt .

=-1.00.5plus##1`##1=12=^^M=12 > M := array(1..4,1..3): > for i to 4 do for j to 3 do M[i,j] := diff(v[i],x[j]) od od: > M; M

> eval(M); [ 0 0 0 ] [ ] [ 1 1 1 ] [ ] [ x[2] + x[3] x[1] + x[3] x[1] + x[2] ] [ ] [ x[2] x[3] x[1] x[3] x[1] x[2] ]

plusplus -100 plus

For further information on arrays see ?array . Note also that one-dimensional arrays indexed from 1 are used to represent vectors and two-dimensional arrays indexed from 1 are used to represent matrices. See the section on Matrices and Vectors.