HOMOGEN Function

HOMOGEN( matrix ) ;

The HOMOGEN function solves the homogeneous system of linear equations for . For at least one solution vector to exist, the matrix , , has to be of rank . The HOMOGEN function computes an column orthonormal matrix with the properties that and . In other words, the columns of form an orthonormal basis for the nullspace of .

If is ill-conditioned, rounding-error problems can occur in determining the correct rank of and in determining the correct number of solutions .

The following statements compute an example from Wilkinson and Reinsch (1971):

a = {22  10   2   3   7,
     14   7  10   0   8,
     -1  13  -1 -11   3,
     -3  -2  13  -2   4,
      9   8   1  -2   4,
      9   1  -7   5  -1,
      2  -6   6   5   1,
      4   5   0  -2   2};
x = homogen(a);
print x;

Figure 23.134 Solutions to a Homogeneous System
x
-0.419095 0
0.4405091 0.4185481
-0.052005 0.3487901
0.6760591 0.244153
0.4129773 -0.802217

In addition, you can use the HOMOGEN function to determine the rank of an matrix where by counting the number of columns in the matrix .

If is an matrix, then, in addition to the memory allocated for the return matrix, the HOMOGEN function temporarily allocates an array for performing its computation.