STABILITY 



20/12 



Numbers, and each matrix was then tested for stability by Hurwitz' 

 rule (S. 20/8 and S. 20/9). Thus a typical 3x3 matrix was 



— 1 - 3 -8' 



- 5 4—2 

 _ 4 _ 4 _ 9^ 



In this case the second determinant is — 86, so it need not be 

 tested further as it is unstable by S. 20/9. The testing becomes 

 very time-consuming when the matrices exceed 3x3, for the time 

 taken increases approximately as /i 5 . The results are summarised 

 in Table 20/12/1. 



Table 20/12/1. 



The main feature is the rapidity with which the probability 

 tends to zero. The figures given arc compatible (x 2 = 4-53, 

 P = 0-10) with the hypothesis that the probability for a matrix 

 of order n x n is l/2 n . That this may be the Correct expression 

 for this particular case is suggested partly by the fact that it 

 may be proved so when n = 1 and n = 2, and partly by the 

 fact that, for stability, the matrix has to pass all of n tests. 

 And in fact about a half of the matrices failed at each test. 

 If the signs of the determinants in Hurwitz' test are statistically 

 independent, then l/2 n would be the probability. 



In these tests, the intrinsic stabilities of the variables, as 

 judged by the signs of the terms in the main diagonal, were 

 equally likely to be stable or unstable. An interesting variation, 

 therefore, is to consider the case where the variables are all 

 intrinsically stable (all terms in the main diagonal distributed 

 uniformly between and — 9). 



The effect is to increase their probability of stability. Thus 

 when n is 1 the probability is 1 (instead of J) ; and when n is 



223 



