20/10 



STABILITY 



defined for it, and then a proper sample space denned. In general, 

 the number of possible meanings of ' probability of stability ' is 

 too large for extensive treatment here. Each case must be 

 considered individually when such consideration is called for. 



A case of some interest because of its central position in the 

 theory is the probability that a linear system shall be stable, 

 when its matrix is filled by random sampling from given distribu- 

 tions. The problem then becomes: 



A matrix of order n x n has elements which are real and are 

 random samples from given distributions. Find the probability 

 that all the latent roots have negative real parts. 



This problem seems to be still unsolved even in the special 

 cases in which all the elements have the same distributions, 

 selected to be simple, as the ' normal ' type e~ x \ or the ' rect- 

 angular ' type, constant between — a and + a. Nevertheless, 

 as I required some indication of how the probability changed with 

 increasing n, the rectangular distribution (integers evenly dis- 

 tributed between — 9 and + 9) was tested empirically. Matrices 

 were formed from Fisher and Yates' Table of Random Numbers, 

 and each matrix was then tested for stability by Hurwitz' rule 

 (S. 20/7). 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. The testing becomes very time-consuming when 

 the matrices exceed 3x3, for the time taken increases approxi- 

 mately as ?i 5 . The results are summarised in Table 20/10/1. 



Table 20/10/1. 



The main feature is the rapidity with which the probability 

 tends to zero. -The figures given are compatible (# 2 = 4-53, 



259 



