Numerical Relativity: On the Fallibility of Computers

Table 1. Jean Francois Colonna. Results of modeling variation in the population of individuals, obtained through successive iterations of the five formulations shown in Xn i l =(R+l)Xn -RXn 2 , where n=60 and R denotes the growing rate. The two sets of results were obtained using the same program on two different IBM computers, an ES9000 and an RS6000. IBM ES9000: (R+l)*Xn-rMXn*Xn) (R+l)*Xn-(R*Xn)*Xn ((R+l)-R*X>Xn R*Xn + (l-R*Xn>*Xn Xn + R*(X„-X„*Xn) IBM RS6000: (R+l)*X„-R*(Xn*Xn) (R+l)*Xn-(R*X„)*Xn ((R+l)-R*Xn)*Xn R*Xn + (l-R*Xn)*X„ X„ + R*(X„-X„*XJ 1.040381 0.846041 0.529794 1.319900 1.214070 0.001145 0.271115 0.616781 0.298613 1.307350 artistic work. I believe that anyone who is able to create should be open to using any tools and methods that can serve the inexhaustible artistic imagination. NUMERICAL RELATIVITY: O N THE FALLIBILITY OF COMPUTERS Jean-Franfois Colonna, GSVLACTAMME , CNET, Ecole Polytechnique, 91128 Palaiseau Cedex, France. Fax: (33 1) 69 33 30 01. E-mail: . Received 24 February 1994. Acceptedfor publication by RogerF. Malina. Is a computer an error-free machine to be trusted? Many would say it is, but what is the actual situation? On the one hand, a computer is a machine of human manufacture and high complexity, and thus every design flaw can be the cause of a wrong result (whether the result is an incorrect numerical value or any other failure in anything a computer can do or produce). On the other hand, one of its components can fail and shut the system down. But what happens if, say, a cosmic ray (or some other unexpected factor) hits a machine ? It could produce a 0 instead of a 1 during an elementary operation and result in, again, a wrong answer. Assuming we have a computer that includes redundancy and is well-protected, can it be really an error-free machine? Fortunately , in most cases (even when they do not involve burying computers under kilometers of protective rock), the answer is yes (otherwise it would be safer to stay home and not risk driving cars or flying planes). But, unfortunately, we can devise situations (most are scientific ones) in which the answer is no. For the study of dynamic systems, the well-known Edward Lorenz's notion of sensitivity to initial conditions should be extended and given the more general name "sensitivity to the accuracy of numerical values." As a matter of fact, the internal accuracy of computers is limited. During each arithmetic computation (multiplication is the worst case), digits are lost; this implies that the results obtained will therefore depend on the order of the operations, and the usual associative mathematical property of multiplication, for example, will no longer exist. The way in which a computer program is written (particularly with regard to parentheses) and the compiler used (and its optimization options) play an important role in the value of the results obtained. Moreover, not all computers (it is important to recall they are made of hardware and, above all, software) represent and handle digits in the same way. Two strictly identical calculations carried out on two different machines can give different results, which, in the case of a dynamic system, may vary increasingly in exponential terms. At the same time, forecasts relating to the development of the state of a system being studied depend largely on the computing tool itself. Is it therefore necessary to introduce the notion of computer subjectivity, or one of numerical relativity? This phenomenon may be observed by studying the Verhulst dynamics, which models the variation of a population of individuals by the following iteration (starting with a given X()): X , = (R+l)-X -R-X2 , with "R" denoting the growing rate. The above iteration, in fact, shows five families of formulation (as far as numerical results are concerned, any other possible form gives the same result as one of these five forms). As Table 1 indicates , these five formulations, obtained under the same conditions (by means of a very...