Introductory Remarks on Reliable Engineering Computing

Examples of the use of interval methods in this connection: 1)A physics problem at Lockheed (ca. 1960): Q: Is the strange behavior of the computer model due to round-off errors? A: After converting the program to run in interval arithmetic with outward rounding, it was determined that round-off error was very small in the original program. Result: the physicist took another look at the model equations and found that there was a missing term. 2)A long controversy between research groups at MIT and Cal Tech concerned whether the observed behavior of computer simulations was due to roundoff error or defects in the mathematical model, in the case of computer solutions of the Birkhoff-Rota complex partial differential/integral equations modeling the onset of turbulence in wind-shears. My graduate student Jeffrey Ely wrote a program for variable-precision interval arithmetic with outward rounding, and finally by using about 300 decimal place (nearly 1000 bits) in outwardly rounded interval arithmetic, was able to settle the controversy. It was NOT roundoff error, but the model itself. The model realistically determined the onset of turbulence at a reproduceable, finite time after the initial appearance of the wind-shear. I have always found it odd to suppose that anything we want to compute can be done carrying only some fixed number of digits or bits. Is 40 bits enough? 80 ? A thousand? It depends on what we are trying to compute. [J. Ely and G. R. Baker. High precision calculations of vortex sheet motion. J. Comp. Phys., 111:275-282, 1994]. The point of these examples is that outwardly rounded interval arithmetic automatically bounds roundoff error in any computation. As a consequence, if the interval results are adequately narrow, then no repetition is needed using higher precision arithmetic, carrying more digits. If the results are not narrow enough, then the computations can be repeated carrying more digits. If roundoff is the only source of error, we may eventually obtain satisfactorily narrow interval results containing the corresponding infinite precision results that would come from using exact real arithmetic. Floating-point arithmetic by itself cannot provide such answers. More important is the fact that interval computational methods can answer questions about the ”appropriateness of the mathematical model” even when roundoff is not the only nor the main source of computational error in simulating the behavior of a proposed