> > You might read more carefully what they mean by verified. > (And report back here). > As I recall, all they did was some random numerical evaluation, > hardly the kind of checking that would confirm behavior at > singularities etc. >
I added the 2 pages from introduction that describes what they did to the above pdf sample. Here is some snippet:
.... " (b) The automatically linearized form of each integral was then processed by a program that numerically differentiated the integral at various values for the variable of integration and at various values for the parameters of the integrand. These derivatives were compared to the respective integrand evaluated at the same points. The truncation errors of the differentiation approximation and the computational rounding error were first estimated for the various evaluation points.
These values were used to adjust the parameters of the differentiation so that the truncation error and the rounding error would be approximately equal. The integral was accepted if the absolute value of the difference between the integrand and the numerical derivative the numerical derivative of the integral (the absolute error) was less than or equal to the estimated total error at aLL points of evaluation.
For situations where the estimated total error could be considered to be negligible, an alternate criterion was to restrict the allowed relative error, that is, the absolute error stated above divided by the value of the integfand.
The program monitored itself to avotd undue loss of precision due to subtraction errors in computing derivatives, and for this purpose made appropriate adjustments in the differentiation parameters. The strictness of the numerical criteria was a function of the word size of the computer (in this case, thirty binary bits for a floating-point mantissa) and of the empirical monitoring of results to minimize the probability of an incorrect integral slipping through the process. These programs were written in the K-M language and compiled by the K-M system. (c) Simultaneously, the results, which included the precision of validation or range of errors for each datum, were recorded via high-speed printer listing. Corrolary data were output via the computer typewriter terminal." ......