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The 17-ketosteroid excretion of a healthy human male is characterized by rhythms with a frequency of one cycle in a) about one week; b) about twenty days (± 3 days); c) about one month, and d) about one year. The terms circaseptan, circavigintan, circatrigintan and circannual are considered as names for these rhythms with a low frequency and a small amplitude. These rhythms, among others, can be resolved as statistically significant components in the frequency domain by so-called least squares spectra, prepared by an electronic computer program. The rhythms then stand out above the general level of the spectrum, as components reproducible during an observation span of over 15 years. Certain displays of the amplitude and phase of a rhythm computed and plotted automatically by the electronic computer also are introduced and illustrated. These techniques quantify objectively the external and internal timing of a rhythm and/or changes of this timing. The crest-time of the seven-day synchronized circaseptan rhythm in 17-ketosteroid excretion and a weekly crest in urine volume are displayed, on the one hand, in a polar amplitude-phase plot. On the other hand, these aspects of physiologic timing are also scrutinized as a function of time in plots on rectangular coordinates involving the fit of a precise frequency, established independently as a characteristic of the series analyzed. For a span covering almost a decade, the circaseptan rhythm in 17-ketosteroid excretion is shown to be clearly synchronized with the seven-day cyclic alternation of professional and other activity of the (one) subject investigated. Moreover, in the face of a continuing precise seven-day routine of living, this circaseptan rhythm can be demonstrated to freerun thereafter during a three year span, presumably as a result of long continued frequent testosterone administration. The computational methods used and, in particular, the displays here introduced are more broadly applicable, inter alia, in endocrinology, to the detection of rhythms and to the objective quantification of their parameters, including the internal and external timing.

For a number of years, formula manipulation has been a prosperous member of the large family of computer applications. It has already facilitated the solution of problems too lengthy and time-consuming for the relatively unreliable human problem-solver. The reader who has not himself encountered such problems need only recall the horrendous exercises in formula manipulation performed by the astronomers of the eighteenth and nineteenth century, e.g., as described in Delaunay’s “Theorie du mouvement de la lune,” or in E. W. Brown’s “Theory of the Motion of the Moon.” Their achievements are hardly surpassable by hand, and continue to challenge the capabilities of current systems.

This chapter shall serve the discussion of the relaxation methods in which all parameters assume invariable values throughout the whole course of the computation. The gradient and Tchebycheff methods described in Chapters II and III are the counterparts of this family of relaxations, because the parameters in the relaxation formula vary from one step to the next. It will be the purpose of this chapter to contrast these two families and thus to permit a comparison of their efficiency in two concrete cases. As starting point for the two systems of equations we deliberately chose boundary value problems which do not lead to elliptical partial differential equations of second order. The latter were excluded partly because there are already different examinations of these systems of equations [7], but principally because these systems of equations may be of a kind which is especially well suited for the solution by means of the overrelaxation method, and therefore the results obtained in this way would not permit a general valuation of the method.