Recurrence relations stability

Dn mx) in the coefficients (4.88) is computed by the downward recurrence relation (4.89) beginning with Z)NMX. Provided that NMX is sufficiently greater than NSTOP and mx, logarithmic derivatives of order less than NSTOP are remarkably insensitive to the choice of Z)NMX this is a consequence of the stability of the downward recurrence scheme for pn. For vastly different choices of Z)NMX, and a range of arguments mx, computed values of DNMX 5 were independent of Z)NMX. Thus, NMX is taken to be Max(NSTOP, mjc ) -I- 15 in BHMIE, and recurrence is begun with Z)NMX = 0.0 + z 0.0. 

For practical calculations one has to choose a sufficiently large enough / = L and set oy 11 and yi+1 to zero. Then, on implementing the backward recurrence relations (4.65) until 1=1, one evaluates the full set of the sweep coefficients. By an upward iteration of (4.63), one can find all the terms bi with the indices l = 1,... L. Due to the stability of both iteration processes, the initial error, caused by truncation of the infinite recurrences (4.65), dissolves in the course of iterations and finally becomes less than computer zero. The desired accuracy is provided by varying the cutoff index number L normally, the higher a or 2, in Eqs. (4.60)-(4.61), the higher L is to be used. In this sense one may call the solutions obtained numerically exact. 

In this chapter we shall show how the observed phenomena may be explained by means of elementary catastrophe theory. In principle, the discussion will be confined to examination of non-chemical systems. However, some of the discussed problems, such as a stability of soap films, a phase transition in the liquid-vapour system, diffraction phenomena or even non-linear recurrent equations, are closely related to chemical problems. This topic will be dealt with in some detail in the last section. The discussion of catastrophes (static and dynamic) occurring in chemical systems is postponed to Chapters 5, 6 these will be preceded by Chapter 4, where the elements of chemical kinetics necessary for our purposes will be discussed. 

Many of the considered problems, such as the problem of stability of soap films, the liquid-vapour phase transition, the diffraction phenomena, descriptions of the heartbeat or the nerve impulse transmission, catastrophes described by non-linear recurrent equations have a close relation to chemical problems. 

Modem psychiatric treattnents were introduced in 1949, when lithium carbonate was discovered as treatment for mania by Australian psychiatrist John F. Cade (Figure 1.45). After Cade s initial report, lithium therapy was principally developed in 1954 by Mogens Schou (Aarhus University, Denmark). In 1969, 20 years after its discovery by John Cade and after a decade of trials, the Psychiatric Association and the Lithium Task Force recommended lithium to the FDA for therapy of mania. A breakthrough had been achieved in the treatment of manic depression, and the genetically related forms of recurrent depression. Bipolar disorders, which afflict about 1% of adults, are now treated with drugs called mood stabilizers, especially lithium and valproic acid, both discovered decades earlier, but nothing better has yet emerged. ... 

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