Logarithmic derivative computation

If the incident light is normal to the cylinder axis the scattering coefficients have their simplest form. However, the coefficients (8.30) and (8.32) are not in the form most suitable for computations. If we introduce the logarithmic derivative... 

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. 

The convergence criterion in BHCOAT is the same as that in BHMIE series are terminated after y + 4y1/3 + 2 terms. Unlike BHMIE, however, all functions, including logarithmic derivatives, are computed by upward recurrence it seemed pointless to compute these derivatives by downward recurrence when they are not the major obstacle to writing a program valid for an arbitrary coated sphere. 

As in the previous programs, series for scattering matrix elements and efficiencies are truncated after NSTOP terms, where NSTOP = x + 4x1/3 + 2. Gn(mx) is computed by (C.l) beginning with CNMX, successive lower-order logarithmic derivatives CNMXGx are computed by downward recurrence. Provided that NMX is sufficiently greater than NSTOP and mx, Gp for... 

Computation of Bessel Functions The Bessel functions Jn and Yn pose more computational problems than the logarithmic derivative. In BHCYL these functions are computed by an algorithm credited to Miller (British Association, 1952, p. xvii), further details of which are given by Stegun and Abramo-witz (1957) and by Goldstein and Thaler (1959) we outline this scheme in the following paragraph. 

Tests of BHCYL Computed values of J (x) and Yn(x) for various n and x were compared with values tabulated by Olver (1964). Computed logarithmic derivatives were also compared with values calculated from tabulated Bessel functions. In all instances there was agreement to as many decimal places as were given in the tables. This gives us some confidence that BHCYL does what it was designed to do. 

The logarithmic derivative of linearly independent solutions of the coupled C( uations (11) arc propagated outwards in each sector [pp-i/2Q- p+i/2] using the Jolmson-Manolopoulos [8] algorithm. At the boundary of each sector / p+i/2, a transformation to the basis of the next sector [Pp+i/2-, Pp+ipi] computed at Pp+i is performed. This is repeated until the last sector (centered at / , ) corresponding to the asymptotic region is readied. 

Because of its simplicity, efficiency and adaptability to the hypercube computer architecture being developed at the California Institute of Technology,we have chosen Johnson s logarithmic derivative method to numerically integrate eq. (6.3). 

In the above equations, E, and are the thermal energies of the reactants and of the transition state, respectively. Such a thermodynamic integration was used within a discrete variable representation of QI approximation to compute the rate constant for several collinear triatomic reactions [33]. In Ref. [46], it is generalized and presented in a form suitable for a path integral evaluation. Unlike gr and Caa, the energies are normalized quantities because they can be written as logarithmic derivatives ... 

If for a scanner a monochromator drive is disposable and controllable by a computer, like for example with the instrument KM 3 Zeiss, spectra and their derivatives can be recorded. A spectrum can be obtained by taking the difference between substance- and background spectrum after a logarithmic transformation or a transformation by the Kubelka-Munk-function. 

For this study, we selected 42 representative FGs from a list of more than 500 substituents compiled by Hansch et a/.128 The electrophilicity index co for the whole series of ethylene derivatives, together with the experimental electronic substituent effects from the reactivity index co was as follows we first compared the computed co values with the experimental crp (see Figure 8). The analysis revealed a poor linear correlation between both quantities, with a regression coefficient R2 = 0.53. A better correlation was found when all the 42 points were fitted to a logarithm curve (.R2 = 0.84, see Figure 8). 

The four tests produce a total of six end-points, because the Ceriodaphnia and SOS Chromotest each have two end-points. Without Ceriodaphnia there are four end-points after biodegradation, so the value of N is 10. The PEEP formula sums the toxic units from each end-point and divides by N to derive an average toxic unit for the effluent. This is multiplied by n to account for the breadth of toxic response and by the discharge volume to compute the total hourly number of toxic units entering the river or receiving water. The result is expressed as a logarithm (adding 1 in the formula merely ensures that the log is computable). This produces a scale of results that would normally fall between 0 and 10. 

A more precise and computationally useful formula for A can be derived. By taking logarithms and noting that <5 = f"(Xg -t-5 )-/"(x ). we obtain... 

A series of computer-simulated chromatograms has been generated to test the validity of a procedure derived from the statistical model for calculating the number of randomly distributed components when many of them are obscured by overlap. Plots of the logarithm of the peak count versus reciprocal peak capacity are used for this purposTI TRese plots are shown to provide reasonable estimates of the total number of components In the synthetic chromatograms. 

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