Recursive procedures

Software to predict the properties of formulated products is made more powerful by a recursive procedure which can use formulas stored in files as raw materials. Particular care must be taken with program flow control and data structures for the recursion to be effective. This paper illustrates these issues using an example derived from a working formulation system for coatings development. 

Fibonacci numbers, where the number sought is the nth number in a series defined in terms of relationships to the n-1, n-2, etc. members of the series. All recursive procedures must have a terminating condition, so that they do not call themselves endlessly. 

Recursive procedures demand special attention to flow control and data structures. For instance, the flow control within the procedure must correctly handle an error, say missing file information or inconsistent data, that is discovered several iterations deep. Should the procedure break and return to the previous level only Should it force return to the level of the original invocation of the procedure Should it allow an interactive user a choice of supplying missing data, and if so, on any level of iteration If files are opened within the procedures, should they be closed when a recursive call is needed, or are new channel numbers to be requested, using up system resources ... 

Figure 1 Flow chart of recursive procedure, accepting both raw material and formulas as inputs.

The appendix shows an example of a recursive procedure to calculate the pigment to binder ratio for a non-reactive coating formulation. We assume that a raw material data base contains raw material code, density, cost per unit weight, solids volume fraction, and pigment solids volume fraction. Formulas are stored in files that contain the identification and amount of... 

Data is passed between levels of recursive procedures in variables cited in the CALL statement for the procedure. As indicated in the section on programming issues, care must be taken in assigning names to the data variables so that values returning from a recursive CALL do not overwrite the values in variables within the calling procedure. Note the difference between the variable names in the SUBROUTINE statement of... 

In Sections 41.2 and 41.3 we applied a recursive procedure to estimate the model parameters of time-invariant systems. After each new measurement, the model parameters were updated. The updating procedure for time-variant systems consists of two steps. In the first step the system state j - 1) at time /), is extrapolated to the state x(y) at time by applying the system equation (eq. (41.15)) in Table 41.10). At time tj a new measurement is carried out and the result is used to... 

In a previous section, we have shown that recursion schemes are more powerful than flowchart schemes. Now we add to the basic model of flowchart of program schemes a mechanism resembling recursive procedures and calls as defined in PL/I, for example. We find that such recursion augmented flowchart schemes are in fact equivalent in expressive power to recursion schemes. Thus we show that recursion schemes are indeed one version of the model we seek to study the power of programs when augmented by recursion, in contrast to programs without this extension tut perhaps with other additions. 

The recursive procedure can be applied repeatedly, with one more atom being bonded to the growing chain at each step, producing GF s in CF form similar to those of (7.29) and (7.34). After (m — 1) iterations, all m (initially isolated) atoms are joined, and the GF at the n = 1 site has CF form... 

A recursive procedure is presented that exhibits some advantages over classical batch processing since it avoids the inversion of large matrices. It is shown that when only one equation is processed at a time, the inversion degenerates into the computation of the reciprocal of a scalar. Furthermore, this sequential approach can also be used to isolate systematic errors that may be present in the data set (Romagnoli and Stephanopoulos, 1981 Romagnoli, 1983). 

Comparing equations (12) and (13) we obtain the recursive procedure for calculating the Cauchy vectors ... 

Based on a perturbation expansion using the KS Hamiltonian [26,27], recently a new systematic scheme for the derivation of orbital-dependent Ec has been proposed [12]. While this representation is exact in principle, an explicit evaluation requires the solution of a highly nonlinear equation, coupling Exc and the corresponding x>xc [19]. For a rigorous treatment of this Exc one thus has to resort to an expansion in powers of e, which allows to establish a recursive procedure for the evaluation of Exc and the accompanying Vxc-... 

When searching very large databases for active compounds, a single-step MP analysis often does not sufficiently reduce database compounds to a small enough number for testing (e.g., fewer than 100). Therefore, we have devised a recursive procedure for MP (called RMP) that proceeds as follows (15) ... 

A recursive procedure to compute factorial / factorial Procedure Arg n... 

The procedure gives a simple recursion because the a matrices are in lower diagonal form. Similar equations can be written for the original RDM p instead of the diadic form a. This recursive procedure has been previously tested by us for density matrix calculations and found to be reliable. [22]... 

The elements of the fraction E and V) are determined from the recursive procedure... 

Mukhopadhyay e t a 1 /122/ generalized the ansatz (7.2.4) still further. They suggested the use of a recursive procedure whereby the wave-operator ft n at the n-valence lavel is recursively generated from the lower valence ft s. Thus, they define... 

In the case of binary alloys, for instance, the direct application of the recursion procedure is physically more transparent than CPA and superior in that detailed local configurations can be included. 

Other self-similar fractals can be generated by changing the recursive procedure. For instance, to obtain a new kind of Cantor set, divide an interval into five equal pieces, delete the second and fourth subintervals, and then repeat this process indefinitely (Figure 11.3.5). 

As I have said, Sekino and Bartlett [31] were the first to show how to proceed to calculate frequency-dependent hyperpolarizabilities within the TDCPHF approximation. They developed an infinite-order recursive procedure, using density matrices, and, by solving the equations iteratively at each order, could, in principle, calculate any non-linear optical property. Their first application was to H2, FH (the work on FH was analysed in detail in another paper [38]), CH4 and the fluoromethanes. The processes SHG, OR, dc-SHG, dc-OR, IDRI and THG were considered but not all hyperpolarizability components were computed (the assumption of Kleinman symmetry was made). 

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